D. Applications of Composition and the Difference Quotient
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Solution
ᮣ
363
a. Since r16002 ϭ 4300, there are currently 4300 rodents in the county. The number
of rodents and small animals is a function of w: r1w2 ϭ 10,000 Ϫ 9.5w, and the
number of wolves w is a function of human population x: w1x2 ϭ 600 Ϫ 0.02x.
To find a function for r in terms of x, we use the composition 1r ؠw21x2 ϭ r3w1x24 .
r1w2 ϭ 10,000 Ϫ 9.5w
r 3w1x2 4 ϭ 10,000 Ϫ 9.53600 Ϫ 0.02x 4
ϭ 10,000 Ϫ 5700 ϩ 0.19x
r1x2 ϭ 4300 ϩ 0.19x
r depends on w
compose r with w (x )
distribute
simplify, r depends on x
b. Evaluating r(35,000) gives 10,950 rodents, showing a decline in the wolf
population will eventually cause the rodent population to flourish, with an adverse
effect on humans. In cases like these, a careful balance should be the goal.
Now try Exercises 63 and 64
ᮣ
You might be familiar with Galileo Galilei and his studies of gravity. According to
popular history, he demonstrated that unequal weights will fall equal distances in equal
time periods, by dropping cannonballs from the upper floors of the Leaning Tower of
Pisa. Neglecting air resistance, the distance an object falls is modeled by the function
d1t2 ϭ 16t2, where d(t) represents the distance fallen after t sec. Due to the effects of
gravity, the velocity of the object increases as it falls. In other words, the velocity or the
¢distance
average rate of change
is a nonconstant (increasing) quantity. We can analyze
¢time
this rate of change using the difference quotient.
EXAMPLE 11
ᮣ
Applying the Difference Quotient in Context
A construction worker drops a heavy wrench from atop a
girder of a new skyscraper. Use the function d1t2 ϭ 16t2 to
a. Compute the distance the wrench has fallen after
2 sec and after 7 sec.
b. Find a formula for the velocity of the wrench (average
rate of change in distance per unit time).
c. Use the formula to find the rate of change in the
intervals [2, 2.01] and [7, 7.01].
d. Graph the function and the secant lines representing
the average rate of change. Comment on what you
notice.
Solution
ᮣ
a. Substituting t ϭ 2 and t ϭ 7 in the given function
yields
d122 ϭ 16122 2
ϭ 16142
ϭ 64
d172 ϭ 16172 2
ϭ 161492
ϭ 784
evaluate d 1t 2 ϭ 16t 2
square input
multiply
After 2 sec, the wrench has fallen 64 ft; after 7 sec, the wrench has fallen 784 ft.
b. For d1t2 ϭ 16t2, d1t ϩ h2 ϭ 161t ϩ h2 2, which we compute separately.
d1t ϩ h2 ϭ 161t ϩ h2 2
ϭ 161t2 ϩ 2th ϩ h2 2
ϭ 16t2 ϩ 32th ϩ 16h2
substitute t ϩ h for t
square binomial
distribute 16
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Using this result in the difference quotient yields
d1t ϩ h2 Ϫ d1t2
h
ϭ
116t2 ϩ 32th ϩ 16h2 2 Ϫ 16t2
h
2
16t ϩ 32th ϩ 16h2 Ϫ 16t2
ϭ
h
32th ϩ 16h2
ϭ
h
h132t ϩ 16h2
ϭ
h
ϭ 32t ϩ 16h
substitute into the difference quotient
eliminate parentheses
combine like terms
factor out h and simplify
result
For any number of seconds t and h a small increment of time thereafter, the
32t ؉ 16h
¢distance
؍
velocity of the wrench is modeled by
.
¢time
1
c. For the interval 3t, t ϩ h 4 ϭ 3 2, 2.01 4, t ϭ 2 and h ϭ 0.01:
32122 ϩ 1610.012
¢distance
ϭ
¢time
1
ϭ 64 ϩ 0.16 ϭ 64.16
substitute 2 for t and 0.01 for h
Two seconds after being dropped, the velocity of the wrench is close to
64.16 ft/sec (44 mph). For the interval 3 t, t ϩ h 4 ϭ 3 7, 7.014 , t ϭ 7 and h ϭ 0.01:
32172 ϩ 1610.012
¢distance
ϭ
substitute 7 for t and 0.01 for h
¢time
1
ϭ 224 ϩ 0.16 ϭ 224.16
Seven seconds after being dropped, the velocity of the wrench is
approximately 224.16 ft/sec (about 153 mph).
d.
y
1000
Distance fallen (ft)
800
600
400
200
0
1
2
3
4
5
6
7
8
9
10
x
Time in seconds
D. You’ve just seen how
we can apply the composition
of functions and the difference
quotient in context
The velocity increases with time, as indicated by the steepness of each
secant line.
Now try Exercises 65 through 68
ᮣ
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365
3.6 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. When evaluating functions, if the input value is a
function itself, the process is called a
of
functions.
