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									FACTORING




            1
College Algebra with Trig                                   Name:__________________________________
Lesson/HW- Factoring Polynomials
                                                            Date:___________________________________

Objective:         factor polynomials

Factor completely. If the polynomial cannot be factored, write simplified.



(1)   8x – 24                       (2)    xy – 17y                     (3)   x2 – 169




(4)   x2 – y2                       (5)    x2 + y2                      (6)   3x3 – 3x




(7)   9x2 – 36y2                    (8)    2x3 – 4x2 – 6x               (9)   5x2 – 13x + 6




                                                                                                       2
(10) x2 – 6x + 2                   (11) 4a2 + 12ab + 9b2               (12) 36w2 – 16




Factor completely. If the polynomial cannot be factored, write simplified.

(13) 6 – 5x + x2                   (14) 40 – 76x + 24x2                (15) 2x2 + 4x – 1




(16) 2x2 + 28x – 30                (17) 6x2 + 7x – 3                   (18) 18x2 – 31xy + 6y2




                                                                                                3
College Algebra with Trig                                    Name:__________________________________
Lesson/HW- Factoring Special Polynomials
                                                             Date:___________________________________

Objective:            factor special polynomials

Guidelines for Factoring:

(1)   Factor out the Greatest Common Factor (GCF) – “Gotta Come First” Monomials

(2)   Factor Binomials – check for special products, for any numbers a and b:
      (a)    Difference of Two Perfect Squares:            a2 – b2 = (a + b)(a – b)
      (b)    Sum of Two Perfect Cubes:                     a3 + b3 = (a + b)(a2 – ab + b2)
      (c)    Difference of Two Perfect Cubes:              a3 – b3 = (a – b)(a2 + ab + b2)

(3)   Factor Trinomials – check for special products, for any numbers a and b:
      (a)    Perfect Square Trinomials:                    a2 + 2ab + b2 = (a + b)2
                                                           a2 – 2ab + b2 = (a – b)2
      (b)    General Trinomials:                           acx2 + (ad + bc)x + bd = (ax + b)(cx + d)

(4)   Factor Polynomials – if there are four or more terms, try factoring by grouping.

Factor completely:

(1)    x3 + 27                                       (2)    x3 – 64




(3)    27x3 – 8                                      (4)    2x3 + 16




(5)    x3 – 4x2 + 3x – 12                            (6)    x3 + 5x2 – 2x – 10




(7)    5a2x + 4aby + 3acz – 5abx – 4b2y – 3bcz




                                                                                                        4
Factor completely:

(8)    35x3y4 – 60x4y   (9)   2r3 + 250




(10) 100m8 – 9          (11) 3z2 + 16z – 35




(12) 162x6 – 98         (13) 4m6 – 12m3 + 9




(14)   x3 – 343         (15) ac2 – a5c




(16) c4 + c3 – c2 – c   (17) ax – ay – bx + by




(18) 64x3 + 1           (19) 3ax – 15a + x – 5




                                                 5
College Algebra with Trig                                  Name:__________________________________
Lesson/HW- Operations with Rational Expressions I
                                                           Date:___________________________________

Objective:           perform operations with algebraic fractions and simplify mixed expressions

Do Now: Factor completely. If the polynomial cannot be factored, write simplified.

(1)   3x2 + 10x + 8                                  (2)    8x3y6 + 27




(3)   4x2 – 12x + 5                                  (4)    4x2 – 4x – 48




Showing all work, perform the indicated operation and simplify your answer.


          x2  9      4 x 2  20 x                                 x 2  3x      x 2  5x  6
(5)                                                       (6)                 
       x 2  x  20   4 x 2  12 x                                2x 2  x  6      x2  4




                                                                                                      6
Showing all work, perform the indicated operation and simplify your answer.

       12a  4   b3                                          8x     8x  16
(7)                                                (8)           
         b       12                                        2x  8
                                                             2
                                                                     32x 2




        3 x 2  12    3 x 2  15 x  18                     9x  45    3x  3
(9)                                                (10)               2
       2x 2  x  6        x 2  3x                        x  4x  5
                                                             2
                                                                        x 1




       x 3  x 2 x 2  2x  1                              2x 2  x  6 4 x  8
(11)                                               (12)               
          3x         x2  1                                 4x 2  9     6x  9




                                                                                  7
Showing all work, perform the indicated operation and simplify your answer.


        a2  64   8a                                      x2  1    3x  3
(13)                                              (14)           2
       8a  64a 9a  72
         2
                                                           3x 2
                                                                  x  2x  1




       2c 2  c  6    12                                 x 2  10 x  25   x 2  25
(15)                                              (16)                   
         4c  8       6c  9                                    x5         5 x  25




(17)
       2x 3  10 x 2  8 x
         x  2x  8
            2
                             2x
                            2
                            x 1
                                                   (18)
                                                          2x 2  x  6
                                                              2x 2
                                                                        
                                                                        4x 2  9   




                                                                                        8
College Algebra with Trig                                  Name:__________________________________
Lesson/HW- Operations with Rational Expressions II
                                                           Date:___________________________________

Objectives:      perform operations with algebraic fractions and simplify mixed expressions
                 simplify mixed expressions and complex fractions

Do Now: Factor completely. If the polynomial cannot be factored, write simplified.

(1)   x2 + 2x + xy + 2y                              (2)    64x2 – 676




(3)   1 – 125y3                                      (4)    3a2 – 2b – 6a + ab




Showing all work, perform the indicated operation and simplify your answer.

         2y        8                                              7k  2    8k
(5)                                                       (6)           
       y  16
        2
                16  y 2                                          4k  3   3  4k




                                                                                                      9
Showing all work, perform the indicated operation and simplify your answer.

           1                                                    5
        x                                                   2
(7)        x                                        (8)         x
           1                                                 x 5
        1                                                    
           x                                                 3 6




       2x  1   1                                           4n 2      9
(9)                                                (10)          
       x x
        2
                x                                          2n  3   3  2n




                                                                              10
Showing all work, perform the indicated operation and simplify your answer.

       3a  1     1                                        3y  4   y2
(11)                                               (12)          
       a 1
        2
                a 1                                         5       4




       x4   x4                                              x2     2
(13)                                               (14)          
        x     4                                            x  16
                                                            2
                                                                    x4




                                                                              11
Showing all work, perform the indicated operation and simplify your answer.

         2a  5      1                                      2b  1      1
(15)                                              (16)              
       a  5a  6
        2
                    a3                                   b  b  12
                                                           2
                                                                       b4




          1                                                 x     z
       x                                                     
(17)      3                                        (18)     z     x
          1                                                 1     1
       3                                                     
          x                                                 z     x




                                                                              12
College Algebra with Trig                                     Name:__________________________________
Lesson/HW- Complex Fractions and Equations
                                                              Date:___________________________________

Objectives:        simplify mixed expressions and complex fractions
                   solve equations with algebraic expressions
                   solve real-world applications with algebraic expressions

                             ON A SEPARATE SHEET OF PAPER,
              ANSWER EACH OF THE FOLLOWING QUESTIONS SHOWING ALL WORK!

Perform the indicated operation and simplify your answer:

                                                                                                 5 6
                                                                       m5                   1    
        a 2
            b       2                       y 1                    m                           y y2
(1)                        (2)    y 1                               m3          (4)
       ab ba                                y         (3)
                                                                                                1
                                                                                                   3
                                                                     m 1
                                                                                                    y


Solve each of the following equations and check:

         x      16    1                                                 1      1       4
(5)          2                                              (7)                  2
       x  8 x  64 x  8                                             2b  6 2b  6 b  9


         1    1     6                                                   x      1     16
(6)             2                                           (8)                 2
       h 1 h 1 h 1                                                 2x  8 x  4 x  16




Show All Work:

(9)    The area of a rectangular patio is represented by the expression (6x2 + 13x – 5). The width of the patio is
       (3x – 1). Write a simplified expression to represent the length of the patio in terms of x.

                                                                            3a  a 2
(10) If the length of a rectangular field is represented by the expression 2          , and the width is represented
                                                                             a 9
          a 2  a  12
     by                , what simplified expression represents the area of the field?
              a4




                                                                                                                  13
                            Unit 1: Algebraic Fractions, Equations & Factoring
                                  Definitions, Properties & Procedures

        Factoring      the process of writing a number or algebraic expression as a product
      Least Common
                    the least common multiple of two or more given denominators
  Denominator (LCD)
                       has the same properties as a numerical fraction, only the numerator and
Algebraic Fraction
                       denominator are both algebraic expressions
         Rational      an algebraic expression whose numerator and denominator are polynomials and
       Expression      whose denominator has a degree of one or greater
                       reducing or simplifying a rational expression means to write the expression in
                       lowest terms, which can only be done with a single fraction, a product of fractions
      Simplifying
                       or a quotient of fractions. If there is an addition or subtraction sign in the
         Rational
                       numerator (or denominator), it must be factored first and then like factors with the
      Expressions
                       denominator (or numerator) can be canceled. Note: you cannot reduce across a
                       sum or difference of two or more fractions!
                       To multiply rational expressions:
                       (1) Factor each numerator and denominator completely
                       (2) Cancel any like factors in any numerator with any like factors in any
                            denominator
    Multiplying &      (3) Multiply the remaining expressions in each numerator
 Dividing Rational     (4) Multiply the remaining expressions in each denominator
       Expressions     (5) Reduce if possible
                    To divide rational expressions:
                    (1) Multiply the first fraction by the reciprocal of the second fraction (KCF)
                    (2) Follow the steps above to multiply rational expressions
                    (1) Find the least common denominator among all fractions (if necessary)
                    (2) Multiply each denominator by an appropriate factor to make it equivalent to the
        Adding &
                         LCD; and multiply each numerator by the same factor that you multiplied its
      Subtracting
                         denominator by (multiply by a “fraction of one”)
         Rational
                    (3) Combine all numerators (make sure the signs are placed appropriately) and
      Expressions
                         simplify; and put over LCD
                    (4) Reduce if possible
                    a fraction that contains one or more fractions in the numerator, the denominator, or
                    both
Complex Fraction    To simplify complex fractions:
                    Combine fractions in the numerator and denominator separately by adding or
                    subtracting. Once there is a simplified fraction above a fraction, use the steps for
                    dividing fractions to further simplify the expression.
                    an equation that contains one or more rational expressions
                    To solve rational equations:
Rational Equation   (1) Find the LCD
                    (2) Multiply each fraction by this LCD
                    (3) Cancel all denominators
                    (4) Solve the remaining equation for the given variable
  Greatest Common the product of the greatest integer and the greatest power of each variable that
       Factor (GCF) divides evenly into each term
                                                                                                              14
                     a polynomial of the form a2 – b2, which may be written as the product (a +
                     b)(a – b)
                     To factor a difference of two perfect squares:
Difference of Two    (1) Create two empty binomials  (              )(      )
  Perfect Squares    (2) Take the square root of the first term of the given binomial and put it in the 1st
                          position in each binomial
                     (3) Take the square root of the last term of the given binomial and put it in the 2nd
                          position in each binomial
                     (4) Make one binomial a sum and the other binomial a difference
                     Sum of Two Perfect Cubes: a polynomial of the form a3 + b3, which may be
                     written as the product (a + b)(a2 – ab + b2)
                     To factor a sum of two perfect cubes:
                     (1) Create an empty binomial and an empty trinomial  (            )(         )
                     (2) Take the cube root of the first term of the given expression
                          (a) put it in the 1st position in the binomial
           Sum of         (b) square it and put it in the 1st position of the trinomial
Two Perfect Cubes    (3) Take the cube root of the last term of the given expression
                &         (a) put it in the 2nd position in the binomial
                          (b) square it and put it in the last position of the trinomial
Difference of Two
                     (4) Find the product of the terms in the binomial and put it in the middle position
    Perfect Cubes
                          of the trinomial
                     (5) Arrange the signs as follows: ( + )( − + )
                     Difference of Two Perfect Cubes: a polynomial of the form a3 – b3, which
                     may be written as the product (a – b)(a2 + ab + b2)
                     To factor a difference of two perfect cubes:
                     Follow above steps and arrange the signs as follows: ( − )( + + )
                     a trinomial whose factored form is the square of a binomial; has the form
                                       a2 – 2ab + b2 = (a – b)2 or a2 + 2ab + b2 = (a + b)2
                     To factor a perfect square trinomial:
                     (1) Create two empty binomials  (              )(      )
   Perfect Square    (2) Take the square root of the first term of the given trinomial and put it in the 1st
       Trinomial          position in each binomial
                     (3) Take the square root of the last term of the given trinomial and put it in the 2nd
                          position in each binomial
                     (4) The signs of each binomial should be the same as the middle term of the given
                          trinomial
                     (1) Find a convenient point in the polynomial to partition (or group)
     Factoring by
                     (2) Factor within each group
       Grouping
                     (3) Factor out the Greatest Common Factor across the groups
                     To factor trinomials in the form ax2 + bx + c:
        Factoring    (1) Multiply the a term by the c term
Trinomials with a    (2) Find the factors of (ac) which will add to the b term
         Leading     (3) Rewrite the b term as the sum of two x terms with coefficients being the
       Coefficient        factors of (ac)
Greater Than One     (4) Group the first two terms and last two terms each in a set of parentheses
                     (5) Factor out the Greatest Common Factor from each group




                                                                                                               15
College Algebra with Trig                                     Name:__________________________________
Review- Rational Expressions Test
                                                              Date:___________________________________

ANSWER EACH ON A SEPARATE SHEET OF PAPER. SHOW ALL WORK!

Factor completely. If the polynomial cannot be factored, write simplified.

(1)    6c2 + 13c + 6                     (4)     y4 – z2                        (7)      3d2 – 3d – 5
(2)    a2b2 + ab – 6                     (5)     x5 + 27x2                      (8)      72 – 26y + 2y2
(3)    t2 – 2t + 35                      (6)     x4 – 81                        (9)      x3 + 7x2 + 2x + 14


Perform the indicated operation and simplify your answer.

        6a 2  2a      9a 2  1                        7
                                                          
                                                             4                                 x
(10)                                          (14)                                          1
       9a 2  6a  1    6a 2                          a3   2a                   (18)         3
                                                                                               x2
                                                                                            3
                                                                                               3
         t 2  6t  9     t 2  t  20                1    1
(11)                    2                     (15)     
       t 2  10 t  25   t  7t  12                  x   1 x
                                                                                                7
                                                                                            1
                                                                                               y2
           x4         3x  12                         2m    18                   (19)
(12)                                          (16)                                            3
                                                      m9   9m                             1
       2x  7 x  3
         2
                      5x 2  45                                                                y2


        x 2  3x      x 2  5x  6                          3      3
(13)                                                                                             4
       2x 2  x  6      x2  4                            xy xy                            1
                                               (17)
                                                                6                 (20)           x 1
                                                                                                    24
                                                             x  y2
                                                              2
                                                                                            x  1
                                                                                                   x 1



Answer the following word problems, showing all               Solve each of the following equations and check:
work to explain your answer:
                                                                       4     5      3x
(21) The area of a rectangle is (x2 – x – 6) square           (23)               
                                                                     x  1 2x  2    4
     meters. The length and width are each
     increased by 9 meters. Write the area of the
     new rectangle as a trinomial in terms of x.                      2      1      1  2a
                                                              (24)              2
(22) The freshman and sophomore classes both                         a4   a  2 a  2a  8
     participated in a fundraiser. The freshman
     class collected (4x2 – 1) and the sophomore
     class collected (6x2 + 7x + 2). Express, in
     simplest form, the ratio of the sophomore’s
     collection to the freshman’s collection.
                                                                                                                 16
FUNCTIONS




            17
College Algebra with Trig                                         Name:____________________________________
Lesson- Relations and functions
                                                                  Date:_____________________________________

Objective:           To know the definitions of relations and functions. To understand the difference between what is
                     a function and what is not. To be able to determine whether a relation is a function.

