Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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320 CHAPTER 3 Quadratic Functions and Operations on Functions 3–40
College Algebra G&M—
While the calculator is a wonderful tool for removing some computational drudgery,
especially when the coefficients are large, noninteger, or not very “pretty,” you are
likely coming to realize that it does not (cannot) replace the analytical thought required
to use it effectively. Most real-world applications still require an attentive analysis of
the context and the question asked, as well as a careful development of the equation
model to be used. This is particularly important in applications like those in Example 7,
where the original equation has more than one independent variable, and a given or
known relationship must be used to eliminate one of them.
EXAMPLE 7
䊳
Due to an increase in consumer demand, a local
nursery is building new chain-link pens for
holding various kinds of mulch. What is the
maximum total area that can be fenced off, if five
open-front pens are needed (see Figure 3.49) and
112 ft of fencing is available?
Solution
䊳
We are asked to maximize the total area, an impossibility right now since the
formula for area has two independent variables: . Knowing that only
112 ft of fencing will be used, we observe that the finished pens will have 1 length
and 6 identical widths, giving the equation . This enables us to
write L in terms of W (solve for L), and substitute the result for L in ,
giving an equation model with the single independent variable W.
known relationship
solve for
L
Using , we substitute for L in the formula .
area formula
substitute for
L
result: a function in one variable
Noting the result is a quadratic function, we enter as Y
1
and
attempt to find an appropriate viewing window. For the maximum value at
, we substitute 112 for b and for a and find this value is near 10.
This shows that would work (with ), since the graph
will be symmetric and 10 is near the “middle” (twice 10 is 20). We reason
further that if the width is 10 ft, is left for the length and
would be a good estimate
for Ymax (the total area). To include a frame
around the viewing window, we set Ymax at
600 with . Using the sequence
(CALC) 4:maximum gives the result
shown in Figure 3.50, which shows a vertex
of . This indicates that a maximum
area of occurs when the width of each
pen is . Note this gives a length of .
Now try Exercises 53 through 56
䊳
56 ftL 112 6 a9
1
3
b9
1
3
ft
522
2
3
ft
2
a9
1
3
, 522
2
3
b
2nd
Ymin 150
10 52 520 ft
2
112 6 10 52 ft
Xmin 0Xmax 20
6x
b
2a
112X 6X
2
A1W2 112W 6W
2
112 6W 1112 6W2W
A LW
Area LWL 112 6W
112 6W L
112 L 6W
A LW
112 L 6W
Area LW
Figure 3.49
20
600
150
0
Figure 3.50
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3–41 Section 3.3 Quadratic Functions and Applications 321
College Algebra G&M—
In a free-market economy, it is well known that if the price of an item is decreased,
more people are likely to buy it. This is why stores have sales and bargain days. But
if the item is sold too cheaply, revenue starts to decline because less money is
coming in—even though more sales are being made. This phenomenon is analyzed in
Example 8 using the standard formula for revenue:
or .
EXAMPLE 8
䊳
Analyzing Retail Sales Revenue
When a popular running shoe is priced at $80, The Shoe Warehouse will sell
an average of 96 pairs per week. Based on sales of similar shoes, the company
believes that for each decrease of $2.50 in price, four additional pairs will be sold.
a. Construct a function that models the store’s revenue at various prices. What is
the maximum possible revenue under these conditions? What price should be
charged to obtain this maximum revenue?
b. What is the cheapest price the manager could charge, while still bringing in a
monthly revenue of at least $7000?
Solution
䊳
a. Based on the given price and sales figures, the store has a current revenue of
. Using these figures, we can develop a general function
model for the revenue, to show a decrease in price and an increase in sales
based on the number of price decreases x.
In standard form, we have
multiply binomials
simplify
To find the maximum revenue, we can
enter the function as Y
1
on the screen
and set a window using the current revenue
as a guide. We chose a window size of
[0, 25] for x and [0, 9000] for y. The result
is shown in Figure 3.51, where the
(CALC) 4:Maximum feature was
used to locate the maximum. It appears
the maximum revenue will be $7840,
which occurs after price decreases
of $2.50. The selling price was
.
b. Since the desired revenue level is $7000, we substitute 7000 for R(x).
substitute 7000 for
R
(
x
)
To solve for x, we’ll use the Intersection-
of-Graphs method with Y
1
as before and
. The result is shown in
Figure 3.52 and indicates that to keep a
revenue of at least $7000 per week, the
price must decrease by no more than about
. The selling price
should be kept near .
