E. Applications Involving Absolute Value
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Section 2.3 Absolute Value Functions, Equations, and Inequalities
EXAMPLE 8
ᮣ
225
Solving Applications Involving Absolute Value Inequalities
For new cars, the number of miles per gallon (mpg) a car will get is heavily
dependent on whether it is used mainly for short trips and city driving, or primarily
on the highway for longer trips. For a certain car, the number of miles per gallon
that a driver can expect varies by no more than 6.5 mpg above or below its field
tested average of 28.4 mpg. What range of mileage values can a driver expect
for this car?
Solution
ᮣ
Field tested average: 28.4 mpg
mileage varies by no more than 6.5 mpg
Ϫ6.5
gather information
highlight key phrases
ϩ6.5
28.4
make the problem visual
Let m represent the miles per gallon a driver can expect.
Then the difference between m and 28.4 can be no more
than 6.5, or Ϳm Ϫ 28.4Ϳ Յ 6.5.
Ϳm Ϫ 28.4Ϳ Յ 6.5
Ϫ6.5 Յ m Ϫ 28.4 Յ 6.5
21.9 Յ m Յ 34.9
assign a variable
write an equation model
equation model
apply Property I
add 28.4 to all three parts
The mileage that a driver can expect ranges from a low of 21.9 mpg
to a high of 34.9 mpg.
E. You’ve just seen how
we can solve applications
involving absolute value
Now try Exercises 61 through 70
ᮣ
2.3 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. When multiplying or dividing by a negative
quantity, we
the inequality symbol to
maintain a true statement.
2. To write an absolute value equation or inequality in
simplified form, we
the absolute value
expression on one side.
3. The absolute value equation Ϳ2x ϩ 3Ϳ ϭ 7 is true
when 2x ϩ 3 ϭ
or when 2x ϩ 3 ϭ
.
4. The absolute value inequality Ϳ3x Ϫ 6Ϳ 6 12 is
true when 3x Ϫ 6 7
and 3x Ϫ 6 6
Describe the solution set for each inequality (assume k > 0). Justify your answer.
5. Ϳax ϩ bͿ 6 Ϫk
6. Ϳax ϩ bͿ 7 Ϫk
.
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CHAPTER 2 More on Functions
DEVELOPING YOUR SKILLS
Solve each absolute value equation. Write the solution in
set notation. For Exercises 7 to 18, verify solutions by
substituting into the original equation. For Exercises
19–26 verify solutions using a calculator.
7. 2Ϳm Ϫ 1Ϳ Ϫ 7 ϭ 3
8. 3Ϳn Ϫ 5Ϳ Ϫ 14 ϭ Ϫ2
9. Ϫ3Ϳx ϩ 5Ϳ ϩ 6 ϭ Ϫ15
10. Ϫ2Ϳy ϩ 3Ϳ Ϫ 4 ϭ Ϫ14
11. 2Ϳ4v ϩ 5Ϳ Ϫ 6.5 ϭ 10.3
12. 7Ϳ2w ϩ 5Ϳ ϩ 6.3 ϭ 11.2
13. ϪͿ7p Ϫ 3Ϳ ϩ 6 ϭ Ϫ5
14. ϪͿ3q ϩ 4Ϳ ϩ 3 ϭ Ϫ5
15. Ϫ2ͿbͿ Ϫ 3 ϭ Ϫ4
16. Ϫ3ͿcͿ Ϫ 5 ϭ Ϫ6
17. Ϫ2Ϳ3xͿ Ϫ 17 ϭ Ϫ5
18. Ϫ5Ϳ2yͿ Ϫ 14 ϭ 6
19. Ϫ3 `
w
ϩ 4 ` Ϫ 1 ϭ Ϫ4
2
20. Ϫ2 ` 3 Ϫ
v
` ϩ 1 ϭ Ϫ5
3
21. 8.7Ϳp Ϫ 7.5Ϳ Ϫ 26.6 ϭ 8.2
35.
Ϳ5v ϩ 1Ϳ
ϩ8 6 9
4
36.
Ϳ3w Ϫ 2Ϳ
ϩ6 6 8
2
37. `
1
7
4x ϩ 5
Ϫ ` Յ
3
2
6
38. `
2y Ϫ 3
3
15
Ϫ ` Յ
4
8
16
39. Ϳn ϩ 3Ϳ 7 7
40. Ϳm Ϫ 1Ϳ 7 5
41. Ϫ2ͿwͿ Ϫ 5 Յ Ϫ11
42. Ϫ5ͿvͿ Ϫ 3 Յ Ϫ23
43.
44.