3. For functions f and g, the domain of 1 f ؠg21x2 is the
set of all x in the
of g, such that
is in the domain of f.
5. Discuss/Explain how and why using the the
difference quotient differs from using the average
rate of change formula.
ᮣ
2. The notation 1 f ؠg21x2 indicates that g(x) is the input
value for f (x), which is written
.
4. The average rate of change formula becomes the
quotient by substituting
for
for x1.
x2 and
6. Discuss/Explain how the domain of 1 f ؠg21x2
is determined, given f 1x2 ϭ 12x ϩ 7 and
2x
g1x2 ϭ
.
xϪ1
DEVELOPING YOUR SKILLS
7. Given f 1x2 ϭ x2 Ϫ 5x Ϫ 14, find f 1Ϫ22, f (7),
f (2a), and f 1a Ϫ 22 .
8. Given g1x2 ϭ x3 Ϫ 9x, find g1Ϫ32, g122, g(3t), and
g1t ϩ 12 .
For each pair of functions below, find (a) h(x) ϭ ( f ؠg)(x)
and (b) H(x) ϭ ( g ؠf )(x), and (c) determine the domain
of each result.
9. f 1x2 ϭ 1x ϩ 3 and g1x2 ϭ 2x Ϫ 5
10. f 1x2 ϭ x ϩ 3 and g1x2 ϭ 29 Ϫ x
11. f 1x2 ϭ 1x Ϫ 3 and g1x2 ϭ 3x ϩ 4
12. f 1x2 ϭ 1x ϩ 5 and g1x2 ϭ 4x Ϫ 1
21. f 1x2 ϭ 1x ϩ 82 2, g1x2 ϭ
22. f 1x2 ϭ
72
xϪ5
1
1
, g1x2 ϭ
2
2x Ϫ 1
x
23. f 1x2 ϭ 24 Ϫ 3x, g1x2 ϭ x2 Ϫ 9
24. f 1x2 ϭ
7
, g1x2 ϭ x2 Ϫ 11
xϩ2
For the functions f(x) and g(x) given, analyze the
domain of (a) ( f ؠg)(x) and (b) ( g ؠf )(x), then (c) find
the actual compositions and comment.
25. f 1x2 ϭ
2x
5
and g1x2 ϭ
x
xϩ3
26. f 1x2 ϭ
Ϫ3
x
and g1x2 ϭ
x
xϪ2
16. f 1x2 ϭ x2 Ϫ 4x ϩ 2 and g1x2 ϭ x Ϫ 2
27. f 1x2 ϭ
1
4
and g1x2 ϭ
x
xϪ5
18. f 1x2 ϭ Ϳ x Ϫ 2 Ϳ and g1x2 ϭ 3x Ϫ 5
28. f 1x2 ϭ
3
1
and g1x2 ϭ
x
xϪ2
13. f 1x2 ϭ x2 Ϫ 3x and g1x2 ϭ x ϩ 2
14. f 1x2 ϭ 2x2 Ϫ 1 and g1x2 ϭ 3x ϩ 2
15. f 1x2 ϭ x ϩ x Ϫ 4 and g1x2 ϭ x ϩ 3
2
17. f 1x2 ϭ Ϳ x Ϳ Ϫ 5 and g1x2 ϭ Ϫ3x ϩ 1
For the functions f and g given, h(x) ϭ ( f ؠg)(x). Use a
1
calculator to evaluate h(Ϫ3), h( 22), ha b, and h(5). If
2
an error message is received, explain why.
19. f 1x2 ϭ x ϩ 3x Ϫ 4, g1x2 ϭ x ϩ 1
2
20. f 1x2 ϭ Ϫx2 Ϫ 15x, g1x2 ϭ x Ϫ 2
29. For f 1x2 ϭ x2 Ϫ 8, g1x2 ϭ x ϩ 2, and
h1x2 ϭ 1 f ؠg21x2, find h(5) in two ways:
a. 1 f ؠg2152
b. f [g(5)]
30. For p1x2 ϭ x2 Ϫ 8, q1x2 ϭ x ϩ 2, and
H1x2 ϭ 1p ؠq21x2, find H1Ϫ22 in two ways:
a. 1p ؠq21Ϫ22
b. p 3q1Ϫ22 4
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31. For h1x2 ϭ 1 1x Ϫ 2 ϩ 12 3 Ϫ 5, find two
functions f and g such that 1f ؠg2 1x2 ϭ h1x2 .
38. Given f 1x2 ϭ 1x, g1x2 ϭ
1
, (a) state
x Ϫ 2x Ϫ 3
the domain of f and g, then (b) use a graphing
calculator to study the graph of h1x2 ϭ 1 f ؠg21x2 .
Finally, (c) algebraically determine the domain of
h, and reconcile it with the graph.
3 2
32. For H1x2 ϭ 2
x Ϫ 5 ϩ 2, find two functions
p and q such that 1p ؠq2 1x2 ϭ H1x2 .
