11.7
Objectives
a
b
Solve maximum–minimum
problems involving
quadratic functions.
Fit a quadratic function
to a set of data to form a
mathematical model, and
solve related applied
problems.
MATHEMATICAL MODELING WITH
QUADRATIC FUNCTIONS
We now consider some of the many situations in which quadratic functions can serve as mathematical models.
a
Maximum–Minimum Problems
We have seen that for any quadratic function f x ax 2 bx c, the value
of f x at the vertex is either a maximum or a minimum, meaning that either
all outputs are smaller than that value for a maximum or larger than that
value for a minimum.
y
y
(x, f (x))
a0
a0
(x, f (x))
x
f (x) at the vertex
a minimum
x
f (x) at the vertex
a maximum
There are many types of applied problems in which we want to find a
maximum or minimum value of a quantity. If a quadratic function can be
used as a model, we can find such maximums or minimums by finding coordinates of the vertex.
EXAMPLE 1 Fenced-In Land. A farmer has 64 yd of fencing. What are the dimensions of the largest rectangular pen that the farmer can enclose?
1. Familiarize. We first make a drawing and label it. We let l the length
of the pen and w the width. Recall the following formulas:
Perimeter: 2l 2w ;
Area:
l
w
A
22.5
20.5
18.5
18.5
12.4
15.5
10.5
12.5
14.5
13.5
19.6
17.5
220.75
240.75
252.75
249.75
243.04
255.75
l w.
w
834
l
To become familiar with the problem, let’s choose some dimensions
(shown at left) for which 2l 2w 64 and then calculate the corresponding areas. What choice of l and w will maximize A?
CHAPTER 11: Quadratic Equations
and Functions
An Addison-Wesley product. Copyright © 2003 Pearson Education, Inc.
2. Translate.
We have two equations, one for perimeter and one for area:
2l 2w 64,
A l w.
Let’s use them to express A as a function of l or w, but not both. To express
A in terms of w, for example, we solve for l in the first equation:
1. Fenced-In Land. A farmer has
100 yd of fencing. What are the
dimensions of the largest
rectangular pen that the
farmer can enclose?
To familiarize yourself with the
problem, complete the
following table.
2l 2w 64
2l 64 2w
l
64 2w
2
32 w.
Substituting 32 w for l, we get a quadratic function Aw, or just A:
A lw 32 ww 32w w 2 w 2 32w.
l
w
A
12.5
15.5
24.5
25.5
26.2
38.5
35.5
26.5
25.5
23.8
456
3. Carry out. Note here that we are altering the third step of our fivestep problem-solving strategy to “carry out” some kind of mathematical manipulation, because we are going to find the vertex rather than
solve an equation. To do so, we complete the square as in Section 11.6:
A w 2 32w
This is a parabola opening down,
so a maximum exists.
1w 2 32w
Factoring out 1
2
1
2 32
1w 32w 256 256







16; 162 256.
We add 0, or 256 256.
1w 2 32w 256 1 256
Using the distributive law
2
w 16 256.
The vertex is 16, 256. Thus the maximum value is 256. It occurs when
w 16 and l 32 w 32 16 16.
4. Check. We note that 256 is larger than any of the values found in the
Familiarize step. To be more certain, we could make more calculations.
We leave this to the student. We can also use the graph of the function
to check the maximum value.
y
300
(16, 256)
250
Maximum: 256
200
A(w) (w 16)2 256
150
100
50
0
10
20
30
40 w
5. State. The largest rectangular pen that can be enclosed is 16 yd by 16 yd;
that is, a square.
Do Exercise 1.