Now try Exercises 57 and 58
䊳
80 ⫺ 33 ⫽ $47
13.21$2.502⫽ $33
Y
2
⫽ 7000
⫺10x
2
⫹ 80x ⫹ 7680 ⫽ 7000
$80 ⫺ 41$2.502⫽ $70
x ⫽ 4
TRACE
2nd
Y=
⫺10x
2
⫹ 80x ⫹ 7680 ⫽ R1x2
7680 ⫹ 320x ⫺ 240x ⫺ 10x
2
⫽ R1x2
1$80 ⫺ 2.5x2196 ⫹ 4x2⫽ R1x2
1$8021962⫽ $7680
R ⫽ P
#
S
revenue ⫽ price
#
number of sales
25
9000
0
0
Figure 3.51
25
9000
0
0
Figure 3.52
for each price decrease of
$2.50, sales increase by 4
D. You’ve just seen how
we can solve applications
involving extreme values
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322 CHAPTER 3 Quadratic Functions and Operations on Functions 3–42
College Algebra G&M—
2. The maximum and minimum values are
called values and can be found using
the formula.
䊳
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. Fill in the blank to complete the square,
given
.
f 1x221x
2
5x
25
4
2 7
f
1x22x
2
10x 7:
3.3 EXERCISES
3. To find the zeroes of ,
we substitute for
and solve.
f 1x2 ax
2
bx c
4. If the leading coefficient is positive and the vertex
(h, k) is in Quadrant IV, the graph will have
x-intercepts.
5. Compare/Contrast how to complete the square on
an equation, versus how to complete the square on
a function. Use the equation
and the function to illustrate.f
1x2 2x
2
6x 3
2x
2
6x 3 0
6. Discuss/Explain why the graph of a quadratic
function has no x-intercepts if a and k [vertex
(h, k)] have like signs. Under what conditions will
the function have a single real root?
䊳
DEVELOPING YOUR SKILLS
Graph each function using end-behavior, intercepts,
and completing the square to write the function in
shifted form. Clearly state the transformations used to
obtain the graph, and label the vertex and all intercepts
(if they exist). If the equation is not factorable, use the
quadratic formula to find the x-intercepts.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20. q1x2 x
2
7x 4
p1x2 x
2
5x 2
g1x22x
2
9x 7
f
1x23x
2
7x 6
q1x2 4x
2
9x 2
p1x2 2x
2
7x 3
g1x23x
2
12x 7
f
1x22x
2
8x 7
v1x2 4x
2
24x 15
u1x2 3x
2
6x 5
H1x2x
2
8x 7
h1x2x
2
2x 3
g1x2 x
2
6x 7
f
1x2 x
2
4x 5
Graph each function using the vertex formula and other
features of a quadratic graph. Label all important
features.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
State the equation of the function whose graph is shown.
33. 34.
x
y
55
5
5
x
y
5
5
5
5
q1x2
1
3
x
2
2x 4
p1x2
1
2
x
2
3x 5
g1x2 3x
2
12x 5
f
1x2 4x
2
12x 3
Y
2
2x
2
8x 3
Y
1
2x
2
10x 7
Y
2
0.2x
2
2x 8
Y
1
0.5x
2
3x 7
H1x2x
2
10x 19
h1x2x
2
4x 2
g1x2 x
2
8x 11
f
1x2 x
2
2x 6
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3–43 Section 3.3 Quadratic Functions and Applications 323
College Algebra G&M—
35. 36.
x
y
6
4
5
5
x
y
55
5
5
37. 38.
x
y
64
5
5
x
y
5
5
5
5
䊳
WORKING WITH FORMULAS
39. Vertex/intercept formula:
As an alternative to using the quadratic formula prior to completing the square, the x-intercepts can easily be
found using the vertex/intercept formula after completing the square, when the coordinates of the vertex are
known. (a) Beginning with the shifted form , substitute 0 for y and solve for x to derive the
formula. (b) Use the formula to find all zeroes, real or complex, of the following functions.
i. ii. iii.
iv. v. vi.
