Chapter 4 Notes Packet on Quadratic Functions and Factoring

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        Chapter 4 Notes Packet on Quadratic Functions and Factoring
Notes #15: Graphing quadratic equations in standard form, vertex form, and intercept form.

A. Intro to Graphs of Quadratic Equations: y  ax 2  bx  c
       A ____________________ is a function that can be written in the form y  ax 2  bx  c where
a, b, and c are real numbers and a  0. Ex: y  5 x 2 y  2 x 2  7 y  x 2  x  3
       The graph of a quadratic function is a U-shaped curve called a ________________. The
maximum or minimum point is called the _____________
Identify the vertex of each graph; identify whether it is a minimum or a maximum.
1.)                                                                                        2.)




Vertex: (   ,   ) _________                        Vertex: (    ,   ) _________
3.)                                                   4.)




Vertex: (   ,   ) _________                         Vertex: (   ,   ) _________
B. Key Features of a Parabola:                  y  ax 2  bx  c

       Direction of Opening: When a  0 , the parabola opens ________:
                              When a  0 , the parabola opens ________:
       Width: When a  1 , the parabola is _______________ than y  x 2
                 When a  1 , the parabola is the ________ width as y  x 2
                    When a  1 , the parabola is ________ than y  x 2
       Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of
                                            b
symmetry. To find the vertex, plug in x       and solve for y. This yields a point (____, ____)
                                            2a
                                                                                                     b
       Axis of symmetry: This is a vertical line passing through the vertex. Its equation is: x 
                                                                                                     2a
       x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y = 0 and
solve for x.
       y-intercept: is the point where the parabola crosses the y-axis. Plug in x = 0 and solve for y.
                                                       1
Without graphing the quadratic functions, complete the requested information:
5.) f ( x)  3x 2  7 x  1                                 5
                                              6.) g ( x)   x 2  x  3
                                                            4
What is the direction of opening? _______     What is the direction of opening? _______
Is the vertex a max or min? _______           Is the vertex a max or min? _______
                             2
Wider or narrower than y = x ? __________     Wider or narrower than y = x2 ? ___________

                    2 2                                                          8.) y  0.6 x 2  4.3x  9.1
7.)         y        x  11
                    3

What is the direction of opening? _______                                        What is the direction of opening? _______
Is the vertex a max or min? _______                                              Is the vertex a max or min? _______
Wider or narrower than y = x2 ? __________                                       Wider or narrower than y = x2 ? ___________


The parabola y = x2 is graphed to the right.

Note its vertex (___, ___) and its width.

You will be asked to compare other parabolas to
this graph.




C. Graphing in STANDARD FORM ( y  ax 2  bx  c ): we need to find the vertex first.
Vertex                                                Graphing
- list a = ____, b = ____, c = ____                    - put the vertex you found in the center of
            b                                        your x-y chart.
- find x =                                             - choose 2 x-values less than and 2 x-values more
            2a
                                                      than your vertex.
- plug this x-value into the function (table)
- this point (___, ___) is the vertex of the parabola - plug in these x values to get 4 more points.
                                                       - graph all 5 points


Find the vertex of each parabola. Graph the function and find the requested information
9.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____
                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                              4
                              3
                              2                                                                              Vertex: _______
                              1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                                                             x                         Max or min? _______
                             -1
                            -2                                                                 Direction of opening? _______
                            -3
                            -4                                                                   Axis of symmetry: ________
                                                                                               Compare to the graph of y = x2
                            -5
                            -6
                            -7
                            -8
                            -9
                                                                                               _________________________
                           -10




                                                                                  2
10.) h(x) = 2x2 + 4x + 1
                                  y
                            10
                              9
                              8
                              7
                              6                                                                    Vertex: _______
                              5
                              4                                                              Max or min? _______
                              3
                              2                                                      Direction of opening? _______
                              1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                                                             x         Axis of symmetry: ________
                                                                                     Compare to the graph of y = x2
                             -1
                            -2
                            -3
                            -4                                                       _________________________
                            -5
                            -6
                            -7
                            -8
                            -9
                           -10


                                              1 2
11.) k(x) = 2 – x –                             x
                                              2
                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                              4
                                                                                                   Vertex: _______
                              3
                              2
                                                                                             Max or min? _______
                              1
                                                                             x
                                                                                     Direction of opening? _______
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                             -1
                            -2
                                                                                       Axis of symmetry: ________
                            -3
                            -4
                                                                                     Compare to the graph of y = x2
                            -5
                            -6
                                                                                     _________________________
                            -7
                            -8
                            -9
                           -10




12.) State whether the function y = 3x2 + 12x  6 has a minimum value or a maximum
value. Then find the minimum or maximum value.




