Get documents about "Course Notes CAT
Shared by: HC120929084014
-
Stats
- views:
- 7
- posted:
- 9/29/2012
- language:
- English
- pages:
- 180
Document Sample
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 x5 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 8k
(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 y2
(11) (12)
a 1
2
a 1 5 4
x4 x4 x2 2
(13) (14)
x 4 x 16
2
x4
11
Showing all work, perform the indicated operation and simplify your answer.
2a 5 1 2b 1 1
(15) (16)
a 5a 6
2
a3 b b 12
2
b4
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
m5 1
a 2
b 2 y 1 m y y2
(1) (2) y 1 m3 (4)
ab ba 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?
a4
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 a3 2a (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
y2
x4 3x 12 2m 18 (19)
(12) (16) 3
m9 9m 1
2x 7 x 3
2
5x 2 45 y2
x 2 3x x 2 5x 6 3 3
(13) 4
2x 2 x 6 x2 4 xy xy 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 a4 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) x3 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. x3 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)
x3
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. T2,3 [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 x6
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 x6
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.
x3 8
2 x3
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 x6
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.
x3 8
2 x2
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 x6
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
x3 8
2 x2
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 x6
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 x2
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.
x3 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 x6
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 x1 42 x2 45 x1 162 x1 3x 9 x4
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 P1
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 90F 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 x1 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. 51 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 P1
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 mA 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 = 6520, 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, mA 45
2. a 9, b 12, and mA 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 1048 with one building and an angle
of 4820 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