40. Surface area of a rectangular box with square ends:
The surface area of a rectangular box with square ends is given by the formula shown, where h is the height
and width of the square ends, and L is the length of the box. (a) If L is 3 ft and the box must have a surface area
of 32 ft
2
, find the dimensions of the square ends. (b) Solve for L, then find the length if the height is 1.5 ft and
surface area is 22.5 ft
2
.
S ⴝ 2h
2
ⴙ 4Lh
r1t20.51t 0.62
2
2s1t2 0.21t 0.72
2
0.8y 31x 22
2
6
y 21x 42
2
7y 1x 42
2
3y 1x 32
2
5
y a1x h2
2
k
x ⴝ h ⴞ
A
ⴚ
k
a
䊳
APPLICATIONS
41. Maximum profit:An automobile manufacturer can
produce up to 300 cars per day. The profit made
from the sale of these vehicles can be modeled by
the function
where P(x) is the profit in dollars and x is the
number of automobiles made and sold. Based on
this model:
a. Find the y-intercept and explain what it means
in this context.
b. Find the x-intercepts and explain what they
mean in this context.
c. How many cars should be made and sold to
maximize profit?
d. What is the maximum profit?
42. Maximum profit: The profit for a manufacturer of
collectible grandfather clocks is given by the
function , where
P(x) is the profit in dollars and x is the number of
clocks made and sold.
a. Find the y-intercept and explain what it means
in this context.
b. Find the x-intercepts and explain what they
mean in this context.
P1x21.6x
2
240x 375
P1x210x
2
3500x 66,000,
c. How many clocks should be made and sold to
maximize profit?
d. What is the maximum profit?
43. Optimal pricing strategy: The director of the
Ferguson Valley drama club must decide what to
charge for a ticket to the club’s performance of The
Music Man. If the price is set too low, the club will
lose money; and if the price is too high, people
won’t come. From past experience she estimates
that the profit P from sales (in hundreds) can be
approximated by where
x is the cost of a ticket and
a. Find the lowest cost of a ticket that would
allow the club to break even.
b. What is the highest cost that the club can
charge to break even?
c. If the theater were to close down before any
tickets are sold, how much money would the
club lose?
d. How much should the club charge to maximize
their profits? What is the maximum profit?
0 x 50.
P1x2x
2
46x 88,
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324 CHAPTER 3 Quadratic Functions and Operations on Functions 3–44
College Algebra G&M—
44. Maximum profit: A kitchen appliance
manufacturer can produce up to 200 appliances
per day. The profit made from the sale of these
machines can be modeled by the function
, where P(x) is the
profit in dollars, and x is the number of appliances
made and sold. Based on this model,
a. Find the y-intercept and explain what it means
in this context.
b. Find the x-intercepts and explain what they
mean in this context.
c. Determine the domain of the function and
explain its significance.
d. How many should be sold to maximize profit?
What is the maximum profit?
45. Cost of production: The cost of producing
a plastic toy is given by the function
where x is the number of
hundreds of toys. The revenue from toy sales is
given by Since
, the profit function must
be (verify). How many
toys sold will produce the maximum profit? What
is the maximum profit?
46. Cost of production: The cost to produce bottled
spring water is given by where x
is the number of thousands of bottles. The total
income (revenue) from the sale of these bottles is
given by the function
Since , the profit function
must be (verify). How
many bottles sold will produce the maximum profit?
What is the maximum profit?
The projectile function: applies
to any object projected upward with an initial velocity v,
from a height k but not to an object under propulsion
(such as a rocket). Consider this situation and answer
the questions that follow.
47. Model rocketry:A member of the local rocketry
club launches her latest rocket from a large field.