                       1 2
                          x  5 x  7 . State whether it is a minimum or maximum. Find that
13.) Find the vertex of y 
                       2
minimum or maximum value.




                                                                                 3
Another useful form of the quadratic function is the vertex form: ________________________________.

GRAPH OF VERTEX FORM y = a(x  h)2 + k
The graph of y = a(x  h)2 + k is the parabola y = ax2 translated ___________ h units and ___________ k
units.
 The vertex is (___, ___).
 The axis of symmetry is x = ___.
 The graph opens up if a ___ 0 and down if a ___ 0.



   Find the vertex of each parabola and graph.
   13.) y   x  1  2
                    2

                                     y
                               10
                                 9
                                 8
                                 7
                                 6
                                 5
                                 4
                                 3
                                 2
                                 1
                                                                                x
   -10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                -1
                               -2
                               -3
                               -4
                               -5
                               -6
                               -7                                                       Vertex: _______
                               -8
                               -9
                              -10


             1
   14.) y    x  1  5
                      2

             3
                                     y
                               10
                                 9
                                 8
                                 7
                                 6
                                 5
                                 4
                                 3
                                 2
                                 1
                                                                                x
   -10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                -1
                               -2
                               -3
                               -4
                               -5
                               -6
                               -7
                               -8
                                                                                        Vertex: _______
                               -9
                              -10




   15.) Write a quadratic function in vertex form for the function whose graph has its vertex
   at (-5, 4) and passes through the point (7, 1).




                                                                                    4
GRAPH OF INTERCEPT FORM y = a(x  p)(x  q):
Characteristics of the graph y = a(x  p)(x  q):
               The x-intercepts are ___ and ___.

               The axis of symmetry is halfway between (___, 0) and ( ___ , 0)

         and it has equation x =
                                                                      2

        The graph opens up if a ___ 0 and opens down if a ___ 0.


    16.) Graph y = 2(x  1)(x  5)
                                      y
                                10
                                  9
                                  8
                                  7
                                  6
                                  5
                                  4
                                  3
                                  2
                                  1
                                                                                 x
    -10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                 -1
                                -2
                                -3
                                -4
                                -5
                                -6                                                                x-intercepts: _______, _______
                                -7
                                -8
                                -9
                               -10                                                                                  Vertex: _______

    Converting between forms:
    From intercept form to standard form                                             Ex:     y  2  x  5 x  8
          Use FOIL to multiply the binomials
    together
          Distribute the coefficient to all 3 terms




                                                                                     Ex: f  x   4  x  1  9
                                                                                                           2
    From vertex form to standard form
          Re-write the squared term as the
    product of two binomials
          Use FOIL to multiply the binomials
    together
          Distribute the coefficient to all 3 terms
          Add constant at the end



    HW #15: Pg. 202: 47-63 odd
    pg. 240 #3-39 x 6’s
    pg. 249 #4-40 x 4’s


                                                                                     5
 Notes 16: Sections 4-3 and 4-4: Solving quadratics by Factoring

 A. Factoring Quadratics
Examples of monomials:_______________________________
Examples of binomials:________________________________
Examples of trinomials:________________________________

 Strategies to use: (1) Look for a GCF to factor out of all terms
                    (2) Look for special factoring patterns as listed below
                    (3) Use the X-Box method



                     (4) Check your factoring by using multiplication/FOIL




 Factor each expression completely. Check using multiplication.
 1.) 3x 2  15 x                 2.) 6 x 2  24                        3.) x 2  5 x  24




 4.) 25x 2  81                     5.) m2  22m  121                 6.) 4 x2  12 x  9