Answer on page A-51
835
11.7 Mathematical Modeling with
Quadratic Functions
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calculator corner
Maximum and Minimum Values We can use a graphing calculator to find the maximum or minimum value of a
quadratic function. Consider the quadratic function in Example 1, A w 2 32w . First, we replace w with x and A with y
and graph the function in a window that displays the vertex of the graph. We choose 0, 40, 0, 300 , with Xscl 5 and
Yscl 20. Now, we press 2nd CALC 4 or 2nd CALC ENTER to select the MAXIMUM feature from the CALC
menu. We are prompted to select a left bound for the maximum point. This means that we must choose an x-value that is to
the left of the x-value of the point where the maximum occurs. This can be done by using the left- and right-arrow keys to
move the cursor to a point to the left of the maximum point or by keying in an appropriate value. Once this is done, we press
ENTER . Now, we are prompted to select a right bound. We move the cursor to a point to the right of the maximum point or
key in an appropriate value.
y x 2 32x
y x 2 32x
300
Y1X2 32X
300
Y1X2 32X
LeftBound?
0 X10.638298
0
Xscl5
Y227.25215
40
Yscl20
RightBound?
0 X20.851064
0
Xscl5
Y232.46718
40
Yscl20
We press ENTER again. Finally, we are prompted to guess the x-value at which the maximum occurs. We move the
cursor close to the maximum or key in an x-value. We press ENTER a third time and see that the maximum function value of
256 occurs when x 16. (One or both coordinates of the maximum point might be approximations of the actual values, as
shown with the x-value below, because of the method the calculator uses to find these values.)
y x 2 32x
y x 2 32x
300
300
Y1X2 32X
Guess ?
0 X15.744681
0
2nd
Y255.93481
40
Maximum
0 X15.999999
0
Y256
To find a minimum value, we select item 3, “minimum,” from the CALC menu by pressing
CALC ENTER .
40
2nd
CALC
3
ENTER
or
Exercises: Use the maximum or minimum feature on a graphing calculator to find the maximum or minimum value of
each function.
1. y 3x 2 6x 4
2. y 2x 2 x 5
3. y x 2 4x 2
4. y 4x 2 5x 1
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CHAPTER 11: Quadratic Equations
and Functions
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b
Fitting Quadratic Functions to Data
As we move through our study of mathematics, we develop a library of
functions. These functions can serve as models for many applications.
Some of them are graphed below. We have not considered the cubic or
quartic functions in detail other than in the Calculator Corners (we leave
that discussion to a later course), but we show them here for reference.
Linear function:
f (x) mx b
Quadratic function:
f (x) ax 2 bx c, a 0
Quadratic function:
f (x) ax 2 bx c, a 0
Absolute-value function:
f (x) x
Cubic function:
f (x) ax 3 bx 2 cx d, a 0
Quartic function:
f (x) ax 4 bx 3 cx 2 dx e, a 0
Now let’s consider some real-world data. How can we decide which type
of function might fit the data of a particular application? One simple way is to
graph the data and look for a pattern resembling one of the graphs above. For
example, data might be modeled by a linear function if the graph resembles a
straight line. The data might be modeled by a quadratic function if the graph
rises and then falls, or falls and then rises, in a curved manner resembling a
parabola. For a quadratic, it might also just rise or fall in a curved manner as
if following only one part of the parabola.
837
11.7
An Addison-Wesley product. Copyright © 2003 Pearson Education, Inc.
Mathematical Modeling with
Quadratic Functions
Choosing Models. For the
scatterplots and graphs in Margin
Exercises 2–5, determine which, if
any, of the following functions might
be used as a model for the data.
Linear, f x mx b;
Let’s now use our library of functions to see which, if any, might fit certain
data situations.
EXAMPLES Choosing Models. For the scatterplots and graphs below, determine which, if any, of the following functions might be used as a model for
the data.
Quadratic, f x ax 2 bx c,
a 0;
Linear, f x mx b;
Quadratic, f x ax bx c,
a 0;
Quadratic, f x ax 2 bx c, a 0;
Polynomial, neither quadratic
nor linear
Polynomial, neither quadratic nor linear
y
Quadratic, f x ax 2 bx c, a 0;
2.
