7 2 Solving Quadratic Equations

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7 2 Solving Quadratic Equations Powered By Docstoc
					7.2 Solving Quadratic
Equations
   Algebra 2
   Mrs. Spitz
   Spring 2007
Objectives

    Solve quadratic equations by graphing,
      Find the equation of the axis of symmetry
       and
      find the coordinates of the vertex of the
       graph of a quadratic function
    solve quadratic equations by factoring
Assignment

    pp. 319-320 #6-45 odd
Definition of a Quadratic Function

    A quadratic function is a function that
     can be described by an equation of the
     form y = ax2 + bx + c, where a ≠ 0.
Generalities
    The path of an object
     when it is thrown or
     dropped is called the
     TRAJECTORY of the
     object. A bouncing
     object like the tennis
     ball in the photo will
     have a trajectory in a
     general shape called
     a parabola.
Generalities
    Equations such as y =
     6x – 0.5x2 and y = x2 –
     4x +1 describe a type of
     function known as a
     quadratic function.
    Graphs of quadratic
     functions have common
     characteristics. For
     instance, they all have
     the general shape of a
     parabola.
Generalities
 The table and graph can be used to illustrate other common
   characteristics of quadratic functions. Notice the matching values
   in the y-column of the table.
                                              6
                                                   y = x2 – 4x + 1
    x        x2 – 4x + 1        y
   -1      (-1)2 – 4(1) + 1     6             4



   0       (0)2 – 4(0) + 1      1                            x=2
   1       (1)2 – 4(1) + 1     -2             2



   2       (2)2 – 4(2) + 1     -3
   3       (3)2 – 4(3) + 1     -2                                      5



   4       (4)2 – 4(4) + 1      1
                                              -2

   5       (5)2   – 4(5) + 1    6
                                                             (2, -3)
                                              -4
Generalities
 Notice in the y-column of the table, -3 does not have a matching value. Also
    notice that -3 is the y-coordinate of the lowest point of the graph. The
    point (2, -3) is the lowest point, or minimum point, of the graph of y = x2 –
    4x + 1.
                                                       6
                                                           y = x2 – 4x + 1
     x          x2 – 4x + 1           y
    -1        (-1)2 – 4(1) + 1        6                4



     0        (0)2 – 4(0) + 1         1                              x=2
     1        (1)2 – 4(1) + 1         -2               2



     2        (2)2 – 4(2) + 1         -3
     3        (3)2 – 4(3) + 1         -2                                       5



     4        (4)2 – 4(4) + 1         1
                                                      -2

     5        (5)2   – 4(5) + 1       6
                                                                     (2, -3)
                                                      -4
Maximum/minimum points

  For the graph of y = 6x – 0.5x2, the point
   (6, 18) is the highest point, or maximum
   point. The maximum point or minimum
   point of a parabola is also called the
   vertex of the parabola.
  The graph of a quadratic function will
   have a minimum point or a maximum,
   BUT NOT BOTH!!!
Axis of Symmetry

  The vertical line containing the vertex of
   the parabola is also called the axis of
   symmetry for the graph. Thus, the
   equation of the axis of symmetry for the
   graph of y = x2 – 4x + 1 is x = 2
  In general, the equation of the axis of
   symmetry for the graph of a quadratic
   function can be found by using the rule
   following.
Equation of the Axis of Symmetry

    The equation of
     the axis of
     symmetry for the        b
     graph of
                         x
     y = ax2 + bx + c,
     where a ≠ 0, is
                             2a
Ex. 1: Find the equation of the axis of symmetry and the
coordinates of the vertex of the graph of y = x2 – x – 6. Then
use the information to draw the graph.

      First, find the axis of       NOTE: for
       symmetry.
                                    y = x2 – x – 6
                 b
          x                       a = 1 b = -1 c = -6
                2a
                  1
          x  (      )
                 2 1
              1
          x
              2
Ex. 1: Find the equation of the axis of symmetry and the
coordinates of the vertex of the graph of y = x2 – x – 6. Then
use the information to draw the graph.