                                                      6
  7.) 5x2  17 x  6                    8.) 3x2  5x  12                 9.) 25t2  110t + 121




  10.) 16 x 2  36                      11.) 9a 2  42a  49              12.) 6 x2  33x  36




  B. Solving quadratics using factoring
  To solve a quadratic equation is to find the x values for which the function is equal to _____. The
  solutions are called the _____ or _______of the equation. To do this, we use the Zero Product
  Property:

  Zero Product Property
  List some pairs of numbers that multiply to zero:

  (___)(___) = 0       (___)(___) = 0             (___)(___) = 0        (___)(___) = 0

  What did you notice? _______________________________________________

ZERO PRODUCT PROPERTY
             If the _________ of two expressions is zero, then _______ or _______ of
             the expressions equals zero.
 Algebra     If A and B are expressions and AB = ____ , then A = _____ or B = __.
 Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is,
         x = __________ or x = _________.


  Use this pattern to solve for the variable:
  1.             get the quadratic = 0 and factor completely
  2.             set each ( ) = 0 (this means to write two new equations)
  3.             solve for the variable (you sometimes get more than 1 solution)

                                                            7
Find the roots of each equation:
13.) x2  7 x  30  0                                                                2   6  4  8             15.) 3x2  2 x  21
                                                                                 14.)  x    x    0
                                                                                      3   7  5  9




Find the zeros of each equation:
16.) 2 x 2  8 x  30  x  34   17.) v(v + 3) = 10                                                               18.) 2 x 2  x  15




Find the zeros of the function by rewriting the function in intercept form:
19.) y  6 x 2  3x  63          20.) f  x   12x2  6x  6      21.) g  x   49x2 16




Graph the function. Label the vertex and axis of symmetry:
22.) y  2  x 1 ( x  3)
                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                              4
                              3                                                                                                 Vertex: _______
                              2
                              1
                                                                             x
                                                                                                            Maximum or minimum value: _______
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
                             -1
                                      1   2   3   4   5   6   7   8   9 10
                                                                                                                           x-intercepts: _______
                            -2
                            -3                                                                                      Axis of symmetry: ________
                            -4
                            -5
                            -6
                                                                                                             Compare width to the graph of y = x2
                            -7                                                                                    _________________________
                            -8
                            -9
                           -10




HW #16:
pg. 255 #4-52 x 4’s, pg. 263 #4-52 x 4’s, Pg. 265: #78-81
    Study for Quiz on Sec. 4.1-4.4 (Quiz is tomorrow!)

                                                                                                8
Notes #17: Section 4-5 Solve Quadratic Equations by Finding Square Roots

A. Simplifying Square Roots:
     Make a factor tree; circle pairs of “buddies.”
     One of each pair comes out of the root, the non-paired numbers stay in the root.
     Multiply the terms on the outside together; multiply the terms on the inside together
Simplify:
1.) 3 120                                     2.) 5 72




B. Multiplying Square Roots:
    Simplify each radical completely by taking out “buddies”
    (outside • outside) inside inside or ( a b )(c d )  ________
    Simplify your answer, if possible
Simplify:
          
3.) 5 12 3 15                                     
                                               4.) 2 5 2 5    



 C. Simplifying Square Roots in Fractions:
                                  a                2     2       2
      Split up the fraction:                             
                                  b               25    25      5
      Simplify first by taking out “buddies” or reducing (you can only reduce two numbers that are
         both under a root or two numbers that are both not in a root)
      Square root top, square root bottom
      If one square root is left in the denominator, multiply the top and the bottom by the square root
         and simplify
     OR If a binomial is left in the denominator, then multiply top and bottom by the conjugate of the
     denominator (exact same expression except with the opposite sign). Remember to FOIL on the
     denominator.
 Reduce if possible
 Simplify:
       7                              6.)
                                            4                               2 5
 5.)                                                                    7.)
       3                                  5+ 3                              3 2




                                                       9
D. Solving Quadratic Equations Using Square Roots

      Isolate the variable or expression being squared (get it ______________)
      Square root both sides of the equation (include + and – on the right side!)
      This means you have _____________ equations to solve!!
      Solve for the variable (make sure there are no roots in the denominator)

