3.1 Quadratic Functions Examples And Worksheet - Chapter 3: Polynomial And Rational Functions Page 6
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Section 3.1
Quadratic Functions
Finding Minimum and Maximum Values
Many applications involve finding the maximum or minimum value of a
quadratic function. By completing the square of the quadratic function
2
f x
ax
bx
c,
you can rewrite the function in standard form.
2
2
b
b
f x
a x
c
Standard form
2a
4a
You can see that the vertex occurs at
x
b 2a ,
which implies the following.
Minimum and Maximum Values of Quadratic Functions
b
1. If
a > 0,
f
has a minimum value at
x
.
2a
b
2. If
a < 0,
f
has a maximum value at x
.
2a
Example 5
The Maximum Height of a Baseball
T E C H N O L O G Y T I P
Note in the graphical solution for
A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per
Example 5, that when using the
second and at an angle of
45
with respect to the ground. The path of the baseball
zoom and trace features, you
2
is given by the function
f x
0.0032x
x
3,
where
f x
is the height of
might have to change the -scale
y
the baseball (in feet) and
x
is the horizontal distance from home plate (in feet).
in order to avoid a graph that is
What is the maximum height reached by the baseball?
“too flat.”
Algebraic Solution
Graphical Solution
2
For this quadratic function, you have
Use a graphing utility to graph
y
0.0032x
x
3
so that you can see the important features of the parabola.
2
2
f x
ax
bx
c
0.0032x
x
3
Use the maximum feature (see Figure 3.8) or the zoom and trace
features (see Figure 3.9) of the graphing utility to approximate
which implies that
a
0.0032
and
b
1.
Because
the maximum height on the graph to be
y
81.125
feet at
the function has a maximum when
x
b 2a ,
you
x
156.25.
can conclude that the baseball reaches its maximum
height when it is feet from home plate, where is
x
x
y = −0.0032x
+ x + 3
2
b
1
81.3
x
100
2a
2
0.0032
156.25
feet.
At this distance, the maximum height is
2
f
156.25
0.0032
156.25
156.25
3
152.26
159.51
0
400
81
0
81.125 feet.
Figure 3.8
Figure 3.9
Now try Exercise 55.
TECHNOLOGY S U P P O R T
For instructions on how to use the maximum,
the minimum, the table, and the zoom and trace features, see Appendix A;
for specific keystrokes, go to this textbook’s Online Study Center.
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