33. Given f 1x2 ϭ 2x Ϫ 1, g1x2 ϭ x2 Ϫ 1, and
h1x2 ϭ x ϩ 4, find p1x2 ϭ f 3 g1 3 h1x2 4 2 4 and
q1x2 ϭ g3 f 1 3 h1x2 4 2 4 .
xϪ3
, find
2
(a) 1 f ؠf 2 1x2 , (b) 1g ؠg21x2 , (c) 1 f ؠg21x2 , and
(d) 1g ؠf 2 1x2 .
35. Reading a graph: Use
the given graph to find the
result of the operations
indicated.
Note f 1Ϫ42 ϭ 5,
g1Ϫ42 ϭ Ϫ1, and
so on.
a. 1 f ؠg21Ϫ42
b. 1 f ؠg2112
c. 1 f ؠg2142
e. 1 f ؠg21Ϫ22
g. 1 f ؠg2162
Exercise 35
y
f(x)
Note p1Ϫ12 ϭ Ϫ2,
q152 ϭ 6, and so on.
a. 1p ؠq21Ϫ42
b. 1p ؠq2112
c. 1p ؠq2142
e. 1p ؠq21Ϫ22
g. 1q ؠq21Ϫ12
6
g(x)
Ϫ4
8 x
Ϫ4
d. 1 f ؠg2102
f. 1g ؠf 2122
h. 1g ؠf 2142
40. g1x2 ϭ 4x ϩ 1
41. j1x2 ϭ x2 ϩ 3
42. p1x2 ϭ x2 Ϫ 2
43. q1x2 ϭ x2 ϩ 2x Ϫ 3
44. r1x2 ϭ x2 Ϫ 5x ϩ 2
45. f 1x2 ϭ
46. g1x2 ϭ
2
x
Ϫ3
x
Use the difference quotient to find: (a) a rate of change
formula for the functions given and (b)/(c) calculate the
rate of change in the intervals shown. Then (d) sketch
the graph of each function along with the secant lines
and comment on what you notice.
48. j1x2 ϭ x2 Ϫ 6x
[Ϫ3.0, Ϫ2.9], [0.50, 0.51]
[1.9, 2.0], [5.0, 5.01]
49. g1x2 ϭ x ϩ 1
[Ϫ2.1, Ϫ2], [0.40, 0.41]
3
y
6
39. f 1x2 ϭ 2x Ϫ 3
47. g1x2 ϭ x2 ϩ 2x
Exercise 36
36. Reading a graph: Use
the given graph to find the
result of the operations
indicated.
2
Compute and simplify the difference quotient
f 1x ؉ h2 ؊ f 1x2
for each function given.
h
34. Given f 1x2 ϭ 2x ϩ 3 and g1x2 ϭ
50. v1x2 ϭ 2x (Hint: Rationalize the numerator.)
p(x)
[1, 1.1], [4, 4.1]
Ϫ4
4
Ϫ4
8 x
q(x)
d. 1p ؠq2102
f. 1q ؠp2122
h. 1p ؠp2172
Ϫ3
, (a) state
xϩ2
the domain of f and g, then (b) use a graphing
calculator to study the graph of h1x2 ϭ 1 f ؠg21x2 .
Finally, (c) algebraically determine the domain of
h, and reconcile it with the graph.
37. Given f 1x2 ϭ 3 2x ϩ 1, g1x2 ϭ
ᮣ
3–86
CHAPTER 3 Quadratic Functions and Operations on Functions
Use the difference quotient to find a rate of change
formula for the functions given, then calculate the rate
of change for the intervals indicated. Comment on how
the rate of change in each interval corresponds to the
graph of the function.
1
52. f 1x2 ϭ x2 Ϫ 4x
x2
30.00, 0.014 , 33.00, 3.014
30.50, 0.514 , 31.50, 1.514
51. j1x2 ϭ
53. g1x2 ϭ x3 ϩ 1
54. r1x2 ϭ 1x
3Ϫ2.01, Ϫ2.004, 30.40, 0.414
31.00, 1.014, 34.00, 4.014
WORKING WITH FORMULAS
55. Transformations via composition: For
f 1x2 ϭ x2 ϩ 4x ϩ 3 and g1x2 ϭ x Ϫ 2, (a) show that
h1x2 ϭ 1 f ؠg21x2 ϭ x2 Ϫ 1, then (b) verify the graph
of h is the same as that of f, shifted 2 units to the right.
56. Compound annual growth: A1r2 ؍P11 ؉ r2 t
The amount of money A in a savings account t yr
after an initial investment of P dollars depends on
the interest rate r. If $1000 is invested for 5 yr, find
f (r) and g(r) such that A1r2 ϭ 1 f ؠg21r2.
