Quadratic Functions: AB rief Overview by s6456f0H

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									Quadratic Functions:
 A Brief Overview
 Definition

A quadratic     function is a function of the form


          f ( x)  ax 2  bx  c,
  where a, b, and c are real numbers and    a  0.

Observe that the highest power of x is 2 for a quadratic function.
Graphs of Quadratic Functions
Consider the quadratic function    f ( x)  ax 2  bx  c.
 Then the graph of this function
 • is called a parabola
 • is symmetric with respect to a line, called its axis of symmetry
 • opens upward if a > 0
 • opens downward if a < 0
 • has a lowest or highest point, called its vertex

                                                          a<0




                    a >0
Extreme Point of a quadratic function


 The amazing thing is we can easily find the x and y coordinates
 of the vertex of a parabola.


  If   f ( x)  ax 2  bx  c,

                                               b
  then the x-coordinate of the vertex is   x    .
                                               2a
  Once we know the x-coordinate of the vertex, we substitute it
  in the function to find the y-coordinate of the vertex.
Example #1
Shown below is the graph of a quadratic function.

  Find:
                                              10
  a)   the coordinates of each                9
       intercept of the graph of               8
       f(x).                                   7


  b)   the coordinates of the                 5
       vertex of the graph of f(x).            4
                                               3
  c)   What is the smallest                    2
       possible value of this                 1

       function?                         -1
                                               0
                                                   1   2   3   4   5
                                              -1
                                              -2
                                              -3
                                              -4
Example #2
Let   f ( x)  3 x 2  12 x  5.
Without graphing the function,
a)    say whether the graph opens upward or downward
b)    find the x and y coordinates of the vertex .




Please verify these results by graphing the function on your
graphing calculator.
Example #3
Consider again       f ( x)  3 x 2  12 x  5.
Determine
a)   the y intercept of the graph of this function.
b)   the x intercepts of the graph of this function.




     Note that:
     • we can use the Zero feature to estimate the x-intercepts.
     • we can use the Quadratic Formula to find the exact
     coordinates of the x-intercepts.
Quadratic Functions:
   Applications
Quadratic functions in real life

  Certain real-life problems can be modeled by linear
  functions.




  However, there are others for which quadratic functions are
  more appropriate.
   Using graphing calculators
Given a problem.


        We will use our knowledge of key features of quadratic
        functions to find a correct window.




        Then we can use the Zero, the Maximum, the Minimum,
        as well as the Intersection features of our graphing
        calculators to address important questions.
Example #4
     The hourly cost of C (in dollars) of manufacturing x cell
     phones at a plant can be modeled by
                C ( x)  5 x 2  200 x  4000.
a)   Determine the hourly fixed cost.
b)   How many cells phones should be manufactured to
     minimize the hourly cost?
c)   What is the minimum hourly cost?
Example #5
     It is known that the population of Roman Catholic nuns in
     the US has been changing. Suppose that for the year that is
     t years after 1970, the number of nuns (in thousands) can
     be modeled by

                P(t )  0.15t 2  2t  123.75.
a)   According to this model, how many nuns were there in
     1990?
b)   According to the model, when would the number of nuns in
     the US be just 60,000?
c)   According to this model, when did the population of Roman
     Catholic nuns in the US reach a peak?
d)   What was the largest population of Roman Catholic nuns in
     the US, rounded to the nearest 1000?
Example #6
     The price p and the quantity sold x of a certain product obey
     the demand equation
                     1
            p( x)   x  100          0  x  600
                     6
a)   Express the revenue as a function of x.
b)   What is the revenue when 200 units are sold?
c)   What is the maximum revenue?
d)   What price should be charged to maximize revenue?

								
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