8
4
0
0
5
10
Population
(in millions)
2.
Sales
(in millions)
2
y
8
4
0
0
x
2
4
6
8
x
Year
Year
y
8
4
5
10
4
0
0
5
10
x
x
The data seem to fit a linear function f x mx b.
4.
Sales
(in millions)
8
Year
0
0
Year
4.
y
y
20
Population
(in millions)
3.
Sales
(in millions)
3.
Population
(in millions)
The data rise and then fall in a curved manner fitting a quadratic function
f x ax 2 bx c, a 0.
y
10
5
0
0
4
2
6 x
Year
10
0
0
4
2
The data rise in a manner fitting the right side of a quadratic function
f x ax 2 bx c, a 0.
6 x
Year
5.
Life expectancy
for women
(in years)
5.
y
100
80
60
40
20
0
0
2
4
6
8
10 12 x
Shoe size
Source: Orthopedic Quarterly
Answers on page A-51
838
The data fall and then rise in a curved manner fitting a quadratic function
f x ax 2 bx c, a 0.
CHAPTER 11: Quadratic Equations
and Functions
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6.
Number of births per 1000 women
U.S. Birth Rate for Women Ages 15 –19
70
60
50
40
30
20
10
0
1970s
1980s
1990s
Source: Centers for Disease Control and Prevention
The data fall, then rise, then fall again. They do not appear to fit a linear
or quadratic function but might fit a polynomial function that is neither
quadratic nor linear.
Do Exercises 2 – 5 on the preceding page.
Whenever a quadratic function seems to fit a data situation, that function
can be determined if at least three inputs and their outputs are known.
EXAMPLE 7 River Depth. The drawing below shows the cross section of a
river. Typically rivers are deepest in the middle, with the depth decreasing
to 0 at the edges. A hydrologist measures the depths D, in feet, of a river at
distances x, in feet, from one bank. The results are listed in the table at
right.
x distance from left bank (in feet)
D(x) depth of river
(in feet)
DISTANCE, x,
FROM THE
RIVERBANK
(in feet)
DEPTH, D,
OF THE RIVER
(in feet)
100
10.2
115
10.2
125
17.2
150
20.2
190
17.2
100
10.2
839
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Mathematical Modeling with
Quadratic Functions
6. Ticket Profits. Valley
Community College is
presenting a play. The profit P,
in dollars, after x days is given in
the following table. (Profit can
be negative when costs exceed
revenue. See Section 3.8.)
PROFIT, P
450
$100
490
$2560
180
$2872
270
$2870
360
$2548
450
$100
c) Use the data points 0, 0, 50, 20, and 100, 0 to find a quadratic function
that fits the data.
d) Use the function to estimate the depth of the river at 75 ft.
a) The scatterplot is as follows.
River Depth
D
20
15
10
5
0
20
40
60
80
100
x
Distance from the river bank (in feet)
b) The data seem to rise and fall in a manner similar to a quadratic function.
P
The dashed black line in the graph represents a sample quadratic function
of fit. Note that it may not necessarily go through each point.
$1400
1200
1000
Profit
b) Decide whether the data seem to fit a quadratic function.
Depth (in feet)
DAYS, x
a) Make a scatterplot of the data.
c) We are looking for a quadratic function
800
Dx ax 2 bx c.
600
400
200
200
100
200
300
400
500 x
Days
We need to determine the constants a, b, and c. We use the three data
points 0, 0, 50, 20, and 100, 0 and substitute as follows:
0 a 0 2 b 0 c,
20 a 50 2 b 50 c,
a) Make a scatterplot of
the data.
0 a 100 2 b 100 c.
After simplifying, we see that we need to solve the system
0 c,
20 2,500a 50b c,
b) Decide whether the data can
be modeled by a quadratic
function.
0 10,000a 100b c.