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367
APPLICATIONS
57. International shoe sizes: Peering inside her
athletic shoes, Morgan notes the following shoe
sizes: US 8.5, UK 6, EUR 40. The function that
relates the U.S. sizes to the European (EUR) sizes
is g1x2 ϭ 2x ϩ 23, where x represents the U.S.
size and g(x) represents the EUR size. The function
that relates European sizes to sizes in the United
Kingdom (UK) is f 1x2 ϭ 0.5x Ϫ 14, where x
represents the EUR size and f (x) represents the UK
size. Find the function h(x) that relates the U.S.
measurement directly to the UK measurement by
finding h1x2 ϭ 1 f ؠg2 1x2. Find the UK size for a
shoe that has a U.S. size of 13.
58. Currency conversion: On a trip to Europe, Megan
had to convert American dollars to euros using the
function E1x2 ϭ 1.12x, where x represents the
number of dollars and E(x) is the equivalent
number of euros. Later, she converts her euros to
Japanese yen using the function Y1x2 ϭ 1061x,
where x represents the number of euros and Y(x)
represents the equivalent number of yen.
(a) Convert 100 U.S. dollars to euros. (b) Convert
the answer from part (a) into Japanese yen.
(c) Express yen as a function of dollars by finding
M1x2 ϭ 1Y ؠE21x2, then use M(x) to convert $100
directly to yen. Do parts (b) and (c) agree?
Source: 2005 World Almanac, p. 231
59. Currency conversion: While traveling in the Far
East, Timi must convert U.S. dollars to Thai baht
using the function T1x2 ϭ 41.6x, where x
represents the number of dollars and T(x) is the
equivalent number of baht. Later she needs to
convert her baht to Malaysian ringgit using the
function R1x2 ϭ 10.9x. (a) Convert $100 to baht.
(b) Convert the result from part (a) to ringgit.
(c) Express ringgit as a function of dollars using
M1x2 ϭ 1R ؠT21x2, then use M(x) to convert $100
to ringgit directly. Do parts (b) and (c) agree?
Source: 2005 World Almanac, p. 231
60. Spread of a fire: Due to a lightning strike, a forest
fire begins to burn and is spreading outward in a
shape that is roughly circular. The radius of the
circle is modeled by the function r1t2 ϭ 2t, where t
is the time in minutes and r is measured in meters.
(a) Write a function for the area burned by the fire
directly as a function of t by computing 1A ؠr2 1t2 .
(b) Find the area of the circular burn after 60 min.
61. Radius of a ripple: As Mark drops firecrackers
into a lake one 4th of July, each “pop” caused a
circular ripple that expanded with time. The
radius of the circle is a function of time t.
Suppose the function is r1t2 ϭ 3t, where t is in
seconds and r is in feet. (a) Find the radius of the
circle after 2 sec. (b) Find the area of the circle
after 2 sec. (c) Express the area as a function of
time by finding A1t2 ϭ 1A ؠr21t2 and use A(t) to
find the area of the circle after 2 sec. Do the
answers agree?
62. Expanding supernova: The surface area of a star
goes through an expansion phase prior to going
supernova. As the star begins expanding, the
radius becomes a function of time. Suppose this
function is r 1t2 ϭ 1.05t, where t is in days and r(t)
is in gigameters (Gm). (a) Find the radius of the
star two days after the expansion phase begins.
(b) Find the surface area after two days. (c)
Express the surface area as a function of time by
finding h1t2 ϭ 1S ؠr2 1t2, then use h(t) to compute
the surface area after two days directly. Do the
answers agree?
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63. Composition and dependent relationships: In
the wild, the balance of nature is often very fragile,
with any sudden changes causing dramatic and
unforeseen changes. With a huge increase in
population and tourism near an African wildlife
preserve, the number of lions is decreasing due to
loss of habitat and a disruption in normal daily
movements. This is causing a related increase in
the hyena population, as the lion is one of the
hyena’s only natural predators. If this increase
remains unchecked, animals lower in the food
chain will suffer. If the lion population L depends
on the increase in human population x according
to the formula L1x2 ϭ 500 Ϫ 0.015x, and the
hyena population depends on the lion population
as modeled by the formula H1L2 ϭ 650 Ϫ 0.5L,
(a) what is the current lion population 1x ϭ 02
and hyena population? (b) Use a composition to
find a function modeling how the hyena population
relates directly to the number of humans, and use
the function to estimate the number of hyenas in
the area if the human population grows by 16,000.
(c) If the administrators of the preserve consider a
population of 625 hyenas as “extremely detrimental,”
at what point should the human population be
capped?
64. Composition and dependent relationships: The
recent opening of a landfill in the area has caused
the raccoon population to flourish, with an
adverse effect on the number of purple martins.
Wildlife specialists believe the population of
martins p will decrease as the raccoon population
r grows. Further, since mosquitoes are the primary
diet of purple martins, the mosquito population
m is likewise affected. If the first relationship is
modeled by the function p1r2 ϭ 750 Ϫ 3.75r and
the second by m1p2 ϭ 50,000 Ϫ 45p, (a) what is
the current number of purple martins 1r ϭ 02 and
mosquitoes? (b) Use a composition to find a
function modeling how the raccoon population
relates directly to the number of mosquitoes,
and use the function to estimate the number of
mosquitoes in the area if the raccoon population
3–88
grows by 50. (c) If the health department considers
36,500 mosquitoes to be a “dangerous level,”
what increase in the raccoon population will
bring this about?