At the moment its fuel is exhausted, the rocket has
a velocity of 240 ft/sec and an altitude of 544 ft
(t is in seconds).
a. Write the function that models the height of
the rocket.
b. How high is the rocket at ? If it took off
from the ground, why is it this high at ?
c. How high is the rocket 5 sec after the fuel is
exhausted?
d. How high is the rocket 10 sec after the fuel is
exhausted?
t 0
t 0
h1t2ⴝⴚ16t
2
ⴙ vt ⴙ k
P1x2x
2
310x 7400
profit revenue cost
R1x2x
2
326x 7463.
C1x2 16x 63,
P1x2x
2
120x 400
profit revenue cost
R1x2x
2
122x 365.
C1x2 2x 35,
P1x20.5x
2
175x 3300
e. How could the rocket be at the same height at
and at ?
f. What is the maximum height attained by the
rocket?
g. How many seconds was the rocket airborne
after its fuel was exhausted?
48. Height of a projectile: A projectile is thrown
upward with an initial velocity of 176 ft/sec. After
t sec, its height h(t) above the ground is given by
the function
a. Find the projectile’s height above the ground
after 2 sec.
b. Sketch the graph modeling the projectile’s
height.
c. What is the projectile’s maximum height?
What is the value of t at this height?
d. How many seconds after it is thrown will the
projectile strike the ground?
49. Height of a projectile: In the movie The Court
Jester (1956; Danny Kaye, Basil Rathbone,
Angela Lansbury, and Glynis Johns), a catapult
is used to toss the nefarious adviser to the king
into a river. Suppose the path flown by the
king’s adviser is modeled by the function
, where h(d) is
the height of the adviser in feet at a distance of
d ft from the base of the catapult.
a. How high was the release point of this
catapult?
b. How far from the catapult did the adviser reach
a maximum altitude?
c. What was this maximum altitude attained by
the adviser?
d. How far from the catapult did the adviser
splash into the river?
50. Blanket toss competition: The fraternities at
Steele Head University are participating in a
blanket toss competition, an activity borrowed
from the whaling villages of the Inuit Eskimos. If
the person being tossed is traveling at 32 ft/sec as
he is projected into the air, and the Frat members
are holding the canvas blanket at a height of 5 ft,
a. Write the function that models the height at
time t of the person being tossed.
b. How high is the person when (i) ,
(ii) ?
c. From part (b) what do you know about when
the maximum height is reached?
t 1.5
t 0.5
h1d20.02d
2
1.64d 14.4
h1t216t
2
176t.
t 10t 5
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3–45 Section 3.3 Quadratic Functions and Applications 325
College Algebra G&M—
d. To the nearest tenth of a second, when is the
maximum height reached?
e. To the nearest one-half foot, what was the
maximum height?
f. To the nearest tenth of a second, how long was
this person airborne?
51. Motorcycle jumps: On December 31, 2007,
Australian freestyle motocross legend Robbie
Maddison set a world record by jumping his
motorcycle 322 ft, 7
1
⁄
2
in. During practice the day
before, the prevailing wind conditions actually
allowed him to jump farther. Suppose his height
on one such jump is given by the equation
, where h represents his
height (in feet) above ground level t sec after takeoff.
a. How high is the top of the takeoff ramp?
b. If he touched down on the landing ramp 15 ft
above ground level, how long was “Maddo” in
the air?
c. What would have been the daredevil’s
maximum height?
52. SuperPipe Finals: In the Winter X Games, one of
the most exciting events to watch is the SuperPipe
Final. The height of a professional snowboarder
during one particularly huge jump is given by the
function , where h
represents her height (in feet) above the pipe base t
sec after leaving the upper edge, or lip.
a. How high is the lip of the superpipe?
b. If the snowboarder lands her trick 11 ft above
the base of the pipe, how long was she in
the air?
c. What was the athlete’s maximum height above
the base of the pipe?
h 16t
2
35t 16
h 16t
2
52t 25
5 ft
32 ft/s
53. Fencing a backyard: Tina and
Imai have just purchased a
purebred German Shepherd, and
need to fence in their backyard
so the dog can run. What is the
maximum rectangular area they can enclose with
200 ft of fencing, if (a) they use fencing material
along all four sides? What are the dimensions of
the rectangle? (b) What is the maximum area if
they use the house as one of the sides? What are
the dimensions of this rectangle?