ͿqͿ
2
ͿpͿ
5
Ϫ
5
1
Ն
6
3
ϩ
3
9
Ն
2
4
45. 3Ϳ5 Ϫ 7dͿ ϩ 9 Ն 15
46. 5Ϳ2c ϩ 7Ϳ ϩ 1 Ն 11
47. 2 6 ` Ϫ3m ϩ
1
4
` Ϫ
5
5
3
5
Ϫ 2n ` Ϫ
4
4
22. 5.3Ϳq ϩ 9.2Ϳ ϩ 6.7 ϭ 43.8
48. 4 Յ `
23. 8.7ͿϪ2.5xͿ Ϫ 26.6 ϭ 8.2
49. 4Ϳ5 Ϫ 2hͿ Ϫ 9 7 11
24. 5.3Ϳ1.25nͿ ϩ 6.7 ϭ 43.8
50. 3Ϳ7 ϩ 2kͿ Ϫ 11 7 10
25. Ϳx Ϫ 2Ϳ ϭ Ϳ3x ϩ 4Ϳ
51. 3.9Ϳ4q Ϫ 5Ϳ ϩ 8.7 Յ Ϫ22.5
26. Ϳ2x Ϫ 1Ϳ ϭ Ϳx ϩ 3Ϳ
52. 0.9Ϳ2p ϩ 7Ϳ ϩ 16.11 Յ 10.89
Solve each absolute value inequality. Write solutions in
interval notation. Check solutions by back substitution,
or using a calculator.
27. 3Ϳp ϩ 4Ϳ ϩ 5 6 8
28. 5Ϳq Ϫ 2Ϳ Ϫ 7 Յ 8
29. Ϫ3 ͿmͿ Ϫ 2 7 4
30. Ϫ2ͿnͿ ϩ 3 7 7
31. Ϳ3b Ϫ 11Ϳ ϩ 6 Յ 9
32. Ϳ2c ϩ 3Ϳ Ϫ 5 6 1
33. Ϳ4 Ϫ 3zͿ ϩ 12 6 7
34. Ϳ2 Ϫ 3uͿ ϩ 5 Յ 4
53. Ϳ4z Ϫ 9Ϳ ϩ 6 Ն 4
54. Ϳ5u Ϫ 3Ϳ ϩ 8 7 6
Use the intersect command on a graphing calculator
and the given functions to solve (a) f 1x2 ϭ g1x2 ,
(b) f 1x2 Ն g1x2 , and (c) f 1x2 6 g1x2 .
55. f 1x2 ϭ Ϳx Ϫ 3Ϳ ϩ 2, g1x2 ϭ 12x ϩ 2
56. f 1x2 ϭ ϪͿx ϩ 2Ϳ Ϫ 1, g1x2 ϭ Ϫ32x Ϫ 9
57. f 1x2 ϭ 0.5Ϳx ϩ 3Ϳ ϩ 1, g1x2 ϭ Ϫ2Ϳx ϩ 1Ϳ ϩ 5
58. f 1x2 ϭ 2Ϳx Ϫ 3Ϳ ϩ 2, g1x2 ϭ Ϳx Ϫ 4Ϳ ϩ 6
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Section 2.3 Absolute Value Functions, Equations, and Inequalities
WORKING WITH FORMULAS
59. Spring Oscillation: Ϳd Ϫ xͿ Յ L
A weight attached to a spring hangs at rest a distance
of x in. off the ground. If the weight is pulled down
(stretched) a distance of L inches and released, the
weight begins to bounce and its distance d off the
ground must satisfy the indicated formula. (a) If x
equals 4 ft and the spring is stretched 3 in. and
released, solve the inequality to find what distances
from the ground the weight will oscillate between.
(b) Solve for x in terms of L and d.
ᮣ
227
60. A “Fair” Coin: `
h Ϫ 50
` 6 1.645
5
If we flipped a coin 100 times, we expect “heads” to
come up about 50 times if the coin is “fair.” In a study
of probability, it can be shown that the number of
heads h that appears in such an experiment should
satisfy the given inequality to be considered “fair.”
(a) Solve this inequality for h. (b) If you flipped a
coin 100 times and obtained 40 heads, is the coin
“fair”?
APPLICATIONS
Solve each application of absolute value.
61. Altitude of jet stream: To take advantage of the jet
stream, an airplane must fly at a height h (in feet)
that satisfies the inequality Ϳh Ϫ 35,050Ϳ Յ 2550.
Solve the inequality and determine if an altitude of
34,000 ft will place the plane in the jet stream.
62. Quality control tests: In order to satisfy quality
control, the marble columns a company produces
must earn a stress test score S that satisfies the
inequality ͿS Ϫ 17,750Ϳ Յ 275. Solve the inequality
and determine if a score of 17,500 is in the passing
range.
63. Submarine depth: The sonar operator on a
submarine detects an old World War II submarine
net and must decide to detour over or under the
net. The computer gives him a depth model
Ϳd Ϫ 394Ϳ Ϫ 20 7 164, where d is the depth in feet
that represents safe passage. At what depth should the
submarine travel to go under or over the net? Answer
using simple inequalities.