8.) x2 = 25                                       9.) 3x2 = 81




                                                         m2
10.)   4x2 – 1 = 0                                11.)      3  2
                                                         15




                                                  13.) 3  x  2  6  34
                                                                 2
12.) (2y + 3)2 = 49




HW #17: Pg. 202: 46-62 even, Pg. 258: 79-89 odd, pg. 269 #3-33 x 3’s




                                                  10
      Notes#18: Sections 4-6 & 4-7 Complex Numbers and Completing the square
Section 4.6: Complex Numbers
      A. Definitions
  Define
  Complex Numbers:
  imaginary unit (i):
  imaginary number :


     B. Solving a quadratic equation with complex roots
         Isolate the expression being squared
         Square root both sides; write two equations Replace                1 with i. Simplify

     Solve
  1.) x2 = 27                             2.) 2x2 + 11 = 37                   3.) 4(2 x  1) 2  8  0




     C. Adding, subtracting, and multiplying complex numbers
         Distribute/FOIL. Combine like terms.
         Replace i 2 with (-1). Simplify.

     Simplify
  4.) (3 + 7i)  (8  2i)        5.) (2 + 5i) + (7  2i)        6.) (2 + i)(5 + 2i)        7.) (3  i)(5  2i)




     B. Dividing complex numbers
                                                                                                   5
                If i is part of a monomial on the denominator, multiply top and bottom by i. ex:
                                                                                                   6i
                If i is part of a binomial on the denominator, multiply top and bottom by the complex
                                                                                                  5
                 conjugate of the denominator (same expression but opposite sign). FOIL. ex:
                                                                                                 6i
                            2
                 Replace i with (-1). Simplify.

        3                              6  4i                          2i                         7  5i
  8.)                            9.)                            10.)                        11.)
        4i                              2i                             7i                         1  4i




                                                                11
Section 4.7: Completing the Square

     B. Review: Solving Using Square Roots
         Factor and write one side of the equation as the square of a binomial
         Square root both sides of the equation (include + and – on the right side; 2 equations!
         Solve for the variable (make sure there are no roots in the denominator)

     1) (k + 2)2 = 12                      2.) x2 + 2x + 1 = 8                   3.) n2 – 14n + 49 = 3




     C. Completing the Square ax 2  bx  c  0
          Take half the b (the x coefficient)
          Square this number (no decimals – leave as a fraction!)
          Add this number to the expression
          Factor – it should be a binomial, squared (     )2

     4.) x2 + 6x + _____                          5.) m2 – 14m + _______
        (          )(          )

        (          )2

     Find the value of c such that each expression is a perfect square trinomial. Then write the expression as
     the square of a binomial.
     6.) w2 + 7w + c                                       7.) k2 – 5k + c




                 Solving by Completing the Square:
                       Collect variables on the left, numbers on the right
                       Divide ALL terms by a; leave as fractions (no decimals!)
                       Complete the square on the left – add this number to BOTH sides
                       Square root both sides (include a ______ and _______ equation!)
                       Solve for the variable (simplify all roots – look for 1  i )


     8.) x2 + 4x – 5 = 0                          9.) m2 – 5m + 11 = 10




                                                           12
10.) 2k 2  9k  3  k  17             11.) 3w2  4w  11  2  2w




12.) 2x2 – 3x – 1 = 0                   13.) 3x 2  x  2 x  6




Cumulative Review: Solving Quadratics
Solve by factoring:
14.) 12k2 – 5k = 2                      15.) 49m2 – 16 = 0




Solve by using square roots:
16.) 4w2 = 18                           17.) 3y2 – 8 = 0




   HW #18: pg. 279 #3-33 x 3’s   pg. 288 #3-36 x 3’s              pg. 1013: 6-32 even
                                                13
Notes #19: Sec. 4-8 Use the Quadratic Formula and the Discriminant
A. Review of Simplifying Radicals and Fractions
      Simplify expression under the radical sign ( 1  i ); reduce
      Reduce only from ALL terms of the fraction.
      (You can’t reduce a number outside of a radical with a number inside of a radical)
      Make sure that you have TWO answers

Simplify:
    6  18                                                5  20
1.)                                                  2.)
       2                                                      2



      4  20                                              8  27
3.)                                                  4.)
         4                                                     2




    9  (5)2  (5)(2)(3)                               9  (6)2  4(3)(3)
5.)                                                  6.)
             4                                                    4




B. Solving Quadratics using the Quadratic Formula

So far, we have solved quadratics by: (1) _______________, (2) ______________, and
(3) ___________________.