65. Distance to the horizon: The distance that a
person can see depends on how high they’re
standing above level ground. On a clear day,
the distance is approximated by the function
d1h2 ϭ 1.2 1h, where d(h) represents the viewing
distance (in miles) at height h (in feet). Use
the difference quotient to find the average rate
of change in the intervals (a) [9, 9.01] and
(b) [225, 225.01]. Then (c) graph the function
along with the lines representing the average
rates of change and comment on what you
notice.
66. Projector lenses: A special magnifying lens is
crafted and installed in an overhead projector.
When the projector is x ft from the screen, the
size P(x) of the projected image is x 2. Use the
difference quotient to find the average rate of
change for P1x2 ϭ x 2 in the intervals (a) [1, 1.01]
and (b) [4, 4.01]. Then (c) graph the function
along with the lines representing the average
rates of change and comment on what you
notice.
67. Fortune and fame: Over the years there have
been a large number of what we know as “one
hit wonders,” persons or groups that published a
memorable or timeless song, but who were
unable to repeat the feat. In some cases, their
fame might be modeled by a quadratic function
as their popularity rose to a maximum, then
faded with time. Suppose the song She’s on Her
Way by Helyn Wheels rode to the top of the
charts in January of 1988, with demand for the
song modeled by d1t2 ϭ Ϫ2t2 ϩ 27t. Here, d(t)
represents the demand in 1000s for month
t 1t ϭ 1 S Jan2 . (a) How many times faster was
the demand growing in March (shortly after
the release) than in June? Use the difference
quotient and the intervals [3, 3.01] for March
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and [6, 6.01] for June. (b) Determine the month
that demand reached its peak using a graphing
calculator. (c) Was the demand increasing or
decreasing in the month of August? At what
rate?
68. Velocity and fuel economy: It has long been
known that cars and trucks are more fuel efficient
at certain speeds, which is why President Richard
Nixon lowered the speed limit on all federal
highways to 55 mph during the oil embargo of
1974. For heavier and less fuel-efficient vehicles,
the miles per gallon for certain speeds can be
modeled by the function m1s2 ϭ Ϫ0.01s2 ϩ s,
where m(s) represents the mileage (in miles per
gallon) at speed s 10 6 s Յ 802 . (a) Use the
difference quotient to find how many times
ᮣ
faster fuel efficiency is growing near s ϭ 30 mph
than near s ϭ 45 mph. Use the intervals [30, 30.1]
and [45, 45.1]. (b) Use a graphing calculator
to determine the speed(s) that maximizes fuel
efficiency for this vehicle. How many miles
per gallon are achieved? (c) Is fuel efficiency
increasing or decreasing at 70 mph? At what
rate?
EXTENDING THE CONCEPT
3
69. Given f 1x2 ϭ x3 ϩ 2 and g1x2 ϭ 2
x Ϫ 2, graph
each function on the same axes by plotting the
points that correspond to integer inputs for
x ʦ 3 Ϫ3, 3 4. Do you notice anything? Next, find
h1x2 ϭ 1 f ؠg21x2 and H1x2 ϭ 1g ؠf 2 1x2. What
happened? Look closely at the functions f and g to
see how they are related. Can you come up with two
additional functions where the same thing occurs?
70. Given f 1x2 ϭ
1
, g1x2 ϭ 1x ϩ 1, and
x Ϫ4
h1x2 ϭ 1 f ؠg21x2 , (a) find the new function rule for
h and (b) determine the implied domain of h. Does
this implied domain include x ϭ 2, x ϭ Ϫ2, and
x ϭ Ϫ3 as valid inputs? (c) Determine the actual
domain for h1x2 ϭ 1 f ؠg21x2 and discuss the result.
ᮣ
369
2
71. Consider the functions f 1x2 ϭ
k
k
and g1x2 ϭ 2 .
x
x
Both graphs appear similar in Quadrant I and both
may “fit” a scatterplot fairly well, but there is a big
difference between them — they decrease as x gets
larger, but they decrease at very different rates.
(a) Assume k ϭ 1 and use the ideas from this
section to compute the rates of change for f and g
for the interval from x ϭ 0.5 to x ϭ 0.51. Were you
surprised? (b) In the interval x ϭ 0.8 to x ϭ 0.81,
will the rate of decrease for each function be
greater or less than in the interval x ϭ 0.5 to
x ϭ 0.51? Why?
MAINTAINING YOUR SKILLS
72. (3.1) Find the sum and product of the complex
numbers 2 ϩ 3i and 2 Ϫ 3i.
74. (3.2) Use the quadratic formula to solve
2x2 Ϫ 3x ϩ 4 ϭ 0.
73. (2.2) Draw a sketch of the functions from memory.
