3 Maxima and Minima: Getting Information from a Model
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CHAPTER 5
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Quadratic Functions and Models
maximum height the rocket reaches, we need to find the maximum value of the quadratic function h.
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The model gives us the height the rocket reaches at any time.
Our goal is to find the maximum height the rocket reaches.
In this section we solve this problem and explore several other real-world problems
that require us to find a maximum (or minimum) value of a quadratic function (see
Example 2). So we begin by developing algebraic formulas for finding these values.
2
■ Finding Maximum and Minimum Values
The maximum value of a function occurs at the highest point on the graph of the
function, and the minimum value occurs at the lowest point on the graph of the function. So to find the maximum value (or minimum value) of a quadratic function
f 1x2 = ax 2 + bx + c, we look at the graph of the function. If a 7 0, then the graph
of f is a parabola that opens upward, so the function has a minimum value at the
vertex (see Figure 1(a)). On the other hand, if a 6 0 , then the graph of f opens downward, so the function has a maximum value at the vertex (see Figure 1(b)). If f is expressed in standard form f 1x2 = a1x - h2 2 + k, then the vertex is (h, k), and the
maximum or minimum value of f occurs at x = h. Also,
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If a 7 0, then the minimum value of f is f 1h2 = k.
If a 6 0, then the maximum value of f is f 1h2 = k.
y
y
Minimum
value is k
Maximum
value is k
(h, k)
0
x
figure 1
(h, k)
0
(a)
x
(b)
If we are interested only in finding the maximum or minimum value, then a formula is available for doing so. This formula is obtained by completing the square for
the general quadratic function as follows:
f 1x 2 = ax 2 + bx + c
= a a x2 +
b
xb + c
a
Factor a from the x-terms
b2
b2
To complete the square, we add 2 inside the parentheses and subtract a a 2 b out4a
4a
side. We get
f 1x 2 = a a x 2 +
= aax +
b
b2
b2
x + 2 b + c - aa 2 b
a
4a
4a
b 2
b2
b + c 2a
4a
Complete the square
Factor
SECTION 5.3
■
Maxima and Minima: Getting Information from a Model
441
b
b2
and k = c - .
2a
4a
Since the maximum or minimum value occurs at x = h, we have the following result.
This equation is in standard form with h = -
Maximum or Minimum Value of a Quadratic Function
The x-coordinate of the maximum or minimum value of a quadratic function
f 1x2 = ax 2 + bx + c occurs at
x=-
b
2a
If a 7 0, then the minimum value is f a -
b
b.
2a
b
If a 6 0, then the maximum value is f a - b .
2a
e x a m p l e 1 Finding Maximum and Minimum Values of Quadratic Functions
Find the maximum or minimum value of each quadratic function.
(a) f 1x2 = x 2 + 4x
(b) f 1x2 = - 2x 2 + 4x - 5
Solution
(a) This is a quadratic function where a is 1 and b is 4. Thus the maximum or
minimum value occurs at
4
b
2a
Formula
4
2#1
Replace a by 1 and b by 4
x =2
_5
=-
= -2
_6
Since a 7 0, the function has the minimum value
The minimum value occurs at
x = - 2.
f 1- 22 = 1- 22 2 + 41- 22 = - 4
(b) This is a quadratic function where a is - 2, b is 4, and c is - 5. Thus the
maximum or minimum value occurs at
1
4
_2
Calculate
x =-
==1
b
2a
Formula
4
21- 22
Replace a by - 2 and b by 4
Calculate
_6
Since a 6 0, the function has the maximum value
The maximum value occurs at
x = 1.
f 112 = - 2112 2 + 4112 - 5 = - 3
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NOW TRY EXERCISES 9 AND 11
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2
CHAPTER 5
■
Quadratic Functions and Models
■ Modeling with Quadratic Functions
Now we study real-world phenomena that are modeled by quadratic functions.
e x a m p l e 2 Maximum Height for a Model Rocket
A model rocket is shot straight upward with an initial speed of 800 ft/s, and the
height of the rocket is given by
h = 800t - 16t 2
where h is measured in feet and t in seconds. What is the maximum height of the
rocket, and after how many seconds is this height attained?
Solution
The function h is a quadratic function where a is - 16 and b is 800. Thus the maximum value occurs when
t =-
=-
b
2a
Formula
800
21-162
Replace a by - 16 and b by 800
= 25
Calculate
The maximum value is h = - 161252 2 + 8001252 = 10,000. So the rocket reaches a
maximum height of 10,000 feet after 25 seconds.
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NOW TRY EXERCISE 23
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e x a m p l e 3 Maximum Gas Mileage for a Car
Most cars get their best gas mileage when traveling at a relatively moderate speed.