Since c 0, the system reduces to a system of two equations in two
variables:
20 2,500a 50b,
0 10,000a 100b.
c) Use the data points 0, 100,
180, 872, and 360, 548 to
find a quadratic function that
fits the data.
(1)
(2)
We multiply equation (1) by 2, add, and solve for a (see Section 8.3):
40 5,000a 100b,
0 10,000a 100b
40 40
a
5000
5000a
Adding
Solving for a
0.008 a.
d) Use the function to estimate
the profits after 225 days.
Answers on page A-51
840
CHAPTER 11: Quadratic Equations
and Functions
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Next, we substitute 0.008 for a in equation (2) and solve for b:
0 10,0000.008 100b
0 80 100b
80 100b
0.8 b.
This gives us the quadratic function:
Dx 0.008x 2 0.8x.
d) To find the depth 75 ft from the riverbank, we substitute:
D75 0.008752 0.875 15.
At a distance of 75 ft from the riverbank, the depth of the river is 15 ft.
y
D(x) 0.008x 2 0.8x
20
D(75) 15
15
10
5
0
20
40
60
80
75
100 x
Do Exercise 6 on the preceding page.
calculator corner
Mathematical Modeling: Fitting a Quadratic Function to Data We can use the quadratic regression
feature on a graphing calculator to fit a quadratic function to a set of data. The following table shows the average number of
live births for women of various ages.
AGE
AVERAGE NUMBER
OF LIVE BIRTHS
PER 1000 WOMEN
16.5
134.1
18.5
186.5
22.5
111.1
27.5
113.9
32.5
184.5
37.5
135.4
42.5
116.8
a) Make a scatterplot of the data and verify that the data can be modeled with a quadratic function.
b) Fit a quadratic function to the data using the quadratic regression feature on a graphing calculator.
c) Use the function to estimate the average number of live births per 1000 women of age 20 and of age 30.
(continued)
841
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Mathematical Modeling with
Quadratic Functions
a) We enter the data on the STAT list editor screen, turn on Plot 1, and plot the points as described in the Calculator Corner
on p. 546. The data points rise and then fall in a manner consistent with a quadratic function.
L1
16
18.5
22
27
32
37
42
L2(7)6.8
L2
34
86.5
111.1
113.9
84.5
35.4
6.8
L3
2
5 VARS 1 1 ENTER . The first three keystrokes select
QuadReg from the STAT CALC menu and display the coefficients a, b, and c of the regression equation y ax 2 bx c .
The keystrokes VARS 1 1 copy the regression equation to the equation-editor screen as y1. We see that the
regression equation is y 0.4868465035x 2 25.94985182x 238.4892193. We can press ZOOM 9 to see the
regression equation graphed with the data points.
b) To fit a quadratic function to the data, we press STAT
QuadReg
yax2bxc
a.4868465035
b 25.94985182
c238.4892193
Plot1 Plot2 Plot3
\Y1 .4868465035
3607X^225.94985
1822606X238.48
921925252
\Y2 \Y3 \Y4 c) Since the equation is entered on the equation-editor screen, we can use a table set in ASK mode to estimate the average
number of live births per 1000 women of age 20 and of age 30. We see that when x 20, y 85.8, and when x 30,
y 101.8, so we estimate that there are about 85.8 live births per 1000 women of age 20 and about 101.8 live births
per 1000 women of age 30.
X
20
30
Y1
85.769
101.84
X
Remember to turn off the STAT PLOT as described on p. 548 before you graph other equations.
Exercise:
1. Consider the data in the table in Example 7.
a) Use a graphing calculator to make a scatterplot of the data.
b) Use a graphing calculator to fit a quadratic function to the data. Compare this function with the one found
in Example 7.
c) Graph the quadratic function with the scatterplot.
d) Use the function found in part (b) to estimate the depth of the river 75 ft from the riverbank. Compare this
estimate with the one found in Example 7.
842
CHAPTER 11: Quadratic Equations
and Functions
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