The final method for solving quadratics is to use the quadratic formula.

                    Solving using the quadratic formula:
                          Put into standard form (ax2 + bx + c = 0)
                          List a = , b = , c =

                                                         b  b 2  4ac
                            Plug a, b, and c into    x
                                                              2a
                            Simplify all roots (look for     1  i ); reduce




                                                       14
Solve by using the quadratic formula:
                                             b  b 2  4ac
1.)             2
               x + x = 12               x
                                                  2a

(std. form):

a = _____
b = _____
c = _____



                                           b  b 2  4ac
       2
2.) 5x – 8x = -3                        x
                                                2a

(std. form):

a = _____
b = _____
c = _____



3.) -x2 + x = -1                               4.) 3x2 = 7 – 2x




5.) -x2 + 4x = 5                               6.) 4( x  1) 2  6 x  2




                                               15
C. Using the Discriminant
 Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine how
   many solutions a quadratic equation has before you solve it by using the ________________.
                                                                                     b  b 2  4ac
 The discriminant is the expression under the radical in the quadratic formula: x 
                                                                                          2a
                                        Discriminant = b2 – 4ac
                        If b2 – 4ac < 0, then the equation has 2 imaginary solutions
                            If b2 – 4ac = 0, then the equation has 1 real solution
                           If b2 – 4ac > 0, then the equation has 2 real solutions


A. Finding the number of x-intercepts
 Determine whether the graphs intersect the x-axis in zero, one, or two points.
1.) y  4 x 2  12 x  9                             2.) y  3x 2  13x  10




B. Finding the number and type of solutions
Find the discriminant of the quadratic equation and give the number and type of solutions of the
equation.
3.) 3x 2  5 x  1                                   4.) x 2  3x  7




5.) 9x2 – 6x = 1                                     6.) 4x2 = 5x + 3




                                                      16
Cumulative Review Problems:
Solve by factoring:
7.) 4m2 +5m – 6 = 0                                                                              8.) 3x3 – 27x = 0




Solve by using square roots:
9.) 4b2 + 1 = 0                                                                                  10.) (3x + 1)2 = 18




Solve by completing the square:
11.) 4m2 + 12m +5 = 0                                                                            12.) x2 – 7x – 18 = 0




For #13, find the vertex of the parabola. Graph the function and find the requested information
13.) g(x) = -2x2 + 8x – 5
                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                              4                                                                                              Vertex: _______
                              3
                              2
                              1                                                                                  Max or min value: _______
                                                                             x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                             -1
                            -2
                                                                                                              Direction of opening? _______
                            -3
                            -4
                            -5
                                                                                                                     Compare width to y = x2 :
                            -6
                            -7                                                                                             _______________
                            -8
                            -9
                           -10
                                                                                                                Axis of symmetry: ________

      HW #19:                         Pg. 265 #1-12                              pg. 296 #4-48 x 4’s       pg. 299 #76-86even



                                                                                                 17
   Notes #20: Section 4-9 Graph and Solve Quadratic Inequalities

GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES
To graph a quadratic inequality, follow these steps:
Step 1 Graph the parabola with equation y = ax2+ bx + c.
   Make the parabola _______________ for inequalities with < or > and ______________ for inequalities
with  or .

Step 2 Test a point (x, y) ______________ the parabola to determine whether the point is a solution of the
inequality.

Step 3 Shade the region ______________ the parabola if the point from Step 2 is a solution. Shade the
       region ________________ the parabola if it is not a solution.