The gas mileage M for a certain new car is modeled by the function
M1s2 = -
1 2
s + 3s - 31
28
15 … s … 70
where s is the speed in miles per hour and M is measured in miles per gallon. What
is the car’s best mileage, and at what speed is it attained?
Solution
The function M is a quadratic function where a is - 281 and b is 3. Thus the maximum
value occurs when
40
s =-
b
2a
Formula
3
=15
0
70
The maximum gas mileage occurs
at 42 mi/h.
2A- 281 B
= 42
Replace a by - 281 and b by 3
Calculate
- 281 1422 2
The maximum value is M1422 =
+ 31422 - 31 = 32. So the car’s best
gas mileage is 32 mi/gal, when it is traveling at 42 mi/h.
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NOW TRY EXERCISE 25
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SECTION 5.3
■
Maxima and Minima: Getting Information from a Model
443
e x a m p l e 4 Fencing a Garden
A gardener has 140 feet of fencing to fence in a rectangular vegetable garden.
(a) Find a function that models the area of the garden in terms of the width of the
garden.
(b) Find the largest area she can fence and the dimensions of that area.
Solution
(a) In Example 5 of Section 1.8 (page 94) we found that the function that models
the area she can fence is
x
A1x2 = 70x - x 2
where x is the width of the garden in feet, and 70 - x is the length in feet.
(See Figure 2.)
70-x
f i g u r e 2 Area is
A1x 2 = x170 - x 2 = 70x - x 2
(b) In Section 1.8 we found the maximum value of the function A graphically.
Here we find the maximum area algebraically.
The function A is a quadratic function where a is - 1 and b is 70. Thus the
maximum value occurs when
x =-
Compare with graphical solution in
Example 5 of Section 1.8
(page 94).
=-
b
2a
Formula
70
21- 12
Replace a by - 1 and b by 70
= 35
Calculate
When the width is 35, the length is 70 - 35 = 35, and the maximum area
is A1352 = 701352 - 1352 2 = 1225. So the largest area she can fence is
1225 square feet, and the dimensions are 35 by 35 feet.
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NOW TRY EXERCISE 27
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e x a m p l e 5 Maximum Revenue from Ticket Sales
A hockey team plays in an arena that has a seating capacity of 15,000 spectators.
With the ticket price set at $14, average attendance at recent games has been 9500.
A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 1000.
(a) Find a function that models the revenue in terms of ticket price.
(b) Find the price that maximizes revenue from ticket sales.
Solution
(a) The model that we want is a function that gives the revenue for any ticket
price.
revenue = ticket price * attendance
There are two varying quantities: ticket price and attendance. Since the
function we want depends on price, we let
x = ticket price
Next, we express attendance in terms of x.
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CHAPTER 5
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Quadratic Functions and Models
In Words
In Algebra
Ticket price
Amount ticket price is lowered
Increase in attendance
Attendance
x
14 - x
1000114 - x 2
9500 + 1000114 - x2
The model that we want is the function R that gives the revenue for a given
ticket price x.
revenue = ticket price * attendance
R1x2 = x * 39500 + 1000114 - x24
150,000
R1x2 = x123,500 - 1000x2
R1x2 = 23,500x - 1000x 2
(b) Since R is a quadratic function where a is - 1000 and b is 23,500, the
maximum occurs at
0
25
x =-
Maximum revenue occurs when the
ticket price is $11.75.
23,500
b
== 11.75
2a
21- 10002
So a ticket price of $11.75 gives the maximum revenue.
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NOW TRY EXERCISE 33
5.3 Exercises
CONCEPTS
Fundamentals
1. The quadratic function f 1x 2 = ax 2 + bx + c is in general form.
(a) The maximum or minimum value of f occurs at x =
(b) If a 7 0, then f has a _______ (maximum/minimum) value.
(c) If a 6 0, then f has a _______ (maximum/minimum) value.
2. (a) The quadratic function f 1x2 = 2x 2 - 12x + 5 has a _______
(maximum/minimum) value of f a
ٗ b ϭ ______.
ٗ
(b) The quadratic function f 1x 2 = - 2x 2 - 12x + 5 has a _______
(maximum/minimum) value of f a
Think About It
ٗ b ϭ ______.
ٗ
3. Consider the quadratic function y = 1x - 22 1x - 42 .
(a) In general form, y = ______________.
(b) The graph is a parabola that opens _______ (up/down).
(c) From the general form we see that the minimum value occurs at x =
_______.
=
SECTION 5.3
■
445
Maxima and Minima: Getting Information from a Model
4. Consider the quadratic function y = 1x - m2 1x - n2 .
(a) In general form, y = ______________.
(b) The graph is a parabola that opens _______ (up/down).
(c) From the general form we see that the minimum value occurs at x =
.
(d) Using the formula in part (c), the minimum value of y = 1x + 32 1x - 52 occurs
when x = _______.