Definitions:

Relation-


Function-


Domain-


Range-


PBLM SET.
1.    State whether the relation is a function or not: Identify the Domain and Range.
   a. {(-2, 0), (3, 2), (4, 5)}



     b.              {(6, -2), (3, 4), (6, -6), (-3, 0)}




2.        Which relation is a function? Why?




            (a)                          (b)                (c)                    (d)



3.        Find the Domain and Range of each choice in exercise #2.



4.        Determine whether each of the following is a function. Justify your answer. Find the Domain and
          Range of each.

a.        f ( x)      x3                                        b.     f(x)  - x 2  2x - 27
                                                                                                                  18
19
20
College Algebra with Trig                                   Name:____________________________________
Lesson- Linear Functions
                                                            Date:_____________________________________


Objectives:    To know the various properties of a linear function. To understand the processes for writing and
               graphing various types of linear functions.


Do Now: State the four different types of slope and give an example for each:




Linear Function:




Forms of Linear Functions:
1.     slope-intercept form:




2.     standard form:




3.     point-slope form




Ex 1: Write the linear equation in slope intercept, standard, and point-slope form given that the line passes
      through (5, 2) and (7, 9)




                                                                                                                21
Ex 2: Write the equation of the horizontal line that passes through (-9, 2)




Parallel & Perpendicular Linear Function Rules:

Parallel                                                  Perpendicular




Ex 3: Write the linear equation in standard form given that the line passes through (-2, 10) and is parallel to
                              4
      the graph of y  3x 
                              5




                                                                                                            2    4
Ex 4: Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of y      x
                                                                                                            3    7




Intercepts
x-intercepts:                                             y-intercepts:




                                               1
Ex 5: Find the x- and y- intercepts of f ( x)  x  2 . Graph the linear function.
                                               3




                                                                                                                  22
College Algebra with Trig                                         Name:____________________________________
Lesson- Evaluating Functions
                                                                  Date:_____________________________________

Objectives:     To know what it means to evaluate a function. To understand how (and be able) to evaluate a
                function algebraically and graphically.


Notation for a function:




What does “evaluate a function” mean?




Evaluating Functions Algebraically

1. find f(-1) if f(x) = x2 – 1                2. find h(3) if h(x) = 3x2           3. find f(-7) if f(w) = 16 + 3w – w2




4. find g(m) if g(x) = 2x6 – 10x4 – x2        5. find k(w + 2) if k(x) = 3x + 4    6. find h(a – 2) if h(x) = 2x2 – x +
+5                                                                                 3




Evaluating Functions Graphically


1.                                       2.                                       3.




                                                                                                                      23
College Algebra with Trig                                 Name:____________________________________
Lesson- Graphing Absolute Value Functions
                                                          Date:_____________________________________

Objective:     To learn how to graph a piecewise and absolute value function

Do Now:
                                x2 1
State the domain for f ( x) 
                                  x




__________________________________________________________________________________________
Absolute Value Functions
                                                                        y
Graph the Following
 f ( x)  x



                                                                                                      x




Now graph each of the following and discuss how each relates to f (x) from above.
g ( x)  2 x                                h( x)  2 x  3                         i( x)  2 x  3




               y                                      y                                  y




                                x                                    x                                    x




                                                                                                          24
College Algebra with Trig                                  Name:____________________________________
Solving Absolute Value Equations
                                                           Date:_____________________________________

Objectives: To learn to solve absolute value equations and absolute inequalities.


Absolute Value Equations
                                       ax  b  c
To solve ax  b  c create 2 equations             and solve each.
                                       ax  b  c

Example:    3x  1  2




Practice:
a. x  1  4
b. 3  y  5
c. 2  3d  4
d . 2m  1  2




                                                                                                   25
College Algebra with Trig                                    Name:____________________________________
Lesson- Solving Absolute Value Inequalities
                                                             Date:_____________________________________

Objectives: To learn to solve absolute value inequalities.

Absolute Value Inequalities

There are three absolute value situations:

Case 1            Case 2             Case 3
ax  b  c        ax  b  c         ax  b  c
 ax  b  c        c  ax  b  c   Either ax  b  c or ax  b  c
 ax  b  c
Examples:
a.    3x 1  2                b.      3x  1  2            c.     3x 1  2        d.    3x  1  2




Practice:
 a. x  1  4
 b. 3  y  5
 c. 2  3d  4
 d . 2m  1  2




                                                                                                        26
27
College Algebra with Trig                              Name:____________________________________
WKST- Mixed equation/inequality and absolute value set
                                                       Date:_____________________________________

Answer each of the following neatly and completely in the space provided.

Solve and graph each inequality:

1.     x  7x  6                                         2.      x  3  3(2 x  1)




3.      x3  4                                           4.      2x  5  x  1




5.      2x  3  5                                        6.      2 x  8




7.     x 2  2 x  24  0                                 8.      x 2  10 x  1  0




                                                                                               28
9.      x2  4  0                      10.    x 2  8x  7  0




11.     3x 2  10x  8                  12.    5d  7  28




13. Explain why the solution set of      14. Explain why the solution set of
     x 2  9  0 is all real numbers.    x 2  16  0 is empty.




                                                                               29
College Algebra with Trig                                    Name:____________________________________
Lesson- Graphing Piecewise Functions
                                                             Date:_____________________________________

Objective:      To learn how to graph piecewise functions.

Do Now: Graph: f ( x)  3x  2 for  3  x  0                              y




                                                                                        x




What is a piecewise function?




Graph the following:                                                            y
         2 x if x  0
f ( x)  
         2 if x  0



                                                                                            x




         2 x  1 if x  0
g ( x)   2                                                                    y
          x if x  0




                                                                                            x




                                                                                                     30
31
College Algebra with Trig                             Name:__________________________________
Mixed Wkst: Graphing Absolute and Piecewise Functions
                                                      Date:___________________________________



                                                                             y
1. Graph the following function:
                        x  2 if x  1
               f ( x)  
                        x  2 if x  1



                                                                                     x




2. Graph each function, and state the domain and range


(1)   f ( x)  x  3                                (2)   f ( x)  3 x  2
                        y                                                        y




                                           x                                             x




                                                                                                 32
College Algebra with Trig                                     Name:__________________________________
Lesson- One-to-One and Onto
                                                              Date:___________________________________

Objectives:    To know what it means for a function to be One-to-one or Onto. To be able to distinguish
               between One-to-one and Onto.


Definitions


Abscissa-




Ordinate-




One-to-one Functions
A function is one-to-one when no two ordered pairs in the function have the same ordinate and different
abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line test.
If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have an
inverse**)




Onto Functions
A function is Onto if each ordinate associated with an abscissa. Multiple abscissas may map onto the same
ordinate. (**Note: if a function does not use all y-values in a Cartesian plane, it cannot be onto)




                                                                                                                  33
Examples:      Determine whether the following refers to a function one-to-one, onto, both or neither.
Explain your reasoning.
1)     f ( x)  2x  1              2)      f ( x)  x 2  1              3)      f ( x)  3  5 x




4)                                  5)                                   6)




7)                                  8)                                   9)




                                                                                                         34
College Algebra with Trig                                       Name:__________________________________
Lesson- Composition & Inverse of Functions
                                                                Date:___________________________________


Objective:        To know how to find the composition and inverse of a function. To understand the process for
                  finding the composition and inverse of a function. To be able to recognize an inverse
                  graphically.

Do Now:           Evaluate f ( x)  x 3  x for x  2




Composition of Functions
“following” one function with another.

Notation:
Both of the following mean “f following g.”
 f ( g ( x))                 and                        ( f  g )(x)


Ex 1:

 f ( x)  x  5
g ( x)  4 x

Find: a) f ( g ( x))             b) f (g (2))           c) ( g  f )(3)        d) g ( f ( x))




Would you say that a composition is a commutative operation? Why/why not?




        h( x )  x 2
Ex 2:                            Find: a) h(r ( x))             b) r (h( x))            c) h(r (5))
        r ( x)  x  3




                                                                                                                 35
Inverse Functions

Definition:


Steps:
   1. Write the equation in terms of x and y.
   2. Switch the x with the y.
   3. Solve for y.

Ex 1: Find the inverse of y  4 x  8




Ex 2: Find the inverse of f ( x)  5x  2




Ex 3: Find the inverse of g ( x)  x 2  4




Ex 4: Graph y  4 x  8 and it’s inverse on the axes below.
                                                                                  y




                                                                                      x




Ex 5: Looking at the graph of a line, can you find a way to graph it’s inverse?



                                                                                          36
College Algebra with Trig                                   Name:__________________________________
Lesson- Operations with Functions
                                                            Date:___________________________________

Operations with Functions: given functions f and g

        sum:      f  g( x)  f ( x)  g( x)
  difference:     f  g( x)  f ( x)  g( x)
     product:     f  g( x)  f ( x)  g( x)
               f       f ( x)
     quotient:  ( x) 
                g             , where g( x)  0
                       g( x)


Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each
(1) f ( x)  3x  1; g ( x)  x




(2) f ( x)  5 x  4 ; g ( x)  x 2  1




(3) f ( x)  5  x ; g ( x)  x  1




                                                                                                       37
College Algebra with Trig                                  Name:__________________________________
Lesson- Function transformations
                                                           Date:___________________________________


Objectives:    To know the rules for various transformations such as: translations, reflections, symmetry,
               rotations, and dilations. To understand the process for transforming coordinates, lines, and
               curves. To be able to conduct various transformations and compositions of transformations.

Do Now: Sketch the graph the following polynomial:          f ( x)  x  1
                                                                                   y




                                                                                                          x




Definitions:
Pre-Image:


Image:


Types of Transformations and their specific rules




                                                                                                              38
Extra Space




              39
                                       Unit 2: Relations & Functions
                                    Definitions, Properties & Formulas

      Relation    a set of ordered pairs (x, y)

       Domain     the set of all x-values of the ordered pairs

        Range     the set of all y-values of the ordered pairs

                  a relation in which each element of the domain is paired with exactly one element
      Function
                  in the range.

                  the slope, m, of the line through (x1, y1) and (x2, y2) is given by the following
         Slope                                y  y1
                  equation, if x1  x2: m  2
                                              x 2  x1


                          y
                                                y                    y                    y
Types of Slope                  x                     x                   x                     x
                     Positive              Negative              Zero                 Undefined
                                                            horizontal line:       vertical line: x
                                                                 y=b                      =a


    y-intercept   where the graph crosses the y-axis

    x-intercept   where the graph crosses the x-axis

                  y = mx + b
Slope-Intercept
          Form    where m represents the slope and b represents the y-intercept of the linear
                  equation

                  Ax + By = C
Standard Form
                  where A, B, and C are constants and A  0 (positive, whole number)

                  y – y1 = m(x – x1)
   Point-Slope
         Form     where m represents the slope and (x1, y1) are the coordinates of a point on the line
                  of the linear equation

                  Two non-vertical lines in a plane are parallel if and only if their slopes are equal
 Parallel Lines
                  and they have no points in common. (Two vertical lines are always parallel.)

 Perpendicular    Two non-vertical lines in a plane are perpendicular if and only if their slopes are
         Lines    negative reciprocals. (A horizontal and a vertical line are always perpendicular.)




                                                                                                         40
Vertical Line Test   If any vertical line passes through two or more points on the graph of a relation,
            (VLT)    then it does not define a function.

  Horizontal Line    If any horizontal line passes through two or more points on the graph of a relation,
      Test (HLT)     then its inverse does not define a function.

      One-to-One     a function where each range element has a unique domain element
       Functions     (use HLT to determine)

  Onto Functions     All values of y are accounted for

Inverse Relations    f -1(x) is the inverse of f(x), but f -1(x) may not be a function
    & Functions      (use HLT to determine)
                     To find f -1(x):
                         (1) let f(x) = y
  Writing Inverse
                         (2) switch the x and y variables
       Functions
                         (3) solve for y
                         (4) let y = f -1(x)

                     sum:             (f + g)(x) = f(x) + g(x)
                     difference:      (f – g)(x) = f(x) – g(x)
 Operations with     product:         (f  g)(x) = f(x)  g(x)
      Functions
                                      f       f ( x)
                     quotient:         ( x) 
                                       g             , where g( x)  0
                                              g( x)

                     Reflections:     rx  axis ( x, y )  ( x, y )   ry  axis ( x, y )  ( x, y ) rorigin ( x, y )  ( x, y )

                     Dilations:        Dk ( x, y )  (kx, ky)
 Transformations
                     Translations: Ta ,b ( x, y )  ( x  a, y  b)

                     Rotations:        R0,90 ( x, y )  ( y, x)




                                                                                                                                      41
College Algebra with Trig                                    Name:__________________________________
Review- Function test
                                                             Date:___________________________________

Objective: To review the material that you will be tested on as part of Test #1-Functions. These topics are in
the outline below:
        Functions
                a. Identifying functions
                b. Domain and Range of functions
                c. Linear Function
                        i.   Finding x and y intercepts
                       ii. Writing and graphing the equation of line in slope intercept form
                      iii. Parallel and perpendicular lines and their graphs
                d. Evaluating functions graphically
                e. Evaluating functions algebraically
                f. Absolute Value Functions
                g. Piecewise Functions
                h. Identifying one-to-one functions
                i. Identifying onto functions
                j. Composition of functions
                k. Inverse functions
                l. Operations with Functions
                m. Transformation of functions


Below you will find a sample of the types of problems you can expect to see on the test.

a.     Which graph of a relation is also a function?


       (a)                    (b)                      (c)                 (d)




b.     Determine the Domain and Range of:
       i.     f ( x)  3 x  4                                      ii.     g ( x)  x 2  9




ci.    Find the x and y intercepts for the following linear equations:
       1. x  3 y  7                                               2. 3x  4 y  12




                                                                                                                 42
cii.      Write and graph the equation of the line given the following information:
          1. m  3, and passes through (3,2)                          2. passes through (5,1) and (2,0)




ciii.     1.      Write & graph the equation of the line that is parallel to y  3x  2 and passes through
                  (4,1).




          2.      Write and graph the equation of the line that is perpendicular to y  3x  2 and passes through
                  its x intercept.




d.      If the following graph is y = f(x), what is the value of f(1)?




       (a) -1                     (b) -2                  (c) 1               (d) 2


                                                                                                                    43
e.     Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)




f.     What are the significance of the a,h,k values in the standard form of an absolute value function?




       Write f ( x)  2 x  2  2
                              2
g.




h.   Which function is not one to one?


       (a)                        (b)                     (c)                      (d)




i.   Which function is not onto?


       (a)                        (b)                     (c)                      (d)




j.     Given f ( x)  3x  4; g ( x)  x 2  9 , find ( f  g )(x) and ( g  f )(x) .