   1.) y  x2  2x + 3
                                                                                                      y
                                                                                                 10
                                                                                                  9
                                                                                                  8
                                                                                                  7
                                                                                                  6
                                                                                                  5
                                                                                                  4
                                                                                                  3
                                                                                                  2
                                                                                                  1
                                                                                                                                                                 x
                                                                  -10 -9 -8 -7 -6 -5 -4 -3 -2 -1          1       2       3       4       5   6   7   8   9 10
                                                                                               -1
                                                                                                 -2
                                                                                                 -3
                                                                                                 -4
                                                                                                 -5
                                                                                                 -6
                                                                                                 -7
                                                                                                 -8
                                                                                                 -9
                                                                                             -10




   2.) y ≥ x2  3x – 4
                                                                                                 y
                                                                                            10
                                                                                             9
                                                                                             8
                                                                                             7
                                                                                             6
                                                                                             5
                                                                                             4
                                                                                             3
                                                                                             2
                                                                                             1
                                                                                                                                                                 x
                                                               -10 -9 -8 -7 -6 -5 -4 -3 -2 -1         1       2       3       4       5       6   7   8   9 10
                                                                                            -1
                                                                                            -2
                                                                                            -3
                                                                                            -4
                                                                                            -5
                                                                                            -6
                                                                                            -7
                                                                                            -8
                                                                                            -9
                                                                                           -10




                                                        18
3.) y > 3x2 +3x 5
    y  x2 +5x + 10




4.) y  -x2 +4
                                                                                         y
                                                                                   10

y  x2 − 2x− 3                                                                       9
                                                                                     8
                                                                                     7
                                                                                     6
                                                                                     5
                                                                                     4
                                                                                     3
                                                                                     2
                                                                                     1
                                                                                                                                    x
                                                       -10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                                                                    -1
                                                                                   -2
                                                                                   -3
                                                                                   -4
                                                                                   -5
                                                                                   -6
                                                                                   -7
                                                                                   -8
                                                                                   -9
                                                                                  -10




HW #20: pg. 304 #4-24 x 4’s   Pg. 323 #1-24 all


                                                  19
Notes# 21: Sec. 4-10 Write Quadratic Functions and Models

A. When given the vertex and a point
    Plug the vertex in for (h, k) in y  a ( x  h) 2  k
    Plug in the given point for (x, y)
    Solve for a. Plug in a, h, k into y  a ( x  h) 2  k

1.) Write a quadratic equation in vertex form
for the parabola shown.




2.) Write a quadratic function in vertex form for the function whose graph has its vertex at (2, 1) and
passes through the point (4, 5).




B. When given the x-intercepts and a third point
    Plug in the x-intercepts as p and q into y = a(x  p)(x  q)
    Plug in the given point for (x, y)
    Solve for a. Plug in a, h, k into y = a(x  p)(x  q)

3.) Write a quadratic function in intercept form
for the parabola shown.




                                                      20
B. When given three points on the parabola
    Label all three points as (x, y)
    Separately, plug in each point into y  ax 2  bx  c
    You now have 3 equations with three variables: a, b, c
    Solve for a, b, and c using elimination (see notes #13). Plug back into y  ax 2  bx  c

4.) Write a quadratic function in standard form for the parabola that passes
through the points (2, 6), (0, 6) and (2, 2).




5.) Write a quadratic function in standard form for the parabola that passes
through the points (1, 2), (1, 4) and (2, 1).




HW #21: pg. 312 #4-14even Pg. 1013: 1-21 odd, 29-39 all, 43, 48
Chapter 4 Test and Notes Check on Friday
                                              21
Notes #22: Chapter 4 Review

To graph a quadratic function, you must FIRST find the vertex (h, k)!!

(A) If the function starts in standard form y  ax 2  bx  c :
                                                  b
        1st: The x-coordinate of the vertex, h, =
                                                  2a
        2nd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function &
             solving for y.

(B) If the function starts in intercept form y  a( x  p)( x  q ) :
        1st: Find the x-intercepts by setting the factors with x equal to 0 & solving for x.
        2nd: The x-coordinate of the vertex is half way between the x-intercepts.
        3rd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function &
             solving for y.