SKILLS
5–8
■ A graph of a quadratic function is shown.
(a) Does the function have a minimum or a maximum value? What is that value?
(b) Find the x-value at which the minimum or maximum value occurs.
5.
6.
y
y
(_3, 4)
2
1 x
1
x
(2, 3)
2
0
7.
0
x
1
8.
y
y
(4, 5)
3
0
4
x
2
0
(_1, _3)
9–14
■
Find the minimum or maximum value of the quadratic function, and find the
x-value at which the minimum or maximum value occurs.
9. f 1x2 = -
x2
+ 2x + 7
3
11. f 1x2 = x 2 + x + 1
13. h1x2 = 12 x 2 + 2x - 6
10. g1x 2 = 2x1x - 4 2 + 7
12. f 1x2 = 1 + 3x - x 2
14. f 1x2 = 3 - x - 21 x 2
15–18 ■ A quadratic function is given.
(a) Sketch its graph.
(b) Find the minimum or maximum value of f, and find the x-value at which the
minimum or maximum value occurs.
15. f 1x2 = x 2 + 2x - 1
17. f 1x2 = - x 2 - 3x + 3
16. f 1x2 = x 2 - 8x + 8
18. f 1x2 = 1 - 6x - x 2
19–22 ■ A quadratic function is given.
(a) Use a graphing device to find the maximum or minimum value of the quadratic
function f, correct to two decimal places.
(b) Find the maximum or minimum value of f, and compare it with your answer to part (a).
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CHAPTER 5
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Quadratic Functions and Models
19. f 1x2 = x 2 + 1.79x - 3.21
20. f 1x2 = x 2 - 3.2x + 4.1
21. f 1x2 = 1 + x - 22x 2
22. f 1x2 = 3 - 2x - 23x 2
23. Height of a Ball If a ball is thrown directly upward with a velocity of 12 m/s, its
height (in meters) after t seconds is given by y = 12t - 4.9t 2. What is the maximum
height attained by the ball, and after how many seconds is that height attained?
CONTEXTS
24. Path of a Ball A ball is thrown across a playing field from a height of 5 ft above the
ground at an angle of 45° to the horizontal and at a speed of 20 ft/s. It can be deduced
from physical principles that the path of the ball is modeled by the function
5 ft
y=x
32 2
x +x+5
1202 2
where x is the distance in feet that the ball has traveled horizontally and y is the height
in feet. What is the maximum height attained by the ball, and at what horizontal
distance does this occur?
25. Agriculture The number of apples produced by each tree in an apple orchard depends
on how densely the trees are planted. If n trees are planted on an acre of land, then each
tree produces 900 - 9n apples. So the number of apples produced per acre is
A1n2 = n1900 - 9n 2
What is the maximum yield of the trees, and how many trees should be planted per acre
to obtain the maximum yield of apples?
26. Agriculture At a certain vineyard it is found that each grape vine produces about 10
pounds of grapes in a season when about 700 vines are planted per acre. For each
additional vine that is planted, the production of each vine decreases by about 1%. So
the number of pounds of grapes produced per acre is modeled by
A1n2 = 1700 + n2 110 - 0.01n 2
where n is the number of additional vines planted. What is the maximum yield of the
grape vines, and how many vines should be planted to maximize grape production?
x
A
x
27. Fencing a Field A farmer has 2400 feet of fencing with which he wants to fence off a
rectangular field that borders a straight river. He does not need a fence along the river
(see the figure).
(a) Find a function A that models the area of the field in terms of one of its sides x.
(b) What is the largest area that he can fence, and what are the dimensions of that area?
[Compare to your graphical solution in Exercise 29, Section 1.8.]
28. Dividing a Pen A rancher with 750 feet of fencing wants to enclose a rectangular
area and then divide it into four pens with fencing parallel to one side of the rectangle
(see the figure).
(a) Find a function A that models the total area of the four pens.
(b) Find the largest possible total area of the four pens and the dimensions of that area.
[Compare to your graphical solution in Exercise 30, Section 1.8.]
29. Fencing a Horse Corral Carol has 1200 feet of fencing to fence in a rectangular
horse corral.
(a) Find a function A that models the area of the corral in terms of the width of the
corral.
x
600 – x
SECTION 5.3
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Maxima and Minima: Getting Information from a Model
447
(b) Find the dimensions of the rectangle that maximize the area of the corral.
30. Making a Rain Gutter A rain gutter is formed by bending up the sides of a 30-inchwide rectangular metal sheet as shown in the figure.
(a) Find a function A that models the cross-sectional area of the gutter in terms of x.
(b) Find the value of x that maximizes the cross-sectional area of the gutter.
(c) What is the maximum cross-sectional area for the gutter?
x
30 in.