                                              1 2 1
      Next, find the vertex.            y  ( )  6
       Since the equation of                  2   2
       the axis of symmetry is x
                                           1 1
       = ½ , the x-coordinate of
       the vertex must be ½ .
                                           6
       You can find the y-
                                           4 2
       coordinate by                       1 2 24
       substituting ½ for x in y           
       = x2 – x – 6 .                      4 4 4
                                             25
   The point ( ½, -25/4) is              
   the vertex of the graph.                   4
   This point is a minimum.
Generalities
 The table and graph can be used to illustrate other common
   characteristics of quadratic functions. Notice the matching values
   in the y-column of the table.                 2


                                               y = x2 – x – 6
    x         x2 – x – 6           y
   -2       (-2)2 – (-2) – 6       0                                  5




   -1       (-1)2 – (-1) – 6       -4            -2
                                                         x=½
   0         (0)2 – (0) – 6        -6
   1         (1)2 – (1) – 6        -6            -4


   2         (2)2 – (2) – 6        -4
   3         (3)2 – (3) - 6        0             -6




        This point is a minimum!
                                                 -8       ½, -25/4)
Solving Quadratic Equations Graphically

          SOLVING QUADRATIC EQUATIONS USING GRAPHS
    The solution of a quadratic equation in one variable x can be solved
    or checked graphically with the following steps:

    STEP 1     Write the equation in the form ax 2 + bx + c = 0.


    STEP 2     Write the related function y = ax 2 + bx + c.


    STEP 3     Sketch the graph of the function y = ax 2 + bx + c.

      The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts.
         Checking a Solution Using a Graph

        1 2
Solve     x = 8 algebraically. Check your solution graphically.
        2



SOLUTION
                  1 2           Write original equation.
                    x = 8
                  2

                   x 2 = 16     Multiply each side by 2.

                    x= 4      Find the square root of each side.


          CHECK    Check these solutions using a graph.
Checking a Solution Using a Graph


     CHECK Check these solutions using a graph.


 1   Write the equation in the form ax 2 + bx + c = 0

          1 2
            x =8        Rewrite original equation.
          2
       1 2
         x –8=0         Subtract 8 from both sides.
       2


 2   Write the related function y = ax2 + bx + c.

             y = 1 x2 – 8
                  2
          Checking a Solution Using a Graph


    CHECK Check these solutions using a graph.


2   Write the related function
                                       – 4, 0    4, 0
       y = ax2 + bx + c.

       y = 1 x2 – 8
           2

                       1
3   Sketch graph of y = x2 – 8.
                           2
     The x-intercepts are  4, which
     agrees with the algebraic solution.
         Solving an Equation Graphically


Solve x 2 – x = 2 graphically.
                                    Check your solution algebraically.

SOLUTION
        1       Write the equation in the form ax 2 + bx + c = 0
                     x2 – x = 2       Write original equation.
                   x2 – x – 2 = 0    Subtract 2 from each side.
                 (x-2)(x+1)=0        Factor and set equal to zero.
 x–2=0                 x+1=0
 x=2                                Solve. These are your x-intercepts.
                       x = -1
            2    Write the related function y = ax2 + bx + c.
                           y = x2 – x – 2
           Solving an Equation Graphically

2   Write the related function y = ax2 + bx + c.

          y = x2 – x – 2


                                             – 1, 0   2, 0



3   Sketch the graph of the function
               y = x2 – x – 2

    From the graph, the x-intercepts
    appear to be x = –1 and x = 2.
           Solving an Equation Graphically



From the graph, the x-intercepts             – 1, 0   2, 0
appear to be x = –1 and x = 2.


   CHECK

You can check this by substitution.



     Check x = –1:               Check x = 2:
           x2 – x = 2                 x2 – x = 2
                    ?                         ?
    (–1) 2 –   (–1) = 2               22 – 2 = 2
           1+1=2                      4–2=2

				
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