54. Building sheep pens: It’s time to drench the sheep
again, so Chance and Chelsea-Lou are fencing off
a large rectangular area to build some temporary
holding pens. To prep the males, females, and kids,
they are separated into three smaller and equal-size
pens partitioned within the large rectangle. If 384 ft
of fencing is available and the maximum area is
desired, what will be (a) the dimensions of the
larger, outer rectangle? (b) the dimensions of the
smaller holding pens?
55. Building windows: Window
World, Inc., is responsible for
designing new windows for
the expansion of the campus
chapel. The current design is
shown in the figure. The metal
trim used to secure the
perimeter of the frame is 126"
long. If the maximum window
area is desired (to let in the
most sunlight), what will be
(a) the dimensions of the
rectangular portion of each
window? (b) the total area of
each window?
56. Building windows: Before
construction on the chapel
expansion begins (see
Exercise 55), a new design is
submitted. The new design is
very different (see figure), but
must still have a perimeter of
126". If the maximum window
area is still desired, what will
be (a) the dimensions of the
rectangular portion of each
window? (b) the total area of
each window?
h 0.866 w
w
l
w
w
Exercise 55
l
w
Exercise 56
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326 CHAPTER 3 Quadratic Functions and Operations on Functions 3–46
College Algebra G&M—
57. Maximizing soft drink revenue: A convenience
store owner sells 20-oz soft drinks for $1.50 each,
and sells an average of 500 per week. Using a
market survey, she believes that for each $0.05
decrease in price, an additional 25 soft drinks
will be sold. (a) What price should be charged to
maximize revenue? What is the maximum revenue?
(b) What is the lowest price the owner could charge
and still bring in at least $700 per week in revenue?
58. Maximizing restaurant revenue: At Figaro’s
Pizzeria, large pizzas usually sell for $12.50 and at
this price an average of 320 large pizzas are sold
each weekend. Using a survey and years of
experience, the manager of Figaro’s believes that
for each $0.25 decrease in price, 16 additional
pizzas will be sold. (a) What price should be
charged to maximize revenue? What is the
maximum revenue? (b) What selling price(s) will
generate exactly $4800 in revenue?
䊳
EXTENDING THE CONCEPT
59. Use the general solutions from the quadratic formula
to show that the average value of the x-intercepts is
. Explain/Discuss why the result is valid even if
the roots are complex.
,
60. Write the equation of a quadratic function whose
x-intercepts are given by
61. Write the equation for the parabola given.
5⫺5
⫺5
5
Ve rt ex 冢q, r冣
冢
⫺
e, 0冣
冢r, 0冣
x
y
x ⫽ 2 ⫾ 3i.
x
2
⫽
⫺b ⫺ 2b
2
⫺ 4ac
2a
x
1
⫽
⫺b ⫹ 2b
2
⫺ 4ac
2a
⫺b
2a
62. Referring to Exercise 39, discuss the nature (real or
complex, rational or irrational) and number of
zeroes (0, 1, or 2) given by the vertex/intercept
formula if (a) a and k have like signs, (b) a and k
have unlike signs, (c) k is zero, (d) the ratio
is positive and a perfect square, and (e) the
ratio is positive and not a perfect square.⫺
k
a
⫺
k
a
䊳
MAINTAINING YOUR SKILLS
63. (1.4) Identify the slope and y-intercept for
Do not graph.
65. (2.2) Given , find the equation of
the function whose graph is that of f(x), shifted
right 2 units, reflected across the x-axis, and then
down 3 units.
f
1x2⫽ 2
3
x ⫹ 3
⫺4x ⫹ 3y ⫽ 9.
64. (R.5) Multiply:
66. (3.2) Given solve
using the x-intercepts and end-behavior of f.
f
1x2ⱕ 0f 1x2⫽ 3x
2
⫹ 7x ⫺ 6,
x
2
⫺ 4x ⫹ 4
x
2
⫹ 3x ⫺ 10
#
x
2
⫺ 25
x
2
⫺ 10x ⫹ 25
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3–47 Reinforcing Basic Concepts 327
College Algebra G&M—
MID-CHAPTER CHECK
1. Find the sum and product of the complex numbers
and . Comment on what you notice.