31. Minimizing Area A wire 10 cm long is cut into two pieces, one of length x and the
other of length 10 - x, as shown in the figure. Each piece is bent into the shape of a
square.
(a) Find a function A that models the total area enclosed by the two squares.
(b) Find the value of x that minimizes the total area of the two squares.
10 cm
x
10-x
32. Light from a Window A Norman window has the shape of a rectangle surmounted
by a semicircle, as shown in the figure. A Norman window with perimeter 30 feet is to
be constructed.
(a) Find a function that models the area of the window.
(b) Find the dimensions of the window that admits the greatest amount of light.
x
33. Stadium Revenue A baseball team plays in a stadium that holds 55,000 spectators.
With the ticket price at $10, the average attendance at recent games has been 27,000. A
market survey indicates that for every dollar the ticket price is lowered, the attendance
increases by 3000.
(a) Find a function R that models the revenue in terms of ticket price.
(b) Find the price that maximizes revenue from ticket sales.
34. Maximizing Profit A community bird-watching society makes and sells simple bird
feeders to raise money for its conservation activities. The materials for each feeder cost
$6, and the society sells an average of 20 feeders per week at a price of $10 each. The
society has been considering raising the price, so it conducts a survey and finds that for
every dollar increase, it loses 2 sales per week.
(a) Find a function P that models weekly profit in terms of price per feeder.
(b) What price should the society charge for each feeder to maximize profits? What is
the maximum weekly profit?
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CHAPTER 5
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Quadratic Functions and Models
5.4 Quadratic Equations: Getting Information from a Model
■
Solving Quadratic Equations: Factoring
■
Solving Quadratic Equations: The Quadratic Formula
■
The Discriminant
■
Modeling with Quadratic Functions
IN THIS SECTION… we study how to solve quadratic equations. Solving quadratic
equations helps us get information from quadratic models.
GET READY… by reviewing how to factor expressions in Algebra Toolkit B.2. Test
your understanding by doing the Algebra Checkpoint at the end of this section.
In Section 4.3 we found that the height a rocket reaches is modeled by a quadratic
function. In this section we use the model to answer the question “When does the
rocket reach a given height?” (see Example 8).
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The model gives us the height the rocket reaches at any time.
Our goal is to find the time at which the rocket reaches a given height.
We can get the information we want from the model; to do so, we need to solve
a quadratic equation. So we start this section by learning how to solve such
equations.
2
■ Solving Quadratic Equations: Factoring
A quadratic equation is an equation of the form
ax 2 + bx + c = 0
where a, b, and c are real numbers with a 0. Some quadratic equations can be
solved by factoring and using the following basic property of real numbers.
Zero-Product Property
AB = 0
if and only if
A=0
or
B =0
This means that if we can factor the left-hand side of a quadratic (or other) equation,
then we can solve it by setting each factor equal to 0 in turn. This method works only
when the right-hand side of the equation is 0.
e x a m p l e 1 Solving a Quadratic Equation by Factoring
Solve the equation x 2 + 5x = 24.
Solution
We must first rewrite the equation so that the right-hand side is 0.
SECTION 5.4
■
Quadratic Equations: Getting Information from a Model
x 2 + 5x = 24
x 2 + 5x - 24 = 0
1x - 32 1x + 82 = 0
x - 3 = 0 or
x + 8 = 0
x = 3
x = -8
449
Given equation
Subtract 24
Factor
Zero-Product Property
Solve
The solutions are x = 3 and x = - 8.
✓ C H E C K We substitute x = 3 into the original equation:
132 2 + 5132 = 9 + 15 = 24 ✓
We substitute x = - 8 into the original equation:
1- 82 2 + 51- 82 = 64 - 40 = 24 ✓
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NOW TRY EXERCISE 7
■
Do you see why one side of the equation must be 0 in Example 1? Factoring the
equation as x1x + 52 = 24 does not help us find the solutions, since 24 can be
factored in infinitely many ways, such as 6 # 4, 12 # 48, A- 25 B # 1- 602 , and so on.
example 2
Graphing a Quadratic Function
Let f 1x2 = x 2 + 5x - 24.
(a) Find the x-intercepts of the graph of f.
(b) Sketch the graph of f, and label the x- and y-intercepts and the vertex.
Solution
x- and y-intercepts are reviewed
in Algebra Toolkit D.2, page T71.
(a) To find the x-intercepts, we solve the equation
x 2 + 5x - 24 = 0
The equation was solved in Example 1. The x-intercepts are 3 and - 8.
(b) To find the y-intercept, we set x equal to 0:
f 102 = 0 + 5 # 0 - 24 = - 24
So the y-intercept is - 24.
The function f is a quadratic function where a is 1 and b is 5. So the
x-coordinate of the vertex occurs at
x =-
b
2a
5
=-
=-
2#1
5
2
Formula
Replace a by 1 and b by 5
Calculate