3
(a) f 1x2 ϭ 1x,
(b) g1x2 ϭ 2x, and
(c) h1x2 ϭ ͿxͿ
75. (1.4) Find an equation of the line perpendicular to
Ϫ2x ϩ 3y ϭ 9, that also goes through the origin.
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CHAPTER 3 Quadratic Functions and Operations on Functions
MAKING CONNECTIONS
Making Connections: Graphically, Symbolically, Numerically, and Verbally
Eight graphs (a) through (h) are given. Match the characteristics shown in 1 through 16 to one of the eight graphs.
y
(a)
Ϫ5
y
(b)
5
Ϫ5
5 x
y
Ϫ5
Ϫ5
5 x
5 x
Ϫ5
1. ____ 1x ϩ 12 2 ϩ 1y Ϫ 22 2 ϭ 4
2
2. ____ y ϭ Ϫ x ϩ 1
5
y
(h)
5
Ϫ5
5 x
5
Ϫ5
Ϫ5
9. ____ center 11, Ϫ22, radius ϭ 3
10. ____ vertex (2, 5), y-intercept (0, 1)
11. ____ y ϭ
5. ____ m 6 0, b 7 0
13. ____ axis of symmetry x ϭ 2,
opens downward
6. ____ m 7 0, b 6 0
14. ____ 4x Ϫ 3y ϭ 3
7. ____ y ϭ Ϫx ϩ 4
2
8. ____ y ϭ 31x Ϫ 22 2 Ϫ 5
5 x
Ϫ5
3. ____ f 1x2T for x ʦ 1Ϫq, 22 ,
f 1x2c for x ʦ 12, q2
4. ____ f 1x2c for x ʦ 1Ϫq, 22 ,
f 1x2T for x ʦ 12, q2
5 x
Ϫ5
y
(g)
5
Ϫ5
5 x
5
Ϫ5
y
(f)
5
Ϫ5
Ϫ5
5 x
y
(d)
5
Ϫ5
Ϫ5
(e)
y
(c)
5
1
1x Ϫ 12 2 Ϫ 3
3
12. ____ 1x Ϫ 12 2 ϩ 1y ϩ 22 2 ϭ 9
15. ____ f 1x2 6 0 for x ʦ 1Ϫq,Ϫ22 ´ 12, q 2 ,
f 1x2 Ն 0 for x ʦ 3Ϫ2, 2 4
16. ____ f 1Ϫ22 ϭ 0, f 112 ϭ Ϫ3
SUMMARY AND CONCEPT REVIEW
SECTION 3.1
Complex Numbers
KEY CONCEPTS
• The italicized i represents the number whose square is Ϫ1. This means i 2 ϭ Ϫ1 and i ϭ 1Ϫ1.
• Larger powers of i can be simplified using i 4 ϭ 1.
• For k 7 0, 1Ϫk ϭ i1k and we say the expression has been written in terms of i.
• The standard form of a complex number is a ϩ bi, where a is the real number part and bi is the imaginary part.
• To add or subtract complex numbers, combine the like terms.
• For any complex number a ϩ bi, its complex conjugate is a Ϫ bi.
• The product of a complex number and its conjugate is a real number.
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Summary and Concept Review
371
• The commutative, associative, and distributive properties also apply to complex numbers and are used to perform
basic operations.
• To multiply complex numbers, use the F-O-I-L method and simplify.
• To find a quotient of complex numbers, multiply the numerator and denominator by the conjugate of the denominator.
EXERCISES
Simplify each expression and write the result in standard form.
1. 1Ϫ72
2. 6 1Ϫ48
4. 13 1Ϫ6
5. i57
3.
Perform the operation indicated and write the result in standard form.
5i
6. 15 ϩ 2i2 2
7.
1 Ϫ 2i
9. 12 ϩ 3i212 Ϫ 3i2
10. 4i1Ϫ3 ϩ 5i2
Ϫ10 ϩ 1Ϫ50
5
8. 1Ϫ3 ϩ 5i2 Ϫ 12 Ϫ 2i2
Use substitution to show the given complex number and its conjugate are solutions to the equation shown.
11. x2 Ϫ 9 ϭ Ϫ34; x ϭ 5i
SECTION 3.2
12. x2 Ϫ 4x ϩ 9 ϭ 0; x ϭ 2 ϩ i 25
Solving Quadratic Equations and Inequalities
KEY CONCEPTS
• The standard form of a quadratic equation is ax2 ϩ bx ϩ c ϭ 0, where a, b, and c are real numbers and a 0. In
words, we say the equation is written in decreasing order of degree and set equal to zero.
• A quadratic function is one that can be written as f 1x2 ϭ ax2 ϩ bx ϩ c, where a, b, and c are real numbers and a 0.
• The following four statements are equivalent: (1) x ϭ r is a solution of f 1x2 ϭ 0, (2) r is a zero of f(x), (3) (r, 0) is
an x-intercept of y ϭ f 1x2 , and (4) 1x Ϫ r2 is a factor of f (x).