2. Determine the quotient:
3. Check by substitution: Is a solution to
?
4. Given f(x) is a quadratic function whose graph has a
vertex of and an x-intercept of (3, 0), find the
two solutions of the equation .
Solve the following quadratic equations using the
method of your choice.
5. 6.
7. Solve the inequality by using a graphing calculator
to locate the x-intercepts and observing the graph.
Round to the nearest hundredth as necessary.
8. Mark Twain’s 1867 book, The Celebrated Jumping
Frog of Calaveras County, has actually spawned an
annual “World Championship” frog jumping contest
in Calaveras County, California. As of 2009, the
f
1x25.1x
2
3.2x 1.9; f 1x2 0
2x
2
7x 431x 22
2
5
f
1x2 0
11, 42
x
2
2x 5 0
x 1 2i
2 3i
3 2i
2 3i2 3i
record for the longest triple jump (the total of three
consecutive jumps) is 21 ft in. set by Rosie the
Ribeter in 1986. In a recent competition, Count
Frogula felt he had beaten the record and was
awaiting the judge’s decision. Suppose the longest
of his three jumps was modeled by the function
, where h(x) represents the
height of the jump in inches, x ft from the starting
point. Use this function to find (a) the maximum
height of the jump and (b) the length of the jump.
(c) If the Count jumped this distance three times,
did he beat Rosie’s record?
9. Graph using the vertex formula
and other features of a quadratic graph. Label all
important features.
10. Find the equation of the
quadratic function whose
graph is shown.
g1x2
1
2
x
2
2x
h1x22.3x
2
16.5x
5
3
4
REINFORCING BASIC CONCEPTS
An Alternative Method for Checking Solutions to Quadratic Equations
To solve by factoring, students will often begin by looking for two numbers whose product is
(the constant term) and whose sum is (the linear coefficient). The two numbers are and 3 since
and In factored form, we have with solutions and When these
solutions are compared to the original coefficients, we can still see the sum/product relationship, but note that while
still gives the constant term, gives the linear coefficient with opposite sign. Although
more difficult to accomplish, this method can be applied to any factorable quadratic equation
if we divide through by a, giving For we divide both sides by 2 and obtain
then look for two numbers whose product is and whose sum is The factors are
and since and showing the solutions are and We again
note the product of the solutions is the constant and the sum of the solutions is the linear coefficient with
opposite sign: No one actually promotes this method for solving trinomials where but it does illustrate
an important and useful concept:
a 1,
1
2
b
a
.
3
2
c
a
,
x
2
1.x
1
3
2
3
2
1
1
2
,a
3
2
b112
3
2
1x 12
ax
3
2
b
1
2
.
3
2
x
2
1
2
x
3
2
0,
2x
2
x 3 0,x
2
b
a
x
c
a
0.
ax
2
bx c 0
5 132 215213215
x
2
3.x
1
51x 521x 32 05 3 2.
1521321552
15x
2
2x 15 0
x
52
y
4
6
cob19545_ch03_314-328.qxd 8/10/10 8:19 PM Page 327
328 CHAPTER 3 Quadratic Functions and Operations on Functions 3–48
College Algebra G&M—
and Although the computation looks impressive, the product can be computed as a binomial
times its conjugate, and the radical parts add to zero for the sum, each yielding the results as already stated.
This observation provides a useful technique for checking solutions to a quadratic equation, even those having
irrational or complex roots! Check the solutions shown in these exercises.
Exercise 1: Exercise 2:
Exercise 3:
x
2
5 213 i
x
1
5 213 i
x
2
10x 37 0
x
2
2 312
2
x
2
1
x
1
2 312
2
x
1
7
2
2x
2
4x 7 02x
2
5x 7 0
x
2
b
2a
2b
2
4ac
2a
.