                                                                                                           44
k.       Find the inverse of the following and state the domain.
                                                                                             4
              a.           f(x) = 5x + 2                                b.       f ( x) 
                                                                                            x3



l.       Perform the four basic operations on f ( x)  3x  4; g ( x)  x 2  9 and determine the domain of the
         result.




m.       Complete the following transformations on graph paper. Label your images.
         a.    rx axis (2,1)                                     b.     D 2 [ f ( x)  x 2  1]

         c.        R0,90 [ g ( x)  2 x  1]                                    d.       T2,3 [h( x)  2x 2  4x  2]
---------------------------------------------------------------------------------------------------------------------------------------
         a.                                                                      b.
                         y                                                                       y




                                                   x                                                                       x




         c.            y                                                         d.               y




                                                  x                                                                        x




                                                                                                                                    45
QUADRATICS




             46
College Algebra with Trig                                         Name:____________________________________
Rational Exponents Review
                                                                  Date:_____________________________________


Objective:     To learn to use the rules of exponents, including zero, negative and fractional exponents, in
multiplication and division of monomials.


Zero Exponents:                     Any value raised to the 0th power is equal to 1.

Example 1:     (5 x 2 ) 0  ?

Negative Exponents:                 When a monomial is raised to a negative exponent, take the reciprocal of the base
                                    to get rid of the negative in the exponent.

                        3
                2x 
Example 2:                 ?
                3 

Fractional Exponents: When a monomial is raised to a fractional exponent, the numerator represents the
                                    power the monomial should be raised to. The denominator represents the root of
                                    the monomial that should be taken.
                        3
                16  2
Example 3:       ?
                25 




                            2
                125  3
Example 4:           =?
                64 




Multiplying:                 If two monomials are being multiplied, you generally:
                                               Multiply the coefficients
                                               Keep the variable and add the exponents.

Example 5:     (3 x 2 )( 2 x 3 )  ?




                                                                                                                   47
But sometimes you are asked to multiply in a different way.

Example 6:         (3 x 3 y ) 3  ?



What is another way that you can write Example 2?




What is a general rule that you can use to simplify this process?




Dividing:          If two monomials are being divided, you generally:
                                            Divide (or simplify) the coefficients
                                            Keep the variable and subtract (or cancel) the exponents.




                    6x 2 y
Example 3:             3
                   3x y 2
                       2




More Practice Exercises
Simplify the following.
1) (2 x 3 y 0 )(3 x 1 y ) 2


2) (2ab 3 )(2a 1b 2 ) 2


     m2n4
3)
     m 2 n
              1
    25  2
4)  
    49 



                                                                                                         48
Some More Complicated Practice Exercises
Simplify each of the following.

                           2                           7                                           1
          x y  6    3   3                    49 x y 3   10                                4
                                                                                          12 a b   3
5)        3  4 
         x y                             6)      1
                                                                                     7)       1
                 
                                                7 x y 12
                                                   3                                       5c 2




Practice:

Simplify
            1
    25  2                                                     12a 4 b 3
1.                                                       2.
    49                                                          5c




                 1
     27  3
3.                                                       4.   49 x 8 y 10
    64 




                  3
    xy 4 z 3                                                                x 2
5.  2 5 
    x yz                                                 6. Challenge:
                                                                            x 2


                                                                                                        49
College Algebra with Trig                                   Name:____________________________________
Lesson- Graphing Quadratic Functions
                                                            Date:_____________________________________

Objective:        To graph a quadratic function using the
                              Roots
                              Axis of symmetry
                              Vertex                                                      y

Do Now:           Graph the following function.
                  f ( x)  3 x  1

                                                                                                          x




Roots:




Axis of symmetry:




Vertex:




__________________________________________________________________________________________

Graph each of the following on the same set of axes by finding the intercepts, axis of symmetry and vertex.
1. f ( x)  x 2  4 x
2. g ( x)   x 2  2 x  5
3. h ( x )  x 2  4 x  4



Based on the graphs above, is there a shortcut for determining if the parabola opens up or down?


                                                                                                              50
51
College Algebra with Trig                                     Name:____________________________________
Challenge Problem Set- Quadratics and their graphs
                                                              Date:_____________________________________

1. Given a quadratic function: y  ax 2  bx  c , determine the value of “b” if the vertex is ( 2, 2) and the y-
       intercept is (0, -2).




2. Given a quadratic function: y  ax 2  bx  c , determine the value of “b” if the vertex is ( p, p) and the y-
       intercept is (0, -p). Hint: Look at your solution to #1.




                                                                                                                    52
College Algebra with Trig                                        Name:____________________________________
Lesson- Zero Product Rule
                                                                 Date:_____________________________________

Objective:       To apply the zero product method for finding the roots of a quadratic function.

Do Now:          Create a table of values to graph the following function:
                 h( x)  x 2  14 x  1




Zero product rule:




Find the roots of the given quadratic using the zero product method:

1. x 2  4 x  3  0      2. x 2  7 x  6  0    3. x 2  10 x  0    4. 3x 2  x          5. x( x  4)  5
                                                             2
                               x   3                 3x x
6. x 2  2( x  12)       7.                     8.                  9. x 2  25  0      10. 16 x 2  64
                               2 x 1                 2   4
                                                      x 24                            13
11. 9 x 2  6 x  1  0   12. 6 x 2  x  2  0   13.                 14. x  6           15. 2 x 2  7 x  4  0
                                                      6 x                            x6




                                                                                                                  53
College Algebra with Trig                                         Name:____________________________________
Lesson- Completing the square
                                                                  Date:_____________________________________


Objective:       To find the roots of a quadratic function by completing the square.

Do Now: Find the roots of the quadratic function by using the zero-product rule.
      x 2  25  200




Completing the square:




Find the roots of each of the following by completing the square (any imaginary answers should be put in
simplest a  bi form).

1. x 2  4 x  3  0      2. x 2  7 x  5  0    3. 2 x 2  10 x  0   4. 3x 2  x         5. x( x  4)  5
                               x   3                 2x x 2
6. x 2  2( x  12)       7.                     8.                   9. x 2  25  0     10. 16 x 2  64
                               2 x 1                3   4
                                                      x 24                             13
11. 9 x 2  6 x  1  0   12. 6 x 2  x  2  0   13.                  14. x  6          15. 2 x 2  7 x  4  0
                                                      6 x                             x6




                                                                                                                  54
College Algebra with Trig                                         Name:____________________________________
Lesson- Quadratic Formula
                                                                  Date:_____________________________________


Objective:       To find the roots of a quadratic function by using the quadratic formula

Do Now: Find the roots of the quadratic function by using the completing the square method.
      x3       8
            
        2     x3




Quadratic Formula Derivation:




Find the roots of each of the following by using the quadratic formula (any imaginary answers should be put in
simplest a  bi form)

1. x 2  4 x  3  0      2. x 2  7 x  5  0    3. 2 x 2  10 x  0   4. 3x 2  x         5. x( x  4)  5
                                                             2
                               x   3                 2x x
6. x 2  2( x  12)       7.                     8.                   9. x 2  25  0     10. 16 x 2  64
                               2 x 1                3   4
                                                      x 24                             13
11. 9 x 2  6 x  1  0   12. 6 x 2  x  2  0   13.                  14. x  6          15. 2 x 2  7 x  4  0
                                                      6 x                             x6
                                                                                                                  55
College Algebra with Trig             Name:____________________________________
Lesson- Quadratic Formula Workspace
                                      Date:_____________________________________

Complete #1-15 in the space below




                                                                              56
College Algebra with Trig                                           Name:____________________________________
Lesson- Standard form of a quadratic function
                                                                    Date:_____________________________________


Objective:       To write and graph a quadratic function in standard form

Do Now: Find the roots of the quadratic function by using the quadratic formula.
      x3      8
           
        2     x2




Standard form of a Quadratic Function:            f ( x )  a ( x  h) 2  k

where (h,k) is the vertex and a determines whether that vertex is a maximum or minimum.

Writing a quadratic in standard form:
Example: x 2  4 x  7




__________________________________________________________________________________________

Write each of the following in standard form and determine the vertex and whether that vertex is a maximum or
minimum.



1. x 2  4 x  3  0      2. x 2  7 x  5  0    3. 2 x 2  10 x  0          4. 3x 2  x         5. x( x  4)  5
                                                               2
                               x   3                 2x x
6. x 2  2( x  12)       7.                     8.                          9. x 2  25  0     10. 16 x 2  64
                               2 x 1                3   4
                                                      x 24                                    13
11. 9 x 2  6 x  1  0   12. 6 x 2  x  2  0   13.                         14. x  6          15. 2 x 2  7 x  4  0
                                                      6 x                                    x6




                                                                                                                         57
College Algebra with Trig                               Name:____________________________________
Lesson- Standard form of a quadratic function Workspace
                                                        Date:_____________________________________

Complete #1-15 in the space below




                                                                                                58
College Algebra with Trig                                       Name:____________________________________
HW- Solving for the Roots of Quadratics
       & Standard Form                                          Date:_____________________________________

Answer each of the following neatly and completely.

Write the following equations in standard form, state the values of a & (h, k).

1. 5x 2  2 x  4                    2. - x 2  4  2 x                            3. 5x  4  6 x 2




Find the roots for each of the following equations using any of the 3 methods (if possible)

4. z 2  5 z  4  0                                       5. x 2  11x  24  0




6. s 2  s  0                                            7. 2 x 2  5 x  2  0




8. x 2  81                                               9. y 2  6 y




      1 2 7
10.     x  x 1
      2    6




                                                                                                        59
College Algebra with Trig                                         Name:____________________________________
Lesson- Describing the nature of quadratic roots
                                                                  Date:_____________________________________


Objective:       To use the discriminant to determine the nature of the roots of quadratics

Do Now: Write the following quadratic in standard form
      x3      8
           
        2    x2




Discriminant:




Determine the nature of the roots of each of the following. In each case, determine which method of finding the
roots of quadratics would be the best to use?




1. x 2  4 x  3  0      2. x 2  7 x  5  0    3. 2 x 2  10 x  0   4. 3x 2  x           5. x( x  4)  5
                               x   3                 2x x 2
6. x 2  2( x  12)       7.                     8.                   9. x 2  25  0       10. 16 x 2  64
                               2 x 1                3   4
                                                      x 24                             13
11. 9 x 2  6 x  1  0   12. 6 x 2  x  2  0   13.                  14. x  6            15. 2 x 2  7 x  4  0
                                                      6 x                             x6




                                                                                                                    60
College Algebra with Trig                                   Name:____________________________________
Lesson- Sum and product of roots
                                                            Date:_____________________________________

Objective:    To find the sum and product of the roots of a quadratic.
              To find the standard form equation of a quadratic given roots or sum and product of roots
              To find shortcuts for different transformations of quadratics

Do Now:       Change f ( x)  x 2  4 x  3 into standard form and identify the vertex




Sum and Product formulas:

Proof:




Writing quadaratic equations given roots

Procedure:




Examples: Write the standard form equation of the quadratic given the following roots:
1.   x= 1,7               2.    x= 4i, -4i           3.     x= e-fi, e+fi




Writing quadaratic equations given the sum and product of the roots
Procedure:




Examples:    Write the standard form equation of the quadratic given the following sum and product of
roots.
1.     sum=12, product= 16                            2.     sum= -7, product= 10
                                                                                                    61
College Algebra with Trig                                     Name:____________________________________
Lesson: Quadratic Word Problems
                                                              Date:_____________________________________

Objective:     To learn to interpret quadratic word problems, write symbolically and solve.

                                 x    x2
Do Now: Solve for x:                
                               x 1    2




__________________________________________________________________________________________
Method:




__________________________________________________________________________________________
1. An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the
object's height s at time t seconds after launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in meters. When does
the object strike the ground?



2 A picture has a height that is 4/3 its width. It is to be enlarged to have an area of 192 square inches. What will
be the dimensions of the enlargement?



3 You have to make a square-bottomed, unlidded box with a height of three inches and a volume of
approximately 42 cubic inches. You will be taking a piece of cardboard, cutting three-inch squares from each
corner, scoring between the corners, and folding up the edges. What should be the dimensions of the cardboard,
to the nearest quarter inch?



4 A factory produces lemon-scented widgets. You know that each unit is cheaper, the more you produce, but
you also know that costs will eventually go up if you make too many widgets, due to storage requirements. The
guy in accounting says that your cost for producing x thousands of units a day can be approximated by the
formula C = 0.04x2 – 8.504x + 25302. Find the daily production level that will minimize your costs.




                                                                                                                 62
College Algebra with Trig                                   Name:____________________________________
HW: Quadratic Word Problems
                                                            Date:_____________________________________


Answer each of the following neatly and completely on separate graph paper. Be sure to include graphs for
each problem.

1. An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. What will be
the object's maximum height? When will it attain this height?




2. The product of two consecutive negative integers is 1122. What are the numbers?




3. You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area.
What are the dimensions of the largest such yard? What is the largest area?




                                                                                                                 63
College Algebra with Trig                                          Name:________________________________________
Wkst- Quadratic Word Problems
                                                                   Date:_____________________________________

A rectangular dog pen is to be made along an
existing fence. The total length of the new fence                                     Existing fence
is to be 100 feet and the width is x.
                                                                               x          pen          x




a) Express, in simplest from, the area of the pen as a function of x.




b) What is the width that gives the maximum area? Show or explain how you arrived at your answer.




c) What is the maximum area?




                                                                                                              64
Another version of the fence problem… The owner takes                                         Existing wall
down the fence and now wants to use the 100 feet of fencing
to make two adjacent dog pens against the existing wall. See                 x         x             x
the diagram at the right.



a) If x is the length of fence perpendicular to the existing wall, express, as a function of x, the length of the fence parallel
   to the wall.




b) Express, as a function of x, the total area of both pens.




c) What is the value of x (to the nearest tenth) that gives the maximum area? Show or explain how you arrived at your
answer.




d) What is the maximum total area(to the nearest whole unit)?




                                                                                                                             65
College Algebra with Trig                                 Name:____________________________________
Lesson- Solving quadratic linear systems
                                                          Date:_____________________________________

Objective:    To learn how to solve systems of equations involving a line and a parabola.

Do Now:       Solve for x:   2 x 2  3x  9  0




__________________________________________________________________________________________
Solving a quadratic linear system
Graphical Process:                              Algebraic Process:




Examples:
      y  x 2  2x  3                             y  x 2  2x  1                         y  x2 1
1.                                           2.                                  3.
      2x  y  2                                   y  2x  3                               y  x 1




                                                                                                        66
College Algebra with Trig                                      Name:____________________________________
Lesson- Quadratic transformations
                                                               Date:_____________________________________

Objective:      To find shortcuts for different transformations of quadratics

Do Now:         Write the standard form equation given: f ( x)  2 x 2  4 x  3




Parent graph: f ( x)  2( x  3) 2  5

Reflections:




                                                                                                       67
Translations:




Dilations:




                68
Shortcut Rules for Quadratic Transformations

Type                                                     f ( x )  a ( x  h) 2  k
Reflection over the x axis                               Negate a,k
Reflection over the y axis                               Negate h
Reflection over the origin                               Negate a,h,k
Tm , n                                                   f ( x )  a ( x  h  m) 2  k  n
Dn                                                       Divide a by n and multiply (h,k) by n


Problems:
For each of the following:
   a.     reflect over the x-axis, y-axis, origin and line y = x.
   b.     translate (4, 2)
   c.     dilate by a factor of 3
   d.     dilate by a factor of 1/3
   e.     graph each on separate graph paper.