• The square root property of equality states that if X2 ϭ k, where k Ն 0, then X ϭ 1k or X ϭ Ϫ 1k.
• Quadratic equations can also be solved by completing the square, or using the quadratic formula.
• If the discriminant b2 Ϫ 4ac ϭ 0, the equation has one real (repeated) root. If b2 Ϫ 4ac 7 0, the equation has
two real roots; and if b2 Ϫ 4ac 6 0, the equation has two nonreal roots.
• Quadratic inequalities can be solved using the zeroes of the function and either an understanding of quadratic
graphs or mid-interval test values.
EXERCISES
13. Solve by factoring.
a. x2 Ϫ 3x Ϫ 10 ϭ 0
b. 2x2 Ϫ 50 ϭ 0
c. 3x2 Ϫ 15 ϭ 4x
d. x3 Ϫ 3x2 ϭ 4x Ϫ 12
14. Solve using the square root property of equality.
a. x2 Ϫ 9 ϭ 0
b. 21x Ϫ 22 2 ϩ 1 ϭ 11
c. 3x2 ϩ 15 ϭ 0
d. Ϫ2x2 ϩ 4 ϭ Ϫ46
15. Solve by completing the square. Give real number solutions in exact and approximate form.
a. x2 ϩ 2x ϭ 15
b. x2 ϩ 6x ϭ 16
c. Ϫ4x ϩ 2x2 ϭ 3
d. 3x2 Ϫ 7x ϭ Ϫ2
16. Solve using the quadratic formula. Give solutions in both exact and approximate form.
a. x2 Ϫ 4x ϭ Ϫ9
b. 4x2 ϩ 7 ϭ 12x
c. 2x2 Ϫ 6x ϩ 5 ϭ 0
17. Solve by locating the x-intercepts and noting the end-behavior of the graph.
a. x2 Ϫ x Ϫ 6 7 0
b. Ϫx2 ϩ 1 Ն 0
c. x2 Ϫ 2x ϩ 2 7 0
18. Solve using the interval test method.
a. x2 ϩ 3x Յ 4
b. x2 7 20 Ϫ x
c. x2 ϩ 4x ϩ 4 Յ 0
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3–92
Solve the following quadratic applications. For 19 and 20, recall the height of a projectile is modeled by
h ϭ Ϫ16t 2 ϩ v0 t ϩ k.
19. A projectile is fired upward from ground level with an initial velocity of 96 ft/sec. (a) To the nearest tenth of a
second, how long until the object first reaches a height of 100 ft? (b) How long until the object is again at 100 ft?
(c) How many seconds until it returns to the ground?
20. A person throws a rock upward from the top of an 80-ft cliff with an initial velocity of 64 ft/sec. (a) To the nearest
tenth of a second, how long until the object is 120 ft high? (b) How long until the object is again at 120 ft?
(c) How many seconds until the object hits the ground at the base of the cliff?
SECTION 3.3
Quadratic Functions and Applications
KEY CONCEPTS
• The graph of a quadratic function is a parabola. Parabolas have three distinctive features: (1) like end-behavior on
the left and right, (2) an axis of symmetry, (3) a highest or lowest point called the vertex.
• By completing the square, f 1x2 ϭ ax2 ϩ bx ϩ c can be written as the transformation f 1x2 ϭ a1x ϩ h2 2 Ϯ k, and
graphed using transformations of y ϭ x2.
• For a quadratic function in the standard form y ϭ ax2 ϩ bx ϩ c,
• End-behavior: graph opens upward if a 7 0, opens downward if a 6 0
• Zeroes/x-intercepts (if they exist): substitute 0 for y and solve for x
• y-intercept: substitute 0 for x S 10, c2
Ϫb
Ϫb
,k ϭ fa
b
• Vertex: (h, k), where h ϭ
2a
2a
• Maximum value: If the parabola opens downward, y ϭ k is the maximum value of f.
• Minimum value: If the parabola opens upward, y ϭ k is the minimum value of f.
• Line of symmetry: x ϭ h is the line of symmetry. If 1h ϩ c, y2 is on the graph, then 1h Ϫ c, y2 is also on the graph.
EXERCISES
Graph p(x) and f (x) by completing the square and using transformations of the parent function. Graph g(x) and h(x)
using the vertex formula and y-intercept. Find the x-intercepts (if they exist) for all functions.
21. p1x2 ϭ x2 Ϫ 6x
22. f 1x2 ϭ x2 ϩ 8x ϩ 15
23. g1x2 ϭ Ϫx2 ϩ 4x Ϫ 5
24. h1x2 ϭ 4x2 Ϫ 12x ϩ 3
25. Height of a superball: A teenager tries to see how high she can bounce her superball by throwing it downward on
her driveway. The height of the ball (in feet) at time t (in seconds) is given by h1t2 ϭ Ϫ16t2 ϩ 96t. (a) How high
is the ball at t ϭ 0? (b) How high is the ball after 1.5 sec? (c) How long until the ball is 135 ft high? (d) What is
the maximum height attained by the ball? At what time t did this occur?