(C) If the function starts in vertex form y  a( x  h)2  k :
        1st: pick out the x-coordinate of the vertex, h. REMEMBER: h will have the OPPOSITE sign
             as what is in the parenthesis!!
         nd
        2 : Pick out the y-coordinate of the vertex, k. It will have the SAME sign as the what is in the
              equation!

AFTER finding the vertex:
    Make a table of values with 5 points: The vertex, plug in 2 x-coordinates SMALLER than
    the x-coordinate of the vertex & 2 x-coordinates LARGER than the x-coordinate of the
    vertex.

Direction of Opening:
       If a is positive, the graph opens up
       If a is negative, the graph opens down.

Width of the function:
       If a  1 , the graph is narrower than y  x2
       If a  1 , the graph is wider than y  x2




                                                      22
Graph each function by making a table of values with at least 5 points. (A) State the vertex. (B)
State the direction of opening (up/down). (C) State whether the graph is wider, narrower, or the
same width as y  x 2 .

            1
1.) f ( x)  ( x  6)2  5
            2
                                  y
                            10
                              9
                              8
                              7
                              6
                              5                                                                                                 Vertex: _______
                              4
                              3
                              2
                              1
                                                                                                                  Direction of opening? _______
                                                                                                x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1           1   2   3   4   5       6       7       8       9 10
                             -1
                            -2
                            -3
                                                                                                             Compare width to the graph of y = x2
                            -4                                                                                   _________________________
                            -5
                            -6
                            -7
                            -8
                            -9
                           -10

                     2
2.) k(x) = x + 2x + 1
                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                              4
                              3
                                                                                                                                Vertex: _______
                              2
                              1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
                             -1
                                         1   2   3   4   5       6       7       8       9 10
                                                                                                x                 Direction of opening? _______
                            -2


                                                                                                             Compare width to the graph of y = x2
                            -3
                            -4
                            -5
                            -6                                                                                   _________________________
                            -7
                            -8
                            -9
                           -10

                    2
3.) f(x) = x – x – 6

                                     y
                             10
                                 9
                                 8
                                 7
                                 6
                                 5
                                                                                                                                Vertex: _______
                                 4
                                 3
                                 2
                                 1
                                                                                                                  Direction of opening? _______
                                                                                                    x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1           1   2   3   4       5       6       7       8    9 10
                             -1
                             -2                                                                              Compare width to the graph of y = x2
                             -3
                             -4
                                                                                                                 _________________________
                             -5
                             -6
                             -7
                             -8
                             -9
                            -10




                                                                                                        23
4.) f ( x)   x2  2x  3


                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                                                                                                         Vertex: _______
                              4
                              3
                              2                                                            Direction of opening? _______
                              1
                                                                             x

                                                                                      Compare width to the graph of y = x2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                             -1
                            -2
                            -3
                            -4
                                                                                          _________________________
                            -5
                            -6
                            -7
                            -8
                            -9
                           -10



5.) h( x)  2 x2  4 x  1


                                  y
                            10
                              9
                              8
                              7
                              6
                              5
                                                                                                         Vertex: _______
                              4
                              3
                              2                                                            Direction of opening? _______
                              1
                                                                             x

                                                                                      Compare width to the graph of y = x2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                             -1
                            -2
                            -3
                            -4
                                                                                          _________________________
                            -5
                            -6
                            -7
                            -8
                            -9
                           -10


              1
6.) g ( x)   ( x  4)( x  6)
              3

                                  y
                            10
                              9
                              8
                              7                                                                          Vertex: _______
                              6
                              5
                              4
                              3
                                                                                           Direction of opening? _______
                              2
                              1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1        1   2   3   4   5   6   7   8   9 10
                                                                             x        Compare width to the graph of y = x2
                             -1
                            -2                                                            _________________________
                            -3
                            -4
                            -5
                            -6
                            -7
                            -8
                            -9
                           -10




                                                                                 24
    Methods for Solving Quadratic Equations:
    A.) Factoring
           1st: Set equal to 0
           2nd: Factor out the GCF
           3rd: Complete the X & box method to find the factors
           4th: Set every factor that contains an x in it, equal to 0 & solve for x.