1.      y  x2




2.      f ( x)  ( x  1) 2  3




                                                                                                 69
College Algebra with Trig                                   Name:____________________________________
Lesson- Quadratic Inequalities
                                                            Date:_____________________________________


Objective:     To solve and graph quadratic inequalities

Do Now: Describe the nature of the roots of the given quadratic.
      x3 8
          
       2    x




Let a = 1st factor and b=2nd factor

                     If        ab=0      ab>0           ab<0
                     Then      a=0       a<0 and b<0    a<0 and b>0
                     Or        b=0       a>0 and b>0    a>0 and b<0



Example:       Solve and graph the solution set for each:
               a. x 2  16  0
                b. x 2  7 x  0
                c. 2 x 2  11x  5  0
                d . x 2  8 x  20




                                                                                                    70
College Algebra with Trig                                         Name:____________________________________
WKST- Quadratic Inequalities
                                                                  Date:_____________________________________


Solve and sketch the graph of the solution set for each:

1. x 2  4 x  3  0      2. x 2  7 x  6  0    3. x 2  10 x  0     4. 3x 2  x         5. x( x  4)  5
                               x   3                 3x x 2
6. x 2  2( x  12)       7.                     8.                   9. x 2  25  0     10. 16 x 2  64
                               2 x 1                2    4
                                                      x 24                             13
11. 9 x 2  6 x  1  0   12. 6 x 2  x  2  0   13.                  14. x  6          15. 2 x 2  7 x  4  0
                                                      6   x                           x6




                                                                                                                  71
College Algebra with Trig                                      Name:____________________________________
Lesson/Activity- Quadratic Exploration
                                                               Date:_____________________________________



1.      f ( x)  x 2  6 x  9
       a.       Use the Quadratic Transformer to determine the vertex h, f (h)  of f ( x)  x 2  6 x  9 .
       b.       Find the following (where “h” is the “x” value of the coordinates of the vertex)
                i.      f (h)
                ii.     f (h 1)
                iii.    f (h  2)
                iv.     f (h  3)
       c.       Find the difference between:
                i.      f (h  3) & f (h)
                ii.     f (h  2) & f (h)
                iii.    f (h 1) & f (h)




2.      f ( x)  x 2  5 x  6
       a.       Use the Quadratic Transformer to determine the vertex h, f (h)  of f ( x)  x 2  5 x  6 .
       b.       Find the following (where “h” is the “x” value of the coordinates of the vertex)
                i.      f (h)
                ii.     f (h 1)
                iii.    f (h  2)
                iv.     f (h  3)
       c.       Find the difference between:
                i.      f (h  3) & f (h)
                ii.     f (h  2) & f (h)
                iii.    f (h 1) & f (h)



3.      f ( x)  x 2  4
       a.       Use the Quadratic Transformer to determine the vertex h, f (h)  of f ( x)  x 2  4 .
       b.       Find the following (where “h” is the “x” value of the coordinates of the vertex)
                i.       f (h)
                ii.      f (h 1)
                iii.     f (h  2)
                iv.      f (h  3)
       c.       Find the difference between:
                i.       f (h  3) & f (h)
                ii.      f (h  2) & f (h)
                iii.     f (h 1) & f (h)

                                                                                                                72
4.   By now you should be noticing a pattern in the differences found in part c of the preceding questions.




5.    f (x)  2x2  8
     a.       Use the Quadratic Transformer to determine the vertex h, f (h)  of f (x)  2x2  8 .
     b.       Find the following (where “h” is the “x” value of the coordinates of the vertex)
              i.      f (h)
              ii.     f (h 1)
              iii.    f (h  2)
              iv.     f (h  3)
     c.       Find the difference between:
              i.      f (h  3) & f (h)
              ii.     f (h  2) & f (h)
              iii.    f (h 1) & f (h)




6.   f ( x)  3 x 2  6 x  5
     a.      Use the Quadratic Transformer to determine the vertex h, f (h)  of f ( x)  3x 2  6 x  5 .
     b.      Find the following (where “h” is the “x” value of the coordinates of the vertex)
             i.      f (h)
             ii.     f (h 1)
             iii.    f (h  2)
             iv.     f (h  3)
     c.      Find the difference between:
             i.      f (h  3) & f (h)
             ii.     f (h  2) & f (h)
             iii.    f (h 1) & f (h)




                                                                                                              73
7.   At first glance, it seems that there is no discernible pattern of consistency between part c of #5,6. But
     closer inspection leads to an interesting discovery. If you are not seeing it yet, try a quadratic function
     in which the leading coefficient is 4, 5, 6…




8.   Let’s prove the relationship:
     a.      Determine the axis of symmetry of: f ( x)  ax 2  bx  c and call it “h.”
             (Hint: You should already know this!)
     b.      Find:
             i.      f (h)
             ii.     f (h 1)
             iii.    f (h  2)
             iv.     f (h  3)
     c.      Find the difference between:
             i.      f (h  3) & f (h)
             ii.     f (h  2) & f (h)
             iii.    f (h 1) & f (h)




                                                                                                                   74
EXPONENTIAL AND LOG FUNCTIONS




                                75
College Algebra with Trig                                             Name:__________________________________
Lesson- properties, equations with exponents and
       power and exponential functions                                Date:___________________________________


Objectives:             use the properties of exponents
                        solve equations containing rational exponents
                        examine power and exponential functions

Do Now: Use the exponential properties to simplify and rewrite the following expressions:

(1)   ax  ay 
(2)   a 
        x y
                 
(3)   ab   x
                 
             x
      a
(4)     
      b
      ax
(5)      
      ay

(6)   a x 

(7)   a0 
__________________________________________________________________________________________
In Small Groups: Use each example in the “Do Now” to arrive at general rules as they apply to monomials with
exponents.




Using Exponential Function Properties to Solve for x:
Process 1                                                             Process 2




Examples (each relates to “Process 1”):
      44 x1  42 x2                              45 x1  162 x1                     3x  9 x4
                                                                                          2
1.                                   2.                                           3.




                                                                                                                 76
More Examples (each relates to “Process 2”):
4.    x 4  81                      5.       x 1  4                      6.      (2 x  1)5  32




Power function:


exponential function:



Small Group Activity

On your graphing calculator, simultaneously graph: y = 0.5x, y = 0.75x, y = 2x, y = 5x

(1)     What is the range of each exponential function?

(2)     What is the behavior of each graph?



(3)     Do the graphs have any asymptotes?



(4)     (a) What point is on the graph of each function?

        (b) Why?


                                   Characteristics of graphs of y = nx
                                           n>1                                    0<n<1

domain

range

y-intercept

behavior

horizontal asymptote

vertical asymptote



Extension: Graph the exponential functions y = 2x, y = 2x + 3, and y = 2x – 2 on the same set of axes.
           Compare and contrast the graphs using a table similar to the one above.
                                                                                                         77
College Algebra with Trig                                   Name:__________________________________
Lesson- Graphing exponential functions, exponential
       growth and decay                                     Date:___________________________________


Objectives:       graph exponential functions
                  use exponential functions to determine growth and decay

Using Exponential Functions for Real World Applications:

Exponential growth:



Exponential decay:



Exponential Growth or Decay: N = N0 (1 + r)t




(1) Write a formula that represents the average growth of the population of a city with a rate of 7.5% per year.
    Let x represent the number of years, y represent the most recent total population of the city, and A is the
    city’s population now. What is the expected population in 10 years if the city’s population now is 22,750
    people? Graph the function for 0  x  20.




                                                                                                               78
__________________________________________________________________________________________
(2) Suppose the value of a computer depreciates at a rate of 25% a year. Determine the value of a laptop
    computer two years after it has been purchased for $3,750.




(3) Mexico has a population of about 100 million people, and it is estimated that the population will double in
    21 years. If population growth continues at the same rate, what will be the population in:
     (a) 15 years
     (b) 30 years
     (c) graph the population growth for 0  time  50




__________________________________________________________________________________________
(4) A researcher estimates that the initial population of honeybees in a colony is 500. They are increasing at a
    rate of 14% per week. What is the expected population in 22 weeks?




                                                                                                              79
(5) In 1990, Exponential City had a population of 700,000 people. The average yearly rate of growth is 5.9%.
    Find the projected population for 2010.




(6) Find the projected population of each location in 2015:
     (a)   In Honolulu, Hawaii, the population was 836,231 in 1990. The average yearly rate of growth is
           0.7%.
     (b)   The population in Kings County, New York has demonstrated an average decrease of 0.45% over
           several years. The population in 1997 was 2,240,384.




                                                                                                           80
College Algebra with Trig                                    Name:____________________________________
Lesson- More exponential function graphs,
       Population growth, half-life


Objectives:       graph exponential functions
                  use exponential functions to determine population growth and half-life decay


(1) The population of Los Angeles County was 9,145,219 in 1997. If the average growth rate is 0.45%, predict
    the population in 2010.
    Graph the equation for 0  time  20.




(2) Radioactive gold 198 (198Au), used in imaging the structure of the liver, has a half-life of 2.67 days. If the
    initial amount is 50 milligrams of the isotope, how many milligrams (rounded to the nearest tenth) will be
    left over after:
        (a) ½ day
        (b) 1 week




                                                                                                                 81
(3) If a farmer uses 25 pounds of insecticide, assuming its half-life is 12 years, how many pounds (rounded to
    the nearest tenth) will still be active after:
        (a) 5 years
        (b) 20 years




(4) In 2000, the chicken population on a farm was 10,000. The number of chickens increased at a rate of 9%
    per year. Predict the population in 2005.
    Graph the equation for 0  time  15.




(5) If Kenya has a population of about 30,000,000 people and a doubling time of 19 years and if the growth
    continues at the same rate, find the population (rounded to the nearest million) in:
       (a) 10 years
       (b) 30 years




                                                                                                             82
College Algebra with Trig                                   Name:__________________________________
Lesson- Compound Interest
                                                            Date:___________________________________


Objectives:       use exponential functions to determine compound interest

Do Now:
(1) A laser printer was purchased for $300 in 2001. If its value depreciates at a rate of 30% a year, determine
    how much it will be worth in 2007.




(2) Rates can be compounded in different increments per year. Exponential growth occurs how often if the
    rate is compounded:

               annually:
               bi-annually:
               quarterly:
               monthly:
               weekly:
               daily:




The general equation for exponential growth is modified for finding the balance in an account that earns
compound interest.

                                  nt
                           r
Compound Interest: A  P1  
                         n




                                                                                                              83
__________________________________________________________________________________________
(1) If Charlie invested $1,000 in an account paying 10% compounded monthly, how much will be in the
    account at the end of 10 years?




(2) Mike would like to have $20,000 cash for a new car 5 years from now. How much should be placed in an
    account now if the account pays 9.75% compounded weekly?




(3) Suppose $2,500 is invested at 7% compounded quarterly. How much money will be in the account in:
     (c) ¾ year
     (d) 15 years




__________________________________________________________________________________________
(4) Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in:
     (e) ½ year
     (f) 10 years




                                                                                                       84
(5) How much money must Cindy invest for a new yacht if she wants to have $50,000 in her account that earns
    5% compounded quarterly after 15 years?




(6) Carol won $5,000 in a raffle. She would like to invest her winnings in a money market account that
    provides an APR of 6% compounded quarterly. Does she have to invest all of it in order to have $9,000 in
    the account at the end of 10 years? Show your work and explain your answer.




                                                                                                          85
College Algebra with Trig                                 Name:__________________________________
Lesson: Exponential Functions with base e
                                                          Date:___________________________________


Objective:       use exponential functions with base e


Euler Savings Bank provides a savings account that earns compounded interest at a rate of 100%. You may
choose how often to compound the interest, but you can only invest $1 over the course of one year.




                                                                                                          86
Exponential Growth or Decay (in terms of e): N = N0 ekt




(1)   According to Newton, a beaker of liquid cools exponentially when removed from a source of heat.
      Assume that the initial temperature Ti is 90F and that k = 0.275.

      (a) Write a function to model the rate at which the liquid cools.




      (b) Find the temperature T of the liquid after 4 minutes (t)




      (c) Graph the function and use the graph to verify your answer in part (b)




                                                                                                        87
(2)   Suppose a certain type of bacteria reproduces according to the model       B = 100 e0.271 t , where t is the
      time in hours.

      (a) At what percentage rate does this type of bacteria reproduce?



      (b) What was the initial number of bacteria?



      (c) Find the number of bacteria (rounded to the nearest whole number) after:
          (i) 5 hours
          (ii) 1 day
          (iii) 3 days




(3)   A city’s population can be modeled by the equation y = 33,430e0.0397 t , where t is the number of years
      since 1950.

      (a) Has the city experienced a growth or decline in population?



      (b) What was the population in 1950?



      (c) Find the projected population in 2010.




                                                                                                                     88
College Algebra with Trig                              Name:__________________________________
HW- Compound Interest
                                                       Date:___________________________________

(1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in
    the account at the end of:
        (a) 6 years 3 months
        (b) 204 months




(2) If you invest $7,500 in an account paying 8.35% compounded continuously, how much money will be in
    the account at the end of:
        (a) 5.5 years
        (b) 12 years




                                                                                                         89
__________________________________________________________________________________________
(3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to
    pay for the note now if the note gains value at a rate of 9% compounded continuously?




(4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously.
    She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be
    enough funds in Niki’s account in 6 years to meet her goal? Explain your answer.




                                                                                                           90
College Algebra with Trig                                 Name:__________________________________
Lesson- Continuous Compound Interest
                                                          Date:___________________________________

Objective:       use exponential functions to determine continuously compounded interest


Continuously Compounded Interest: A = Pert




(1) Tim and Kerry are saving for their daughter’s college education. If they deposit $12,000 in an account
    bearing 6.4% interest compounded continuously, how much will be in the account when she goes to college
    in 12 years?




(2) Paul invested a sum of money in a certificate of deposit that earns 8% interest compounded continuously.
    If Paul made the investment on January 1, 1995, and the account was worth $12,000 on January 1, 1999,
    what was the original amount in the account?




                                                                                                               91
(3) Compare the balance after 30 years of a $15,000 investment earning 12% interest compounded
    continuously to the same investment compounded quarterly.




(4) Given the original principal, the annual interest rate, the amount of time for each investment, and the type
    of compounded interest, find the amount at the end of the investment:


       (a)   P = $1,250;      r = 8.5%;      t = 3 years;                   compounded semi-annually




       (b)   P = $2,575;      r = 6.25%;     t = 5 years 3 months;          compounded continuously




                                                                                                               92
College Algebra with Trig                              Name:__________________________________
HW- Compound Interest
                                                       Date:___________________________________

(1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in
    the account at the end of:
        (a) 6 years 3 months
        (b) 204 months




(2) If you invest $7,500 in an account paying 8.35% compounded continuously, how much money will be in
    the account at the end of:
        (a) 5.5 years
        (b) 12 years




                                                                                                         93
__________________________________________________________________________________________
(3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to
    pay for the note now if the note gains value at a rate of 9% compounded continuously?