26. Theater Revenue: The manager of a large, 14-screen movie theater finds that if he charges $2.50 per person for
the matinee, the average daily attendance is 4000 people. With every increase of 25 cents the attendance drops an
average of 200 people. (a) What admission price will bring in a revenue of $11,250? (b) How many people will
purchase tickets at this price?
SECTION 3.4
Quadratic Models; More on Rates of Change
KEY CONCEPTS
• Regardless of the form of regression chosen, obtaining a regression equation uses these five steps: (1) clear out old
data, (2) enter new data, (3) set an appropriate window and display the data, (4) calculate the regression equation,
and (5) display the data and equation, and once satisfied the model is appropriate, apply the result.
• The choice of a nonlinear regression model often depends on many factors, particularly the context of the data,
any patterns formed by the scatterplot, some foreknowledge on how the data might be related, and/or a careful
assessment of the correlation coefficient.
• Applications of quadratic regression are generally applied when a set of data indicates a gradual decrease to some
minimum value, with a matching increase afterward, or a gradual increase to some maximum, with a matching
decrease afterward.
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Summary and Concept Review
373
• For nonlinear functions, the average rate of change gives an average value for how changes in the independent
variable cause a change in the dependent variable within a specified interval.
• The average rate of change is given by the slope of a secant line through two points (x1, y1) and (x2, y2) on the
f 1x2 2 Ϫ f 1x1 2
¢y
graph, and is computed as:
ϭ
, x2 x1.
x2 Ϫ x1
¢x
EXERCISES
Year
Amount
27. While the Internet has been with us for over 20 years, its use continues to grow at a
(2000 S 0)
(billions)
rapid pace. The data in the table gives the amount of money (in billions of dollars)
1
31
consumers spent online for retail items in selected years (amounts for 2010 through
5
84
2012 are projections). Use the data and a graphing calculator to (a) draw a scatterplot
6
108
and decide on an appropriate form of regression, then (b) find the regression equation
and use it to estimate the amount spent by consumers in 2003, (c) the projected
7
128
amount that will be spent in 2014 if this rate of growth continues, and (d) the year
10
267
that $591 billion is the projected amount of retail spending over the Internet.
11
301
28. The drag force on a compact car driving along the highway on a windless day,
12
335
depends on a constant 2k and the velocity of the car, where k is determined using the
density of the air, the cross-sectional area of the car, and the drag coefficient of the
Velocity
vehicle. The data shown in the table gives the magnitude of the drag force Fd, at given
Fd
(mph)
velocity v. Use the data and a graphing calculator to (a) draw a scatterplot and decide
10
32
on an appropriate form of regression, then (b) find a regression equation and use it to
estimate the magnitude of the drag force for this car at 60 mph. Finally, (c) estimate the
30
306
speed of the car if the drag force has a magnitude of 2329 units.
50
860
29. The graph and accompanying table show the number N of active Starbucks outlets for
70
1694
selected years t from 1990 to 2008. Use the graph and table to (a) find the average rate of
change for the years 1994 to 1996 (the interval [4, 6]). (b) Verify that the rate of growth between the years 2000
and 2002 (the interval [10, 12]) was about 4 times greater than from 1994 to 1996. (c) Show that the average rate
of change for the years 2002 to 2004 was very close to the rate of change for the years 2006 to 2008.
200
Outlets (N)
160
120
Year t
(1990 → 0)
Outlets N
(100s)
Year t
(1990 → 0)
Outlets N
(100s)
0
0.84
10
35.01
2
1.65
12
58.86
80
4
4.25
14
85.69
6
10.15
16
124.40
8
18.86
18
150.79
40
0
2
4
6
8
10
12
14
16
18
20
Year (t)
30. According to Torricelli’s law for tank draining, the volume (in ft3) of a full 5 ft ϫ 2 ft ϫ 2 ft bathtub t sec after
the plug is pulled can be modeled by the function V1t2 ϭ 1Ϫ0.2t ϩ 1202 2. (a) What is the volume of the bathtub
at t ϭ 0 sec? (b) What is the volume of the bathtub at t ϭ 1 sec? (c) What is the average rate of change from t ϭ 0
to t ϭ 1? (d) What is the average rate of change from t ϭ 20 to t ϭ 21? (e) When is the bathtub empty?
SECTION 3.5
The Algebra of Functions
KEY CONCEPTS
• The notation used to represent the basic operations on two functions is
• 1 f ϩ g21x2 ϭ f 1x2 ϩ g1x2
• 1 f Ϫ g2 1x2 ϭ f 1x2 Ϫ g1x2
f 1x2
f
; g1x2 0
• 1 f # g21x2 ϭ f 1x2 # g1x2
• a g b1x2 ϭ
g1x2
• The result of these operations is a new function h(x). The domain of h is the intersection of domains for f and g,
f
excluding values that make g1x2 ϭ 0 for h1x2 ϭ a b1x2 .
g