3.4 Quadratic Models; More on Rates of Change
So far in our study, we’ve seen real data sets that were best modeled by linear functions
(Section 1.6) and power functions (Section 2.4). These models enabled us to make mean-
ingful decisions and comparisons, as well as reasonable projections for future occurrences.
In this section, we explore data relationships that are best modeled by quadratic functions,
and look at how the “rate of change” concept can be applied to these and other functions.
A. Quadratic Equation Models
After a set of data is collected and organized, any patterns or relationships that exist
may not be readily known or easily seen. The regression model chosen for the data will
depend on a number of factors, including the context of the data, any patterns noted in
the scatterplot, any foreknowledge of how the data might be related, along with a care-
ful assessment of the correlation coefficient. Most graphing calculators have the abil-
ity to find regression models for a large number of functions, and the more we are
familiar with each family of functions, the better we’ll be able to use the technology.
Regardless of the form of regression used, the steps for inputting the data, setting a
window size, viewing the scatterplot, and so on, are identical.
LEARNING OBJECTIVES
In Section 3.4 you will see how we can:
A. Develop quadratic
function models from
a set of data
B. Calculate the average
rate of change for
nonlinear functions using
points from a graph
C. Calculate average rates
of change using the
average rate of change
formula
If and are the two roots of
then and
Justification for this can be found by taking the product and sum of the general solutions x
1
b
2a
2b
2
4ac
2a
x
1
x
2
b
a
x
1
x
2
c
a
x
2
b
a
x
c
a
0,x
2
x
1
Exercise 4: Verify this sum/product check by computing
the sum and product of the general solutions
for x
1
and x
2
.
cob19545_ch03_314-328.qxd 8/10/10 8:19 PM Page 328
College Algebra G&M—
EXAMPLE 1
䊳
Growth of Online Sales of Pet Supplies
Since the year 2000, there has been a tremendous increase in the
online sales of pet food, pet medications, and other pet supplies.
That data shown in the table shows the amount spent (in billions
of dollars) for the years indicated.
Source: 2009 Statistical Abstract of the United States, Page 646, Table 1016.
a. Input the data into a graphing calculator, set an appropriate
window and view the scatterplot, then use the context and the
scatterplot to decide on an appropriate form of regression.
b. Determine the regression equation, and use it to
find the amount of online sales projected for the
year 2012.
c. If the predicted trends continue, in what
year will online sales of pet supplies surpass
$25 billion dollars?
Solution
䊳
a. Begin by entering the years in L1 and the dollar
amounts in L2. From the data given, a viewing
window of , , ,
and seems appropriate. The
scatterplot in Figure 3.53 shows a definite
increasing, non-linear pattern and we opt for
a quadratic regression.
b. Using (CALC) 5:QuadReg
places this option on the home screen.
Pressing at this point will give us the
quadratic coefficients, but we can also have
the calculator paste the equation itself
directly into Y
1
on the screen, simply
by appending Y
1
to the QuadReg command (Figure 3.54). The resulting
equation is (to three decimal places), and the
graph is shown in Figure 3.55. Using the home screen, we find that
(Figure 3.56), indicating that in 2012, about $19.2 billion is
projected to be spent online for pet supplies.
c. To find the year when spending will surpass $25 billion, we set and
use the Intersection-of-Graphs method (be sure to increase Ymax so this graph
can be seen: ). There result is shown in Figure 3.57 and indicates
that spending will surpass $25 billion late in the year 2013.
Ymax ⫽ 30
Y
2
⫽ 25
Y
1
1122⬇ 19.2
y ⬇ 0.118x
2
⫹ 0.136x ⫹ 0.537
Y=
ENTER
STAT
Ymax ⫽ 15
Ymin ⫽⫺2Xmax ⫽ 15Xmin ⫽ 0
Year Sales
(2001 S 1) (billions)
1 0.8
5 4.1
6 5.6
7 7.4
8 9.1
15
15
⫺2
0
Figure 3.53
3–49 Section 3.4 Quadratic Models; More on Rates of Change 329
15
15
⫺2
0
Figure 3.54, 3.55
Figure 3.56
15
30
⫺2
0
Figure 3.57
Now try Exercises 7 through 10
䊳
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