(4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously.
    She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be
    enough funds in Niki’s account in 6 years to meet her goal? Explain your answer.




                                                                                                           94
College Algebra with Trig                                   Name:____________________________________
Lesson- Properties of a logs, rewriting
       Exponential functions as logarithms, log graphs      Date:_____________________________________

Objective:
    To learn what a logarithm is
    To learn the properties of logs
    To learn to rewrite an exponential function as a logarithm
    Graphing logs
   
Do Now:     Solve for x: 3x  9 x1 and check.


_________________________________________________________________________________________
What is a logarithm?

Logarithms are inverses of exponential functions. Logarithms are functions because exponential functions are
one-to-one functions.

We cannot solve an equation like: y  2 x using the algebraic techniques we have learned so far. Therefore, we
must try an alternative technique.

Rule: x  b y is equivalent to y  log b x
The log to the base b is the exponent to which b must be raised to obtain x.

                                               Properties of Logs
                                       logb 1  0
                                       logb b  1
                                       logb b x  x

                                       blog b x  x , where x > 0
                                       log b MN  log b M  log b N
                                             M
                                       logb     logb M  logb N
                                             N
                                       log b Mp  p log b M




Example:
Convert each into logarithmic form                          Convert each into logarithmic form
                                                                          1
1. y  2 x                                                  4. log 25 5 
                                                                          2
2. 3  9                                                    5. log a b  c
     1                                                               1
3.      51                                                6. log 3    2
     5                                                               9
                                                                                                             95
_________________________________________________________________________________________
What is a Natural Logarithm?




Rule: x  b y is equivalent to y  log b x
The log to the base b is the exponent to which b must be raised to obtain x.

                                                  Properties of Logs
                                       ln 1  0
                                       ln b  1
                                       ln e x  x
                                       eln x  x , where x > 0
                                       ln MN  ln M  ln N
                                          M
                                       ln      ln M  ln N
                                           N
                                       ln M p  p ln M




Example:
Convert each into logarithmic form                               Convert each into logarithmic form
1. y  e x                                                       4. ln 5  x
2. e  x                                                         5. ln b  c
   1
3.    e 1                                                      6. ln y  2
   e


Example:
Graph each of the following on the same set of axes using the graphing calculator. y
1. y  2 x
2. x  2 y
3. log 2 y  x
4. log 2 x  y
5. y  e x
6. x  e y                                                                                            x
7. ln y  x
8. ln x  y




                                                                                                          96
College Algebra with Trig                              Name:__________________________________
Lesson/HW- Simplify log expressions, common logs, evaluate
                                                       Date:___________________________________

Objectives:              simplify expressions using the properties of logarithmic functions
                         define common logarithms
                         evaluate expressions involving logarithms


Problem Set: write the following expressions in simpler logarithmic forms:

                                                                          1
(1)   log b u2 v 7                                           (2)   logb
                                                                          a2




                  2
              m3                                                           u
(3)   log b       1
                                                             (4)   logb
                  2
                                                                          vw
              n




                                                                          3
                                                                            n
(5)   logb x                                                 (6)   logb    2
                                                                          p q3




(7)   Use logarithmic properties to find the value of x (without using a calculator):
               1         2
      logb x  logb 9  logb 8  logb 6
               2         3




                                                                                                  97
Write each expression in terms of a single logarithm with a coefficient of one:                                          u2
                                                                                      ie : 2 log b u  log b v  log b
                                                                                                                         v

(8)    5 logb x  4 logb y                                 (9)    2 logb x  logb y




                             1                             (11)  8 logb c
(10) 3 logb x  2 logb y      logb z
                             4




       3                                                          1
(12)     logb w  2 logb u                                 (13)     logb (a2  b3 )
       2                                                          3




Common Logarithm:

log 10  log x




Change of Base Formula:

            log a ln a log p a
log b a             
            log b ln b log p b




Given loga n, evaluate each logarithm to four decimal places:

(14) log 8 172                          (15) log 6 1.258                       (16) log13 0.0065




Extension: Given y = logb n, what can you determine about the log value (y) based on b and n?

                                                                                                                              98
College Algebra with Trig                                       Name:____________________________________
Lesson/HW- Properties of Logarithmic Functions,
       Simplifying logarithmic expressions                      Date:_____________________________________

Objective:           examine properties of logarithmic functions
                     simplify expressions using the properties of logarithmic functions

Use the properties of logarithmic functions to solve for x:

(1)    log 5 x  2                                      (2)     log 4 64  x




       log x 8  3                                                           2
(3)                                                      (4)     log8 x 
                                                                             3




Use the properties of logarithmic functions to simplify each expression:

(5)    log8 8                                            (6)     log 0.5 1




(7)           ,
       log10 1000                                        (8)     log 2 64




(9)    log7 343                                          (10)    log10 0.001




(11)   loge e                                            (12)    log5 3 5




                                                                                                        99
Write the following expressions in simpler logarithmic forms:


(13)   logb x 6 y 9                                                v7
                                                     (14)   logb
                                                                   u8




               mn                                                  1
(15)   log b                                         (16)   logb
               pq                                                  a4




(17)   logb 5 x                                      (18)   logb 3 x 2  y 2




                                                                               100
College Algebra with Trig                                    Name:__________________________________
Lesson- Natural Log Word Problems
                                                             Date:___________________________________

Objectives:       solve real-world applications with natural logarithmic functions


Do Now:
Laura won $2,500 on a game show. She would like to invest her winnings in an account that earns an interest
rate of 12% compounded continuously. Does she have to invest all of it in order to have $4,000 in the account
at the end of 4 years to put a down payment on a new sailboat? Show your work and explain your answer.




(1)   Ana is trying to save for a new house. How many years, to the nearest year, will it take Ana to triple the
      money in her account if it is invested at 7% compounded annually?




(2)   At what annual percentage rate (to the nearest hundredth of a percent) compounded continuously will
      $6,000 have to be invested to amount to $11,000 in 8 years.




                                                                                                              101
__________________________________________________________________________________________
(3) In 1990, Exponential City had a population of 142,000 people. In what year will the city have a
     population of about 200,000 people if it was growing at an exponential rate of k = 0.014?




(4)   If $5,000 is invested at an annual interest rate of 5% compounded quarterly, how long will it take the
      investment to double?




(5)   What was the annual interest rate (to the nearest hundredth of a percent) of an account that took 12 years
      to double if the interest was compounded continuously and no deposits or withdrawals were made during
      the 12-year period?




                                                                                                               102
College Algebra with Trig                                    Name:__________________________________
Lesson- More natural log word problems
                                                             Date:___________________________________

Objective:        solve real-world applications with natural logarithmic functions


(1) If a car originally costs $18,000 and the average rate of depreciation is 30%, find the value of the car to the
    nearest dollar after 6 years.




(2) How many years, to the nearest year, will it take for the balance of an account to double if it is gaining 6%
    interest compounded semiannually?




(3) When Rachel was born, her mother invested $5,000 in an account that compounded 4% interest monthly.
    Determine the value of this investment when Rachel is 25 years old.




                                                                                                                103
__________________________________________________________________________________________
(4) The decay of carbon-14 can be described by the formula A  A 0 e 0.000124 t . Using this formula, how many
    years, to the nearest year, will it take for carbon-14 to diminish to 1% of the original amount?




(5) In 2002, a farmer had 400 pigs on his farm. He estimated that this population of pigs will double in 15
    years. If population growth continues at the same rate, predict the number of pigs in:
           a. 2010
           b. 2030




(6) If the world population is about 6 billion people now and if the population grows continuously at an annual
    rate of 1.7%, what will the population be (to the nearest billion) in 10 years from now?




                                                                                                              104
__________________________________________________________________________________________
(7) If $100 is invested in an account that has an interest of 7% compounded quarterly, how long will it take for
    the balance to reach a value of $1,000?




(8) What interest rate (to the nearest hundredth of a percent) compounded monthly is required for an $8,500
    investment to triple in 5 years?




(9) An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision.
    However, even optical instruments have their limitations. The limiting magnitude L of any optical
    telescope with lens diameter D, in inches, is given by the equation L  8.8  5.1 logD . Use this equation
    to find the following to the nearest tenth:
           a.     the limiting magnitude for a homemade 6-inch reflecting telescope.
           b. the diameter of a lens that would have a limiting magnitude of 20.6.




                                                                                                             105
                Unit 6: Exponential & Logarithmic Functions
                            Definitions, Properties & Formulas
                                    Properties of Exponents
                Property                                                Definition
                 Product                                              x a xb  x a  b
                                                              xa
                 Quotient                                       b
                                                                   x a  b , where x  0
                                                              x
         Power Raised to a Power                                        (xa)b = xab
        Product Raised to a Power                                      (xy)a = xa ya
                                                                  a
                                                              x  xa
       Quotient Raised to a Power                                a , where y  0
                                                              y
                                                                 y
               Zero Power                                        x0 = 1, where x  0
                                                                        1
             Negative Power                                    x n       , where x  0
                                                                        xn
                                                                           1
                                                                    x n x n

            Rational Exponent
                                                  for any real number x  0 and any integer n > 1
                                                            and when x < 0 and n is odd


                                                         N = N0 (1 + r)t
     Exponential       where: N is the final amount, N0 is the initial amount, t is the number of time
   Growth/Decay        periods, and r is the average rate of growth(positive) or decay(negative) per
                       time period
                                                                         nt
                                                                r
                                                        A  P1  
       Compound                                               n
Interest (Periodic)    where: A is the final amount, P is the principal investment, r is the annual
                       interest rate, n is the number of times interest is compounded each year, and t is
                       the number of years

                                                           N = N0 ekt
     Exponential
   Growth/Decay        where: N is the final amount, N0 is the initial amount, t is the number of time
                       periods, and k (a constant) is the exponential rate of growth(positive) or
    (in terms of e)
                       decay(negative) per time period

    Continuously                                            A = Pert
    Compounded         where: A is the final amount, P is the principal investment, r is the annual
         Interest      interest rate, and t is the number of years




                                                                                                            106
             Logarithmic              are inverses of exponential functions
               Functions               a logarithm is an exponent!

                                      when no base is indicated, the base is assumed to be 10
                   Common                 log x  log10 x
                 Logarithms
                                       log x  y  10 y  x

                                                      log b n
    Change of Base                        log a n 
                                                      log b a
         Formula
                                      where a, b, and n are positive numbers, and a 1, b 1

                                      instead of log, ln is used; these logarithms have a base of e

                    Natural           ln x  log e x
                 Logarithms
                                       ln x = y  e y  x
                                      all properties of logarithms also hold for natural logarithms


                                                  Properties of Logarithmic Functions

                    If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
                    Definition                                                              Examples
logb 1  0                                              written exponentially: b0 = 1
logb b  1                                              written exponentially: b1 = b
logb b x  x                                            written exponentially: bx = bx
blog b   x
              x , where x > 0                          10 log 10   7
                                                                        7
                                                        log 3 9 x  log 3 9  log 3 x
log b MN  log b M  log b N                            log 1 yz  log 1 y  log 1 z
                                                             5               5    5


                                                            2
                                                        log4   log4 2  log4 5
         M                                                  5
logb        logb M  logb N
         N                                                  7
                                                        log8  log8 7  log8 x
                                                            x
                                                        log 2 6 x  x log 2 6
log b M  p log b M
             p

                                                        log 5 y 4  4 log 5 y
                                                        log 6 (3 x  4)  log 6 (5 x  2)
log b M  logb N         if and only if   M=N
                                                           (3x  4)  (5x  2)




                                                                                                           107
                                 Properties of Logarithmic Functions
                     If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
                     Definition                                                      Examples
logb 1  0                                       written exponentially: b0 = 1
logb b  1                                       written exponentially: b1 = b
logb b x  x                                     written exponentially: bx = bx

blog b   x
              x , where x > 0                   10 log 10   7
                                                                 7
                                                 log 3 9 x  log 3 9  log 3 x
log b MN  log b M  log b N                     log 1 yz  log 1 y  log 1 z
                                                      5               5    5


                                                     2
                                                 log4   log4 2  log4 5
         M                                           5
logb        logb M  logb N
         N                                           7
                                                 log8  log8 7  log8 x
                                                     x
                                                 log 2 6 x  x log 2 6
log b M  p log b M
             p

                                                 log 5 y 4  4 log 5 y
                                                 log 6 (3 x  4)  log 6 (5 x  2)
log b M  logb N          if and only if   M=N
                                                    (3x  4)  (5x  2)


                                                             Common Errors:
                                                                           M
                                                 logb M  logb N  logb
log b M                                                                    N
         log b M  log b N
log b N                                          log b M
                                                         cannot be simplified
                                                 log b N
                                                 log b M  log b N  log b MN
logb (M  N)  logb M  logb N
                                                 log b (M  N) cannot be simplified
                                                 p log b M  log b Mp
(log b M)  p log b M
                 p

                                                 (log b M)p cannot be simplified




                                                                                                            108
College Algebra with Trig                                       Name:____________________________________
Review- Exponential and Logarithmic Functions part 1
                                                                Date:_____________________________________

       ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK!

Write each expression in terms of simpler logarithmic forms:

                                             4

                                            s5                          1                              m5n3
(1)   log b x 5 y             (2)   log b                 (3)   log b                     (4)   logb
                                            u7                          c8                               p


Given loga n, evaluate each logarithm to four decimal places:

(5)   log 3 42                         (6)       log1 5                          (7)   log 6 0.00098
                                                    2




Solve each equation and round answers to four decimal places where necessary:

(8)   log 2 x  3                                        (9)   log 5 4  log 5 x  log 5 36


(10) 1000  75e0.5 x                                      (11) log 6 x  2


             1                                                               1
(12) log7      x                                         (13) logx 4 
            49                                                               2


(14) 10x  27.5                                           (15) log x  log5  log 2  log(x  3)


(16) log x  log 2  1                                    (17) log 4 x  3


(18) log 9 (5  x )  3 log 9 2                           (19) log 20  log x  1


(20) 2  1.002 4 x                                        (21) e25 x  1.25


(22) log(x  10)  log(x  5)  2                                                 1
                                                          (23) log 6 216           log 6 36  log 6 x
                                                                                  2




                                                                                                              109
College Algebra with Trig                                  Name:____________________________________
Review- Exponential and Logarithmic Functions part 2
                                                           Date:_____________________________________
SHOW ALL WORK:
(1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22
    million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he
    need for this investment?




(2)   The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially:
      c.   Express the number of guppies as a function of the time t.
      d.   Use your answer from part (a) to find how many guppies are present after 1 week?
      e.   Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies?




                                                                                                               110
SHOW ALL WORK:
(3)    The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by
           i
      log     0.00235x . What is the intensity, to the nearest tenth, at a depth of 40 feet?
          12




(4)    Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula
                   i
      D  10 log     , where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per
                  i0
      square meter is a standardized sound level. Use this information and formula to find the intensity of the
      sound at the concert.




                                                                                                                    111
SHOW ALL WORK:
(5)    How many years, to the nearest year, will it take the world population to double if it grows continuously at
      an annual rate of 2%.




(6)    Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If
      you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two
      banks?




(7)    Determine how much time, to the nearest year, is required for an investment to double in value if interest
      is earned at the rate of 5.75% compounded quarterly.




