11.4 Graphing Quadratic Functions by bfk20410

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									11.4 Graphing Quadratic Functions

Parabolas: the graphs of all quadratic functions; they are cup-shaped and
symmetric with respect to a vertical line (Axis of Symmetry).
Vertex: the minimum or maximum point of a parabola
Axis of Symmetry: a vertical line that passes through the vertex.


             The Graph of f ( x ) = ax
                                       2


Using a graphing calculator to compare f ( x ) = 2x and f ( x ) = −2x
                                                   2                  2


Determine the vertex and the axis of symmetry.
    Vertex:
    AOS:

For a > 0 , the parabola opens upward.
For a < 0 , the parabola opens downward.


Using a graphing calculator to compare f ( x ) = x , f ( x ) = 2x and
                                                  2              2



f (x ) = x 2
        1
        2
Determine the shape of each function.


If a is greater than 1, the parabola is narrower than y = f ( x ) = x .
                                                                     2


If a is between 0 and 1, the parabola is wider than y = f ( x ) = x .
                                                                   2




             The Graph of f ( x ) = a( x − h )
                                                    2




Using a graphing calculator to compare f ( x ) = x , f ( x ) = ( x − 3) and
                                                  2                 2


f (x ) = (x + 3)
                    2


Determine the h value, the vertex, and the axis of symmetry.


If h is positive, the graph of y = ax is shifted h units to the right.
                                     2


If h is negative, the graph of y = ax is shifted h units to the left.
                                      2




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             The Graph of f ( x ) = a(x − h ) + k
                                                         2



Using a graphing calculator to compare f ( x ) = ( x − 3) , f ( x ) = ( x − 3) + 2
                                                             2               2


and f ( x ) = ( x − 3) − 2
                             2


Determine the k value, the vertex, and the axis of symmetry.


If k is positive, the graph of y = a (x − h ) is shifted k units up.
                                                         2


If k is negative, the graph of y = a (x − h ) is shifted k units down.
                                             2




Vertex: (h, k)                           Axis of symmetry: x = h


Ex. Without graphing, find the vertex and the axis of symmetry.
(a) f (x ) = 2( x − 5) − 3        (b) f ( x ) = −( x + 1) + 4
                      2                                  2




                                 1
Ex. Graph: f ( x ) = −             ( x + 5) .
                                           2

                                 2

Vertex:

AOS:




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Ex. Graph: f ( x ) = 2 x − 5 .
                        2



Vertex:

AOS:




         The Graph of f ( x ) = ax + bx + c
                                  2


                         b
Axis of Symmetry: x = −
                        2a

The Vertex of a Parabola
The vertex of the parabola given by f ( x ) = ax + bx + c is
                                                2


                             ⎛ b        ⎛ b ⎞⎞
                             ⎜−
                             ⎜ 2a ,   f ⎜ − ⎟⎟ .
                                               ⎟
                             ⎝          ⎝ 2a ⎠ ⎠

Ex. f ( x ) = − x − 2 x + 3
                 2


  (a) Find the x- and y-intercepts.




    (b) State whether the parabola opens upwards or downwards.


    (c) Find the vertex and the axis of symmetry.




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    (d) Graph the function by hand.




             Solve Applications Involving Parabolas
Ex. (#56) A ball is drop-kicked straight up with an initial velocity of 36 feet per
second. The equation h = −16 t + 36 t describes the height, h, of the ball in
                                   2


feet t seconds after being kicked.
   (a) After how many seconds does the ball reach its maximum height?




    (b) What is the maximum height the ball reaches?




    (c) How long is the ball in the air?




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