    B.) Completing the Square
          1st: Move the constant (number with no variable) to the right so that all variables are on the left
               & all constants are on the right.
           nd
          2 : Divide every term in the equation by the value of a, if it is not already 1.
                                                                                   2
                                                                             b
           3rd: Create a perfect square trinomial on the left side by adding   to both sides.
                                                                             2
                                                   2
                                               b
           4 : Factor the left side into a  x   and simplify the value on the right side.
             th

                                               2
           5 : Take the square root of both sides of the equation. REMINDER: Don’t forget the 
            th

           6th: Solve for x

    C.) Finding Square Roots
           1st: Isolate the term with the square.
           2nd: Take the square root of both sides of the equation. REMINDER: Don’t forget the 
           3rd: Solve for x.

    D.) Quadratic Formula
          1st: Set the equation equal to 0.
          2nd: Find the values of a, b, and c & plug them into the Quadratic Formula:
                        b  b2  4ac
                     x
                               2a
            rd
           3 : Simplify the radical as much as possible.
           4th: If possible, simplify the numerator into integers.
                                                                               46 3
           5th: Divide. REMINDER: If you have 2 terms in the numerator (ex:             ), divide BOTH
                                                                                   2
                  terms by the number in the denominator (the example would result in 2  3 3 )

     Examples: Solve each equation by the method stated.
By Square Roots:
            2                                  2
1.) (2y + 3) = 49                  2.) (3m – 1) = 20                           3.) (3r  5)2 1  11




                                                            25
By Factoring:
4.) x2 – 4x – 5 = 0              5.) 3 x 2  2 x  21            6.) 2 x 2  x  15




By Completing the Square:
     2                                 2                               2
7.) x – 6x – 11 = 0              8.) 2y + 6y – 18 = 0            9.) 2x – 3x – 1 = 0




By Quadratic Formula:
      2                                    2                               2
10.) x + x = 12                  11.) 5x – 8x = -3               12.) 2x = 4 – 7x




    HW #22—Chapter 4 Review Sheet all (check answers online!!)
    Chapter 4 Test and Notes Check on Friday

                                                        26
                                          Chapter 4 Review Sheet

Please complete each problem on a separate sheet of paper. Show all of your work and please use
graph paper for all graphs.

For questions 1-3, solve by factoring.
1. x 2  12 x  32  0       2. 3x 2  8 x  5  0              3. 2 x 2  4 x  30  0

For questions 4-5, solve by finding square roots.
  1
                             5. 2  x  1  3  6
                                           2
4. ( x  1)2  16  0
  4

For questions 6-8, solve by completing the square.
                                                                    1
6. x 2  16 x  15  0         7. 5 x 2  10 x  20  0        8.  x 2  4 x  6  0
                                                                    2

For questions 9-10, solve by using the quadratic formula.
9.  x 2  3x  5  0        10. 3x 2  2 x  x 2  5x  1

For questions 11-13, simplify each expression.
    48  32                       5
11.                         12.                                 13.   252  3
        4                         48

For questions 14-16, write each function in vertex form, graph the function, and label the vertex
and axis of symmetry.
                                                                1
14. y  x 2  16 x  2      15. y  2 x 2  4 x  7     16. y  ( x  4)2  8
                                                                3

For question 17-19, graph each inequality.
17. y  2( x  1) 2  5     18. y   x 2  8 x  2             19. y  2 x 2  12 x  16

For questions 20-21, write a quadratic function in vertex form whose graph has the given vertex
and passes through the given point.
20. Vertex (2, 3) and passes through (-3, 7)
21. Vertex (-1, -5) and passes through (2, -1)

For questions 22-23, write a quadratic function in standard form whose graph passes through
the given points.
22. (1, 1), (0, -2), (2, 8)        23. (-1, -7), (1, -5), (2, -1)

For questions 24-28, write the expression as a complex number in standard form.
24. (7  3i)  (9  4i)      25. (6  2i)(8  3i)      26. (3  7i)  (9  2i)

      2  5i                         4  7i
27.                            28.
      7i                              3i

                            Chapter 4 Test and Notes Check Tomorrow!


                                                           27

				
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