                                                                                                               112
TRIGONOMETRY UNIT NOTES
        PART 1




                          113
College Algebra with Trig                                       Name:____________________________________
Lesson- Intro to Trigonometry
                                                                Date:_____________________________________

Objective:       To review special right triangles and learn the basics about trig ratios.

DO NOW:
1)   Find the measure of the hypotenuse of a right triangle if the legs are 1 and 2.



2)     Given equilateral triangle ABC, find the length of the altitude to AB if BC=4.



3)     What is the length of the diagonal of square ABCD if AB=5?




__________________________________________________________________________________________
The Three Basic Trig Functions



ex: Find the length of the missing side of the triangle and the exact value of the three trigonometric functions of
the angle theta (  ) in the figures below:


1)                                                              2)
                                                5                                            7
                                                                          
                                   12                                            11




3)                                                             4)
                     3
                                                                                             7
                                                                           
             2
                                                                                 1



5)     Find the values of the three trigonometric functions for angle  in standard position if a point with the
       coordinates (-3, -5) lies on its terminal side.



                                                                                                               114
College Algebra with Trig                              Name:____________________________________
Lesson: The building blocks of trig functions
                                                       Date:_____________________________________

Building Blocks of the Unit Circle
Graph set up:




Quadrantal Angles:




Coterminal Angles:




Examples:
Find an angle that is COTERMINAL with each. *Note: it is often helpful to draw a diagram when solving
these types of problems.

1.     100º                                            2.     650º




3.     405º                                            4.     400º




                                                                                                        115
Reference Angles:




Examples:
Find an angle that is the REFERENCE ANGLE of each. *Note: it is often helpful to draw a diagram when
solving these types of problems.

5.     100º                                                  6.      650º




7.     405º                                                  8.      400º




Function of a positive acute angle:




Examples:
Express the given function as a function of a positive acute angle and, if possible, find the exact function value.
1.     tan 225º                       2.      cos 100º                       3.       cos 405º




4.     sin 650 º                      5.      tan (-120º)                    6.     cos 400º




                                                                                                                116
College Algebra with Trig                                    Name:____________________________________
Lesson: Finding one trig function exactly given another
                                                             Date:_____________________________________

Objective:      To be able to find one trig function exactly given another without solving for the angle.

                             3             7
DO NOW:         If sin       and cos     determine the quadrant in which  lies.
                             4            4




__________________________________________________________________________________________
Solving trig functions exactly
Process:




Examples:

Given the value of sin or cos and the quadrant in which  lies, find the value of the other function.

                1                                                            4
1.     sin    , Quadrant IV                               2.      cos   , Quadrant II
                2                                                            5




                24                                                              5
3.     cos       , Quadrant I                              4.      sin        , Quadrant II
                25                                                             12




                                                                                                            117
College Algebra with Trig                                   Name:____________________________________
Lesson- The Unit Circle
                                                            Date:_____________________________________

Objective:    To learn the meaning of a radian and to learn to create the unit circle.

DO NOW:       Find the reference angle that corresponds to an angle of 240º.




__________________________________________________________________________________________
What is a unit circle?




__________________________________________________________________________________________
Special Right Triangles
30-60-90 Right Triangles                        45-45-90 Right Triangles



__________________________________________________________________________________________
Use the area below to create the unit circle with degrees that are multiples of 30 and 45. Note the sine and
cosine result for each.




                                                                                                        118
Find the exact value of each expression without using a calculator:
1.      tan 135                                           2.     sin 150




3.      cos 210                                             4.        cos315




Practice:
Express the given function as a function of a positive acute angle and, if possible, find the exact function value.
5.     tan 225º                               6.     cos 100º                        7.      cos 405º




8.     sin 650º                               9.      tan (-120º)                   10.     cos 400º




Find the exact value of the given expression.
11.    tan 135º + sin 330º            12.     sin 300º + sin (-240º)         13.    (sin 60º)(cos 150º) – tan (-45º)




                                                                                                                119
Deg 0        30    45    60    90    120         135         150          180         210         225         240         270         300          315         330         360
Sin      0     1     2     3     4     3           2           1            0              1           2           3            4              3          2          1       0
                                                                                                                                                         
        2     2     2     2     2     2           2           2            2              2           2           2            2              2          2          2       2


Cos      4     3     2     1     0         1            2             3         4          3           2           1        0           1            2           3           4
                                                                                                        
        2     2     2     2     2         2            2             2         2          2           2           2        2           2            2           2           2


Tan     0     1     2      3    4            3           2           1            0       1           2           3               4           3            2           1    0
                                                                                                                                                        
        4      3    2     1     0            1           2           3            4       3           2           1               0           1            2           3    4


Csc     2     2     2     2     2     2           2           2            2              2           2           2            2              2          2          2       2
                                                                                                                                                         
        0     1     2      3    4        3           2           1            0               1           2           3           4           3            2           1    0


Sec     2     2     2     2     2         2            2             2         2          2           2           2        2           2            2           2           2
                                                                                                        
        4      3    2     1     0            1           2           3            4           3           2           1       0           1            2           3        4


Cot     4      3    2     1     0            1           2           3            4       3           2           1               0           1            2           3    4
                                                                                                                                                        
        0     1     2      3    4            3           2           1            0       1           2           3               4           3            2           1    0




                                                                                                                                                                                 120
College Algebra with Trig                                      Name:____________________________________
Review- Basic Trig Test
                                                               Date:_____________________________________

Objective:         To review the following concepts in preparation for a test
                   I.   SOHCAHTOA
                   II. Coterminal, reference, and quadrantals
                   III. Finding exact trig values
                   IV. Unit Circle

MIXED PBLM SET
Answer each of the following neatly and completely and show all work. NO CALCULATORS.



1.       Find the exact value of each expression:
                a.      cos 270º                                        b.     sin 90º




                                                                        7
3.       Without finding , find the exact value of tan  if cos       and sin   0 .
                                                                        8




4.       Find the values of the three trigonometric functions for angle  in standard position if a point with the
         coordinates (-3, -5) lies on its terminal side.




5.       Given the following triangle find the measure of angle  exactly

         
     3       6




                                                                                                                 121
                                                                       8
6.   Without finding , find the exact value of cos  if sin          and tan  0 .
                                                                       9




7.   Find the values of the three trigonometric functions for angle  in standard position if a point with
     the coordinates (5, -4) lies on its terminal side.




8.    Find the length of the missing side and the exact value of the three trigonometric functions of the angle 
      in each figure:

a)                                           b)                                          c)
                                                               3                                   
                                5                                                              8
                                                      2
                       12                                                                              7




                                                                  7
d)                                           e)
                                                  
                            9
                  13                              11




9.    Find the exact value of each expression without using a calculator:
      a.      sin 150                     b.     cos210                                 c.   cos315




                                                                                                             122
TRIGONOMETRY UNIT NOTES
        PART 2




                          123
College Algebra with Trig                                   Name:____________________________________
Activity- What is a radian?
                                                            Date:_____________________________________

Objective:    To discover what a radian is.

Follow the directions below and be sure to round each answer to the nearest ten-thousandth. Do all work
on separate paper.

1.     Use the paper provided to cut as many horizontal strips as you can. For convenience purposes, make
       sure each is about ¾ inch wide.

2.     Use a piece of tape to tape the edges together to form a quasi-cylinder.

3.     Trace the circular edge on another sheet of paper and estimate the center.

4.     Measure the distance from the center to the circumference of the circle.

5.     Unfurl the paper and measure its length.

6.     Determine how many times the length of the radius goes into the distance found in step 5.

7.     Make a table comparing your width of paper, distance from center to edge, and quotient.

8.     Find the mean of all of your quotients. What does the mean represent?

9.     How many radians are in the circumference of a circle?

10.    1 radian is approximately equal to ______________ degrees.

11.    There are ________ radians in 180 degrees and there are ________ radians in 360 degrees.

12.    In your own words, a radian is…?

PRACTICE: As you do the practice problems, see if an equation for the conversion from radians to degrees
and degrees to radians becomes apparent.

Find the radian measure                                             Find the degree measure
for each degree measure:                                            for each radian measure:

1.     720                                                          1.      2

                                                                            
2.     90                                                           2.
                                                                            2

                                                                            
3.     45                                                           3.
                                                                            6

                                                                            2
4.     60                                                           4.
                                                                             3


                                                                                                            124
Deg 0        30    45    60    90      120         135         150          180         210         225         240         270         300          315         330         360
Rad   0     1    1    2    3      4          3          5           6          7          5          8          9          10          7          11         12
                                   ,                                            ,                                              ,                                                 ,
       6      6     4     6     6       6           4           6            6           6           4           6           6           6            4           6           6
                               2                                           4                                              6                                               8
                                4                                            4                                               4                                                4
Sin      0     1     2     3     4       3           2           1            0               1           2           3         4                3          2          1       0
                                                                                                                                                           
        2     2     2     2     2       2           2           2            2               2           2           2         2                2          2          2       2


Cos      4     3     2     1     0           1            2             3         4           3           2           1       0           1            2           3           4
                                                                                                          
        2     2     2     2     2           2            2             2         2           2           2           2       2           2            2           2           2


Tan     0     1     2      3     4             3           2           1            0       1           2           3               4           3            2           1    0
                                                                                                                                                          
        4      3    2     1      0             1           2           3            4       3           2           1               0           1            2           3    4


Csc     2     2     2     2     2       2           2           2            2               2           2           2           2              2          2          2       2
                                                                                                                                                           
        0     1     2      3     4         3           2           1            0               1           2           3           4           3            2           1    0


Sec     2     2     2     2     2           2            2             2         2           2           2           2       2           2            2           2           2
                                                                                                          
        4      3    2     1      0             1           2           3            4           3           2           1       0           1            2           3        4


Cot     4      3    2     1      0             1           2           3            4       3           2           1               0           1            2           3    4
                                                                                                                                                          
        0     1     2      3     4             3           2           1            0       1           2           3               4           3            2           1    0




                                                                                                                                                                                     125
College Algebra with Trig                                     Name:____________________________________
Lesson- Reciprocal Trig Functions
                                                              Date:_____________________________________

Objective:     To learn about the reciprocal trig functions csc, sec, cot.

DO NOW:        Construct the unit circle in the space provided.




__________________________________________________________________________________________
Reciprocal Trig Functions




Examples:
Find the exact values of the following trig functions.
1.     sec 300                        2.      cot 270                        3.   csc (-210)




4.     sec π                          5.      (sec 2π/3)(sin 2π/3)           6.   cot (π/4) +csc (3π/4)



                                                                                                          126
College Algebra with Trig                                    Name:____________________________________
Lesson: Using radians to solve trig functions
                                                             Date:_____________________________________


Objective:     Use radians to solve trig functions

DO NOW:        Determine the exact value of sin (-45) without using the calculator.




Sketch a figure and find the coordinates for each circular point:
        8                                                         5 
1.                                                         2.        
        3                                                          6 




         7                                                        11 
3.                                                        4.          
         6                                                         3 




Find the sine, cosine, and tangent of each radian measure:
                                                    3
5.                                            6.                                      7.   2
        2                                             4




                                                                                                    127
College Algebra with Trig                                    Name:____________________________________
Lesson: Radians and Arc Length
                                                             Date:_____________________________________

Objective:     Discuss the relationship among central angles, radii and arc lengths.

DO NOW:        Determine the exact value of tan (-45) without using the calculator.




__________________________________________________________________________________________
What is a central angle?                        What is an arc length?




Relationship among central angle, radius and arc length:




Examples:
1.    Find the measure of a positive central angle that intercepts an arc of 14 cm on a circle of radius 5 cm.




2.     Find the length of the arc intercepted by a central angle of 3.5 radians on a circle of radius 6 m.




3.     A wheel of radius 18 cm is rotating at a rate of 90 revolutions per minute.
       a.    How many radians per minute is this?
       b.    How many radians per second is this?
       c.    How far does a point on the rim of the wheel travel in one second?
       d.    Find the speed of a point on the rim of the wheel in centimeters per second.




                                                                                                                 128
College Algebra with Trig                                    Name:____________________________________
Lesson- Inverse Trig Functions
                                                             Date:_____________________________________

Objective:    To learn to use inverse trig functions to solve for an angle or angles

                                            5
DO NOW:       Find the exact value of sec
                                             4




__________________________________________________________________________________________
What is an inverse trig function? What is it used for?




Examples:

1.     Write in the form of an inverse function:     cos  




                                                                   2
2.     Write in the form of an inverse function:     cos 45  
                                                                  2




3.     Solve by finding the value of x to the nearest degree:          Sin 1 (1)  x




                                                                              1
4.     Solve by finding the value of x to the nearest degree:          Arc cos  x
                                                                              2




Find each value (put angles in radian measure). Round any decimals to the nearest hundredth.
                3                                                          3
5.      Arc tan
                3 
                                                         6.     cos 2Sin 1
                                                                     
                                                                                 
                                                                           2 




                                                                                                    129
College Algebra with Trig                                  Name:____________________________________
Lesson: Law of Cosines
                                                           Date:_____________________________________

Objective:       solve triangles by using the Law of Cosines



Law of Cosines:




1.     Suppose a triangle ABC has side a = 4, side b = 7, and angle C = 54º. What is the measure of side C?




2.     Suppose a triangle XYZ has sides of x = 5, y = 6, and z = 7. What is the measure of the angle across
       from the side of measure 6?




3.     Suppose a triangle ABC has side b = 2, side a = 5, and angle B = 27º. Find the measure of side c.




4.     Suppose a triangle ABC has side b = 4, side a = 5, and angle B = 27º. Find the measure of side c.




Exit Ticket: Complete on separate paper and hand in when finished.

1.     In a triangle PQR we have p = 8 and r = 11. Angle Q is 47º. What is the length of side q?
2.     A triangle XYZ has sides x = 1, y = 2, and z = 2.5. What is the measure of angle Y?
                                                                                                              130
College Algebra with Trig                                  Name:____________________________________
Lesson- Forces and the Law of Cosines
                                                           Date:_____________________________________

Objective:    To determine the resultant force vector when given two force vectors and an included angle.

DO NOW:       If mA  30 , AC=5, and AB=7, solve the triangle. Find all sides to the nearest tenth and
              angles to the nearest degree.




__________________________________________________________________________________________
Force- push or pull upon an object resulting from the object's interaction with another object.

Vector- a quantity of force having both magnitude and direction.




Examples:

1.     Two forces separated by 52 degrees acts on an object at rest. The magnitude of the two forces are 32
       Newtons and 17 Newtons. Find the resultant force vector to the nearest Newton.




                                                                                                              131
2.     A game of “Three Way Tug-O-War” is being played by a group of students. Two of the students are
       trying to gang up on the other. They believe that it will be easier to win if they increase the angle they
       create with the third person. Is that true? Justify your answer by providing examples.




3.     Two fisherman have hooked the same fish and they are trying to cooperatively reel it in. The angle the
       fisherman make with the fish is 87 degrees. If the first fisherman’s line has a maximum tensile strength
       223 Newtons and the second fisherman’s line has a maximum tensile strength of 401 Newtons and the
       fishermans’ lines are at maximum strain, what is the resultant force applied to the fish?




4.     What is the angle separating two component force vectors whose magnitude are 15N and 17N
       respectively if the resultant vector is 21N?




Exit Ticket

Make up your own real-life Force scenario. Solve it and turn in on a separate piece of paper. THE MORE
CREATIVE THE BETTER!
                                                                                                               132
College Algebra with Trig                                       Name:____________________________________
Lesson: Law of Sines, area of a triangle
                                                                Date:_____________________________________

Objective:         solve triangles by using the Law of Sines
                   find the area of a triangle


Law of Sines:




Area of Triangles:




(1)   Given DEF where D = 29, E = 112, and d = 22:
       (a)   Solve DEF, rounding answers to the nearest tenth
       (b)   Find the area of  DEF to the nearest tenth




                                                                                                       133
(2)   Given ABC where A = 13, B = 6520, and a = 35:
      (a)   Solve ABC such that:
              (i)  C is in DMS form
             (ii)  b is rounded to the nearest tenth
            (iii)  c is rounded to the nearest tenth
                                                                                 1
      (b)   Find the area of ABC, to the nearest tenth, using the formula K      bc sin A
                                                                                 2




(3)   Given GHJ where g = 45.7, H = 111.1, and J = 27.3:
      (a)   Solve GHJ, rounding answers to the nearest tenth
      (b)   Find the area of  GHJ (to the nearest tenth)




                                                                                              134
College Algebra with Trig                                   Name:____________________________________
WKST- Law of Sines/Cosines WP
                                                            Date:_____________________________________


1.    A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle
      from the tip of the shadow to the top of the lamppost is 45. Find the length of the lamppost to the
      nearest tenth of a foot.




                                                                                     2




                                                                                                        45


                                                                                                25 ft




2.    A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the
      dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To
      the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened?




                                                                                      40 m

                                                                       25 m


                                                                                             122




3.    Using the picture seen to the right, and rounding to the nearest tenth of a
      meter, find the height of the tree.
                                                                                                110


                                                                                      23

                                                                                             120 m



                                                                                                              135
College Algebra with Trig                                     Name:____________________________________
Lesson:       Determining the number of
              Distinct triangles (ambiguous case)             Date:_____________________________________

Objective:     To determine the number of distinct triangles that can be formed given an angle and two
               consecutive sides

DO NOW:        The sides of a triangle measure 6, 7, and 9. What is the largest angle in the triangle?




__________________________________________________________________________________________
Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct
                  triangles for which you can solve.


There is a shortcut method to finding the number of distinct triangles that exist:

Assume: Given two sides and one opposite angle:

If a is acute:                                                If a is obtuse:
a  b sin a  no solution
a  b sin a  one solution                                    a  b  no solution
b  a  b sin a  two solutions                               a  b  one solution
a  b  one solution



Examples:
How Many distinct triangles can be formed from the given information?
1. a  2 , b  3, mA  45 




2. a  9, b  12, and mA  35




                                                                                                         136
College Algebra with Trig                                    Name:____________________________________
Lesson: Law of Sines- The Ambiguous Case
                                                             Date:_____________________________________

Objective:         solve triangles by using the Law of Sines (ambiguous case)
                   find the area of a triangle

DO NOW:         The sides of a triangle measure 6, 7, and 9. What is the measure of the smallest angle in the
                triangle?




Law of Sines:



__________________________________________________________________________________________
Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct
                  triangles for which you can solve.

             Showing all work, find all solutions for each ABC. If no solutions exist, write none.
                                    Round all answers to the nearest tenth.

1.     A = 42, a = 22, b = 12                               2.     b = 50, a = 33, A = 132




3.     a = 125, A = 25, b = 150                             4.     a = 32, c = 20, A = 112




5.     b = 15, c = 13, C = 50                               6.     a = 12, b = 15, A = 55




                                                                                                                137
College Algebra with Trig                                     Name:____________________________________
Review- Trig Test #2
                                                              Date:_____________________________________

Objective:    To prepare for a test on the following topics

              I.       Reciprocal Trig Functions
                       a.     Secant
                       b.     Cosecant
                       c.     Cotangent
              II.      Radians and Arc Length
              III.     Inverse trig functions
              IV.      Law of Sines
                       a.     Including Ambiguous Case
              V.       Law of Cosines
                       a.     Including Forces
              VI.      Area of a triangle

Practice Problem Set

1.     Two adjacent apartment buildings in Geometry Garden Estates share a triangular courtyard. They plan
       to install a new gate to close the courtyard that forms an angle of 1048 with one building and an angle
       of 4820 with the second building, whose length is 527 feet.
       a.      Find, to the nearest tenth, the area of the courtyard.
       b.      Find, to the nearest tenth, the length of this new gate.




2.     A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle
       from the tip of the shadow to the top of the lamppost is 45. Find the length of the lamppost to the
       nearest tenth of a foot.




                                                                                       2




                                                                                                      45


                                                                                              25 ft




                                                                                                               138
3.   A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the
     dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To
     the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened?




                                                                                          40 m

                                                                        25 m


                                                                                                 122




4.   Using the picture seen to the right, and rounding to the nearest tenth of a
     meter, find the height of the tree.
                                                                                                    110


                                                                                          23

                                                                                                 120 m




5.   Solve ABC if c = 49, b = 40, and A = 53 (round each answer to the nearest tenth)




6.   Solve ABC, to the nearest tenth, if A = 50, b =12, and c = 14 & find the area.




                                                                                                           139
7.   Two forces act upon a body at rest. The first force, 35N, is separated from the second force, 52N, by 63
     degrees. Determine the resultant force to the nearest Newton.




8.   The resultant force acting on a body at rest is 90 pounds. If one of the component forces is 70 pounds
     and the other is 110 pounds, find the angle separating the resultant force from the larger of the
     component forces to the nearest ten minutes.




9.   Find each of the following exactly:
                                                   5 
     a.     sec                        b.       cot                         c.     csc( )
                                                                                          300
               6                                    4 




                                                                                                          140
TRIGONOMETRY UNIT NOTES
        PART 3




                          141
College Algebra with Trig                                      Name:____________________________________
Lesson- Graphing sine and cosine functions
                                                               Date:_____________________________________

Objectives:    To construct graphs of the sine and cosine functions

Definitions:

Amplitude:




Frequency:




Period:




Graphing Trigonometric Functions:

(1)   Graph y  sin x in the interval -2  x  2

                                                     y
                                                     1


                                                                                               x
                                                                          



                                                     -1


Period:                                                   x-intercepts:

Domain:                                                   y-intercepts:

Range:                                                    Maximum point:

                                                          Minimum point:
                                                                                                      142
(2)   Graph y  cos x in the interval -2  x  2

                                                            y
                                                            1


                                                                                     x
                                                                                 



                                                            -1


Period:                                                          x-intercepts:

Domain:                                                          y-intercepts:

Range:                                                           Maximum point:

                                                                 Minimum point:



(3)   Graph y  2 sin 2 x in the interval -  x  2


                                                        y




                                                                                    x




                      1                             3
(4)   Graph y  3 cos  x  in the interval    x 
                      2                              2
                                                        y




                                                                                    x




                                                                                         143
College Algebra with Trig                                           Name:______________________________
HW- Sine and cosine graphs
                                                              Date:_______________________________

Objective:         find the amplitude and period to graph sine and cosine functions

(1)   y  2 sin x              2  x  2

                                                          y




                                                                                               x
                                                                          




(2)   y  4 cos
                    x
                                3  x  3
                    3

                                                      y




                                                                                              x




(3)   y
             1
               cos 2x             x  2
             2

                                                      y




                                                                                              x




                                                                                                     144
(4)   y  2 sin
                  x
                        2  x  2
                  4


                                         y




                                                                x




(5)   y  1.5 cos 4x      x  


                                                 y




                                                                    x
                                                         2




(6)   y  3 sin
                   x
                        2  x  2
                   2



                                             y




                                                                 x
                                                             




                                                                         145
                                   3 
(7)   y   sin x           x     
                                    2 

                                               y
                                               1


                                                                x
                                                            



                                               -1


(8)   y  2 cos 4x        x  


                                           y




                                                               x
                                                        2




           1     1 
(9)   y     sin  x     4  x  4
           2     2 



                                           y




                                                               x




                                                                    146
College Algebra with Trig                                    Name:____________________________________
Lesson: Phase shifts and translations
                                                             Date:_____________________________________

Objectives:       find the phase shift for sine and cosine functions
                  graph translations of sine and cosine functions

Phase Shift:




Translation:




(1)   Graph y  sin x   in the interval 0  x  4


                   y




                                                                                           x




                         
(2)   Graph y  cos  2x   in the interval 0  x  2
                         2


                   y




                                                                                           x




                                                                                                    147
                          
(3)   Graph y  3 cos  x   in the interval -2  x  2
                          2



                                                  y




                                                                x




                       x 
(4)   Graph y  2 sin    in the interval -2  x  2
                      2 8



                                                  y




                                                                x




(5)   Graph y  cos 2x   in the interval -2  x  2



                                                  y




                                                                x




                                                                     148
(6)   Graph y  2 cos x  4 in the interval 0  x  4

                     y

                                                                    x
                                        




(7)   Graph y  2 sin 2x  2 in the interval -2  x  2

                                                     y




                                                                   x




                      x    
(8)   Graph y  4 cos      6 in the interval 0  x  4
                      2    
                     y

                                                                    x
                                        




                                                                        149
College Algebra with Trig                                      Name:______________________________
HW- Phase shift and translations
                                                       Date:_______________________________

            GRAPH THE FOLLOWING TRIGONOMETRIC FUNCTIONS WITHIN THE GIVEN INTERVAL:

(1)   y  cos ( x  )  1               x  3

                                           y




                                                                                      x




(2)   y  sin
                x 1
                                      2  x  3
                2 2


                                           y




                                                                                      x




                    
(3)   y  2 sin  x    2            2  x  2
                    2
                                            y




                                                                                       x




                                                                                               150
             cos 4x        x  
           1
(4)   y
           2



                                y




                                                     x




                    
(5)   y  3 sin  x        2  x  2
                    4




                                    y




                                                         x




                                                              151
College Algebra with Trig                                     Name:____________________________________
WKST- More practice with phase shift and translations
                                                              Date:_____________________________________

Objectives:       find the phase shift and vertical translation for sine and cosine functions
                  graph translations of sine and cosine functions



(1)   Graph y  2 cos x  1 in the interval 0  x  4




                     y

                                                                                                 x
                                         




(2)   Graph y  2 sin x  2 in the interval -2  x  2



                                                       y




                                                                                                x




                                                                                                     152
                      x    
(3)   Graph y  4 cos      6 in the interval 0  x  4
                      2    




                     y

                                                                    x
                                        




                       x
(4)   Graph y  3 cos    1 in the interval -  x  2
                      2




                                                     y




                                                                   x




                                                                        153
College Algebra with Trig                                    Name:_______________________________
Lesson- Writing equations of sine and cosine functions
                                                             Date:________________________________

Objectives:        write the equations of sine and cosine functions given the amplitude, period, phase shift, and
                    vertical translation

Writing Trigonometric Functions:




Write an equation of the sine function with each given amplitude and period:

(1)   amplitude = 4,           period = 2



                                          
(2)   amplitude = 35.7,        period =
                                          4



(3)   amplitude = 0.8,         period = 10




Write an equation of the cosine function with each given amplitude and period:

                    5                     
(4)   amplitude =     ,        period =
                    8                     7



(5)   amplitude = 0.5,         period = 0.3



(6)   amplitude = 17.9,        period = 16




                                                                                                               154
Write a sine function with each given period, phase shift, and vertical translation:

(7)   period = 2,     phase shift = 0,         vertical translation = -6



                                      
(8)   period =     ,   phase shift =     ,      vertical translation = 0
                 2                     8



                                          
(9)   period = ,      phase shift =       ,   vertical translation = 3
                                          4




Write a cosine function with each given period, phase shift, and vertical translation:

(10) period = 3,      phase shift = ,         vertical translation = -1



(11) period = 5,      phase shift = -,        vertical translation = -6



                                         
(12) period =      ,   phase shift =       ,   vertical translation = 10
                 3                        2




State the amplitude, period, phase shift, and vertical translation for each function:

                       x
(13) y  7.5 cos                       A=                  P=              PS =         VT =
                       3

           1
(14) y      sin 6x  8                 A=                  P=              PS =         VT =
           4

             3          
(15) y       sin  2x    9         A=                  P=              PS =         VT =
             5          4

                   
(16) y  cos  3 x                    A=                  P=              PS =         VT =
                   2




                                                                                                155
College Algebra with Trig                                                       Name:______________________________
HW- Writing the equation of sine and cosine functions
                                                                                Date:_______________________________

Write an equation of the sine function with each given amplitude and period:

(1) amplitude = 6.7,            period = 6



(2)   amplitude = 0.5,          period = 




Write an equation of the cosine function with each given amplitude and period:

                     3
(3)   amplitude =      ,        period = 0.2
                     7



                     1                         2
(4)   amplitude =      ,        period =
                     5                         5




Write a sine function with each given period, phase shift, and vertical translation:

                                         
(5)   period = 2,     phase shift =      ,        vertical translation = 12
                                         2



(6)   period = 8,     phase shift = -,            vertical translation = -2




Write a cosine function with each given period, phase shift, and vertical translation:

                                         
(7)   period = ,      phase shift =      ,        vertical translation = -1
                                         4



                                       
(8)   period = 4,     phase shift =     ,          vertical translation = 5
                                       8




                                                                                                                 156
SHOW ALL WORK:

State the amplitude, period, phase shift, and vertical translation for each function:


(9)   y  2 cos 0.5x  3            A=                 P=                  PS =        VT =




           2     3
(10) y      cos    x                A=                 P=                  PS =        VT =
           3     7




                   
(11) y  3 sin 2x                 A=                 P=                  PS =        VT =
                   2




          1  x 
(12) y   sin    6              A=                 P=                  PS =        VT =
          3 3 6




                     
(13) y  4 sin 4 x    4         A=                 P=                  PS =        VT =
                     4




               x 
(14) y  8 sin    1              A=                 P=                  PS =        VT =
              2 8




                                                                                               157
College Algebra with Trig                                                           Name:__________________________________________________
Lesson- Graphing Tangent
                                                                                    Date:___________________________________________________

Objective:       To learn the proper technique for graphing the tangent function.

The tangent curve is unlike the sine and cosine curves.
       1.     It is not a smooth continuous curve
       2.     There is no maximum or minimum height (goes on to  )
       3.     There are values that are undefined

What are the tangent values in the interval 0    2 ?

Radians      0                                    2      3      5             7    5     4    3     5    7     11   2
                      6       4       3       2        3       4       6               6     4      3     2      3     4      6
Degrees

Tan
(fraction)
Tan
(decimal)


Which values for tan are undefined in the interval 0    2 ?



What happens graphically when there is an undefined value?



Graph y  tan x in the interval 0    2 on the graph on the reverse:




                                                                                                                                         158
College Algebra with Trig                                Name:______________________________
Lesson- Graphing Reciprocal Trig Functions
                                                         Date:_______________________________

Objective- To learn how to graph the reciprocal of sine, cosine, and tangent.

1.     Graph the reciprocal of a sine function:




2.     Graph the reciprocal of a sine function:




3.     Graph the reciprocal of a sine function:




                                                                                                159
College Algebra with Trig                                   Name:______________________________
Review for test- Trig Graphs
                                                            Date:_______________________________
Objective:     To review for a test on:
               I.     Graphing sine (including phase shift and translation)
               II.    Graphing cosine (including phase shift and translation)
               III.   Graphing tangent from [2 ,2 ]
               IV.    Writing the equations of sine and cosine
               V.     Graphing Reciprocal Trig Functions


                     x 
1.   Graph y  2 sin    in the interval -2  x  2
                     2 4
                                                 y




                                                                                         x




                         
2.   Graph y  3 cos  x   in the interval -2  x  2
                         2                     y




                                                                                         x




                    
3.    y  2 sin  x    2     2  x  2
                    4                          y




                                                                                         x




                                                                                                   160
4.   y  cos x     4        x  
                                                y




                                                                            x




5.   y  2 sin 4 x  2       x  

                                                y




                                                                            x




               x 
6.   y  2 cos     1         x  
               2 2
                                                y




                                                                            x




7.    Graph, on separate paper, the reciprocal of y=sin x and y=cos x.




                                                                                 161
TRIGONOMETRY UNIT NOTES
          PART 4




                          162
College Algebra with Trig                                          Name:_____________________________
HW- Rational Expressions
                                                            Date:_______________________________

Simplify each of the following and put all answers in simplest factored form.

        5x       3x                                                      x 2  x  30      2 x 2  11 x  12
1)              2                                                 2)                    
     x  5x  6 x  4
      2
                                                                        2 x 2  11 x  6    4x 2  4x  3




                                                                          x     x2
                                                                             
       3 a2  b2 5                                                      x 1 x2 1
3)                                                               4)
     a  b 2a  b a                                                       3x   2x 2
                                                                             
                                                                         x 1 x 1




                                                                                                               163
  Trigonometry Unit: Formulas & Identities




      Pythagorean and Quotient Identities
sin2 A + cos2 A = 1                sin A
tan2 A + 1 = sec2 A       tan A 
                                   cos A
cot2 A + 1 = csc2 A                cos A
                          cot A 
                                   sin A



      Functions of the Sum of Two Angles
    sin (A + B) = sin A cos B + cos A sin B
    cos (A + B) = cos A cos B – sin A sin B
                        tan A  tan B
          tan(A  B) 
                       1  tan A tan B


   Functions of the Difference of Two Angles
    sin (A – B) = sin A cos B – cos A sin B
    cos (A – B) = cos A cos B + sin A sin B
                         tan A  tan B
         tan(A  B) 
                        1  tan A tan B



        Functions of the Double Angle
           sin 2A = 2 sin A cos A
          cos 2A = cos2 A – sin2 A
           cos 2A = 2 cos2 A – 1
            cos 2A = 1 – 2 sin2 A
                         2 tan A
            tan 2A 
                       1  tan2 A




       Functions of the Half Angle
            1         1  cos A
        sin A  
            2              2
              1     1  cos A
        cos     A
              2          2
              1     1  cos A
        tan     A
              2     1  cos A




                                               164
College Algebra with Trig                                                 Name:______________________________
Lesson- Proving Trig Identities
                                                                Date:_______________________________

Objective:      To prove the validity of pythagorean, reciprocal and quotient identities.

                    5x       3x
DO NOW:                     2
                 x  5x  6 x  4
                  2




____________________________________________________________________________________
Pythagorean Identities

Identity:       an equation that is true for all values of the variable
                                                    2
                                                a  a2
Ex:     a 2  b 2  (a  b)( a  b)     or         2
                                                b  b

Pythagorean Identities
1st : cos2   sin 2   1
This identity is from the equation of a circle centered at (0,0) and whose radius is 1.




From the previous identity, others can be obtained.
2nd : 1  tan 2   sec2 



3rd:    cot2   1  csc2 



Proving Trig Identities
The idea is to show both sides of the equation can be written in the same form.

Helpful hints:
1.     Always start with the most complicated side
                                                                          sin 
2.      Look for algebraic identity that can be applied (ex: tan                    or      cos2   sin 2   1 )
                                                                          cos
3.      Try writing the expression in terms of sine and/or cosine.
4.      If you get stuck on one side, try the other!
5.      Never cross over the equal sign. Work with each side independently.

*To prove that an equation is an identity, you must show that it is true for all values of the variable for each side
of the equation.
Examples:

                                                                                                                   165
                 1    1        1
1.   Prove:             
              sin  cos  sin  cos2 
                  2    2     2




2.   Prove: sin 4   cos4   sin 2   cos2 




3.   Prove: tan   cot  csc sec




              csc x  cot x
4.   Prove:                  cot x csc x
              tan x  sin x




                                                  166
College Algebra with Trig                                  Name:____________________________________
CW/HW: Trig Identity Mixed Problem Set
                                                           Date:_____________________________________

Objective:     To use algebraic/proportion techniques in conjunction with Pythagorean, reciprocal, sum,
difference, double angle and half angle rules to verify identities.



Verify each identity:

1.     sec4 x  2 sec2 x tan 2 x  tan 4 x  1             2.     (1  cos x)(cscx  cot x)  sin x




       1  tan y sec y                                             cos 2 x  cot x cos 2 x tan x  1
3.                                                        4.                     
       1  cot y csc y                                             cos 2 x  cot x cos 2 x tan x  1




                                                                   tan y  cot y
5.     cos 2 x(1  sec 2 x)   sin 2 x                    6.                     sec y
                                                                       csc y




       1  cos 2 x                                                    1         1
7.                  cot x                                 8.                         2 sec 2 x
         sin 2 x                                                  1  sin x 1  sin x




                                                                                                          167
College Algebra with Trig                                    Name:____________________________________
HW: Trig Identity Mixed Problem Set
                                                             Date:_____________________________________

Objective:     To use algebraic/proportion techniques in conjunction with Pythagorean, reciprocal, sum,
difference, double angle and half angle rules to verify identities.


HW Verify each of the following neatly on separate paper. Be sure to show all steps for full credit. WILL BE
COLLECTED AND GRADED! [30 point quiz]

1. cos x tan x  sin x                              2. cot x cos x  sin x  csc x




     1  sin x     cos x                                 sin 2 x  2 sin x  1 1  sin x
3.                         2 sec x                4.                        
       cos x     1  sin x                                      cos2 x          1  sin x




     tan x  cot x                                       3 cos2 z  5 sin z  5 3 sin z  2
5.                  1  2 cos2 x                   6.                         
     tan x  cot x                                              cos2 z           1  sin z




                                                                                                          168
College Algebra with Trig                                   Name:_______________________________
CW/HW- Proving Trig Identities #1
                                                            Date:________________________________

Verify each of the following identities:
1.      sin x cot x  cos x                                 2.      cos x  sin x 2    1  2 sin x cos x




3.      cos x(tan x  sin x cot x)  sin x  cos 2 x        4.      cot x cos x  sin x  cscx




5.
        1  cos x 1  cos x   tan 2 x                  6.      cscx  cos x cot x  sin x
               cos2 x




7.      tan 2 x  sin 2 x  tan 2 x sin 2 x            8.   sec4 x  2 sec2 x tan 2 x  tan 4 x  1




        sin 2 x  2 sin x  1 1  sin x                                                   cos x
9.                                                         10.     tan x  sec x 
               cos 2 x         1  sin x                                                1  sin x




             tan x         1                                        1  sin x     cos x
11.                                                        12.                           2 sec x
        sin x  2 tan x cos x  2                                     cos x     1  sin x




                                                                                                              169
College Algebra with Trig                                  Name:_______________________________
Lesson: Sum, difference and ½ angle identities
                                                           Date:________________________________

Objective:    use the sum, difference, and half-angle identities to evaluate trigonometric expressions


                         cos2 y
DO NOW:       Prove:               1  sin y
                        1  sin y




Sum and Difference Identities




Half Angle Identities




Double Angle Identities




                                                                                                         170
Showing all work, complete the following chart to find the exact value of each trigonometric expression using
the specified trigonometric identity:

            use a sum or difference identity                use a half-angle identity




cos 105




sin 75




tan 165




                                                                                                           171
College Algebra with Trig                             Name:____________________________________
CW/HW- Trig Identities #2- Sum and Difference
                                                      Date:_____________________________________
Verify each of the following identities:

1.      sin( x  y)  sin( x  y)  2 sin x cos y     2.    cos(x  y)  cos(x  y)  2 cos x cos y




                                 tan 2 x  tan 2 y          cos(x  y )
3.      tan(x  y) tan(x  y)                        4.                 cot x  tan y
                                1  tan 2 x tan 2 y         sin x cos y




                          sin( x  y )                      sin( x  y )
5.      tan x  tan y                                6.                  cot x  cot y
                          cos x cos y                       sin x sin y




        cos(x  y )                                                          sin( x  y )
7.                   1  tan x tan y                 8.    tan( x  y ) 
        cos x cos y                                                          cos(x  y )




                                                                                                      172
College Algebra with Trig                       Name:_______________________________
CW/HW- Trig Identities #3- Double and ½ angle
                                                Date:________________________________
Verify each of the following identities:
                                                         4 tan x
1.      sin 4x  2 sin 2x cos2x                 2.                   tan 2 x
                                                      2  2 tan 2 x




                                                                  1
3.      (sin x  cos x) 2  1  sin 2 x         4.    csc2 x       sec x csc x
                                                                  2




5.      2 csc2 x  csc2 x tan x                 6.    cot x  tan x  2 csc2x




                          4 cos 2 x  2
7.      cot x  tan x                          8.    cot x  tan x  2 cot 2x
                             sin 2 x




                      sec2 x                                      1  tan 2 x
9.      sec 2 x                                10.   cos 2 x 
                    2  sec2 x                                    1  tan 2 x




                                                                                        173
                                     Trigonometric Identities & Equations
                                           Definitions & Identities



                      The following trigonometric identities hold for all values of  where each
                      expression is defined:
                                 1                           1                           1
Reciprocal             sin                     cos                         tan  
                               csc                        sec                        cot 
Identities
                                 1                           1                           1
                       csc                      sec                        cot  
                               sin                        cos                        tan 

                      The following trigonometric identities hold for all values of  where each
                      expression is defined:
Quotient Identities     sin                                    cos 
                               tan                                     cot 
                       cos                                      sin 

                      The following trigonometric identities hold for all values of  where each
Opposite-Angle
                      expression is defined:
Identities
                       sin ()   sin         cos ()  cos               tan ( )   tan 
                      The following trigonometric identities hold for all values of  where each
Pythagorean           expression is defined:
Identities             sin2   cos2   1        tan2   1  sec2           1  cot2   csc 2 

                      If A and B represent the measures of two angles, then the following identities hold
                      for all values of A and B:
                      sin (A  B)  sin A cos B  cos A sin B
                               sin (A  B)  sin A cos B  cos A sin B
Sum & Difference      cos (A  B)  cos A cos B  sin A sin B
Identities
                               cos (A  B)  cos A cos B  sin A sin B
                                       tan A  tan B                       tan A  tanB
                      tan(A  B)                            tan(A  B) 
                                      1  tan A tan B                     1  tan A tanB

                      If  represents the measure of an angle, then the following identities hold for all
                      values of :
Double-Angle                                                                               2 tan 
                                                   cos 2  cos2   sin2  tan 2 
Identities
                        sin 2  2 sin  cos      cos 2  2 cos2   1                 1  tan2 
                                                   cos 2  1 2 sin2 
                      If  represents the measure of an angle, then the following identities hold for all
                      values of :
                                  1  cos                 1  cos               1  cos 
Half-Angle              sin                   cos                     tan                , cos   1
Identities                  2           2            2           2           2       1  cos 
                                                                                             
                      where the sign of the radical is determined by the quadrant in which       lies
                                                                                             2




                                                                                                              174
College Algebra with Trig                                         Name:______________________________
Review- Trig Identity Test
                                                          Date:_______________________________

Show All Work!


Use the sum or difference identities to find the   Use the half-angle identities to find the exact
exact values of each trigonometric expression:     values of each trigonometric expression:

1. cos74 cos44  sin 74 sin 44                 5. cos112.5

      tan 110   tan 50                          6. sin 165
2.
     1  tan 110  tan 50 
                                                   7. tan105
3. cos345

4. sin 195




Verify Each Identity

             1         1                                          cos x  sin x
1.                           2 sec2 x                   2.                     1  tan x
         1  sin x 1  sin x                                         cos x



         1  cos 2 x
3.                    2 csc2 x  1                       4.      ln tan x   ln cot x
           sin 2 x



         2 tan x  sin 2 x
5.                          sin 2 x                      6.      cos4 x  sin 4 x  cos 2 x
              2 tan x


                                                                  1  sin x     cos x
                                                                            
7.       2 sin x cos x  2 sin x cos x  sin 2 x
                      3            3
                                                          8.        cos x     1  sin x




                     1  tan 2 x                                                    4 cos 2 x  2
         cos 2 x                                                 cot x  tan x 
9.                   1  tan 2 x                          10.                          sin 2 x

                                                                                                     175
       1  cot x 
                          2

                               1  sin 2 x
       csc x                                                      ln sec x  tan x   ln sec x  tan x
11.                                                           12.


      1  sin x     cos x
                
13.     cos x     1  sin x                                   14.   2 sin 2  cos 2   cos 4   1  sin 4 




      cos( u  v)                                                    tan 3t  tan t    2 tan t
                   tan u  cot v                                                   
15.   cos u sin v                                             16.   1  tan 3t tan t 1  tan 2 t




                                                                                     cot x  cot y
                                                                    tan( x  y ) 
                      sin(m  n)                                                     cot x cot y  1
17.   tan m  tan n                                          18.
                      cos m cos n



                                                                    tan y  sin y         y
19.   log(cos x  sin x)  log(cos x  sin x)  log cos 2 x   20.                  cos 2
                                                                       2 tan y            2



      1  sin x
21.              (sec x  tan x) 2                           22.   sec4 s  tan 2 s  tan 4 s  sec2 s
      1  sin x



      tan x  cot x       1
23.                                                          24.   tan x  cot x  (sec x  csc x)(sin x  cos x)
      sec x  csc x cos x  sin x



      tan x  sin x         x                                       cos 3   sin 3  2  sin 2 
25.                  sin 2                                   26.                    
         2 tan x            2                                        cos   sin          2



27.   1  cos5x cos3x  sin 5x sin 3x  2 sin 2 x             28.   cos2  cot2   cot2   cos2 




                                                                                                                 176
TRIGONOMETRY UNIT NOTES
              PART 5




                          177
College Algebra with Trig                                  Name:____________________________________
Lesson- Solving Trig Equations
                                                           Date:_____________________________________

Objective:    To learn to solve trig equations in both degree and radian measure.

DO NOW:       Find the exact value of csc (2π/3)




__________________________________________________________________________________________
Solving Trig Equations
There are two types of problems that we could deal with.
1.     Trig Equations of a linear form.                  2.     Trig Equations of a quadratic form.
Process:                                                 Process:




Example:      4 cos  3  4                               Example:       2 sin 2   sin x  1  0




                                                                                                      178
Practice Problems
Solve each from 0    2

                                                                           2 sin 2    3
1.     2 sin   5  4 sin   6   2.    Tan2  Tan                 3.            
                                                                               2      2




4.     cos2   5  8 cos         5.   2 sin 2   5 sin   2  0   6.   3 sin 2   cos  1




7.     2 cos  1  sec           8.   sin 2   2  cos tan       9.   sin 2   1  0




                                                                                                  179
180

								
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