64. Optimal fishing depth: When deep-sea fishing,
the optimal depths d (in feet) for catching a
certain type of fish satisfy the inequality
28Ϳd Ϫ 350Ϳ Ϫ 1400 6 0. Find the range of depths
that offer the best fishing. Answer using simple
inequalities.
For Exercises 65 through 68, (a) develop a model that
uses an absolute value inequality, and (b) solve.
65. Stock value: My stock in MMM Corporation
fluctuated a great deal in 2009, but never by more
than $3.35 from its current value. If the stock is
worth $37.58 today, what was its range in 2009?
66. Traffic studies: On a
given day, the volume
of traffic at a busy
intersection averages
726 cars per hour
(cph). During rush hour
the volume is much
higher, during “off
hours” much lower.
Find the range of this
volume if it never
varies by more than
235 cph from the
average.
67. Physical training for recruits: For all recruits in the
3rd Armored Battalion, the average number of sit-ups
is 125. For an individual recruit, the amount varies
by no more than 23 sit-ups from the battalion
average. Find the range of sit-ups for this battalion.
68. Computer consultant salaries: The national
average salary for a computer consultant is
$53,336. For a large computer firm, the salaries
offered to their employees vary by no more than
$11,994 from this national average. Find the range
of salaries offered by this company.
69. Tolerances for sport balls: According to the official
rules for golf, baseball, pool, and bowling, (a) golf
balls must be within 0.03 mm of d ϭ 42.7 mm,
(b) baseballs must be within 1.01 mm of
d ϭ 73.78 mm, (c) billiard balls must be within
0.127 mm of d ϭ 57.150 mm, and (d) bowling balls
must be within 12.05 mm of d ϭ 2171.05 mm. Write
each statement using an absolute value inequality,
then (e) determine which sport gives the least
width of interval
b for the diameter
tolerance t at ϭ
average value
of the ball.
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70. Automated packaging: The machines that fill
boxes of breakfast cereal are programmed to fill
each box within a certain tolerance. If the box is
overfilled, the company loses money. If it is
underfilled, it is considered unsuitable for sale.
ᮣ
Suppose that boxes marked “14 ounces” of cereal
must be filled to within 0.1 oz. Find the acceptable
range of weights for this cereal.
EXTENDING THE CONCEPT
71. Determine the value or values (if any) that will
make the equation or inequality true.
x
a. ͿxͿ ϩ x ϭ 8
b. Ϳx Ϫ 2Ϳ Յ
2
c. x Ϫ ͿxͿ ϭ x ϩ ͿxͿ
d. Ϳx ϩ 3Ϳ Ն 6x
e. Ϳ2x ϩ 1Ϳ ϭ x Ϫ 3
72. The equation Ϳ5 Ϫ 2xͿ ϭ Ϳ3 ϩ 2xͿ has only one
solution. Find it and explain why there is only one.
73. In many cases, it can be helpful to view the solutions
to absolute value equations and inequalities as
follows. For any algebraic expression X and positive
ᮣ
2–42
CHAPTER 2 More on Functions
constant k, the equation ͿXͿ ϭ k has solutions X ϭ k
and ϪX ϭ k, since the absolute value of either
quantity on the left will indeed yield the positive
constant k. Likewise, ͿXͿ 6 k has solutions X 6 k
and ϪX 6 k. Note the inequality symbol has not
been reversed as yet, but will naturally be reversed
as part of the solution process. Solve the following
equations or inequalities using this idea.
a. Ϳx Ϫ 3Ϳ ϭ 5
b. Ϳx Ϫ 7Ϳ 7 4
c. 3Ϳx ϩ 2Ϳ Յ 12
d. Ϫ3Ϳx Ϫ 4Ϳ ϩ 7 ϭ Ϫ11
MAINTAINING YOUR SKILLS
74. (R.4) Factor the expression completely:
18x3 ϩ 21x2 Ϫ 60x.
76. (R.7) Simplify
Ϫ1
by rationalizing the
3 ϩ 23
denominator. State the result in exact form and
approximate form (to hundredths).
75. (1.5) Solve V2 ϭ
2W
for (physics).
CA
77. (R.3) Solve the inequality, then write the solution
set in interval notation:
Ϫ312x Ϫ 52 7 21x ϩ 12 Ϫ 7.
MID-CHAPTER CHECK
1. Determine whether the following function is even,
ͿxͿ
odd, or neither. f 1x2 ϭ x2 ϩ
4x
2. Use a graphing calculator to find the maximum and
minimum values of
f 1x2 ϭ Ϫ1.91x4 Ϫ 2.3x3 ϩ 2.2x Ϫ 5.12 . Round to
the nearest hundredth.
3. Use interval notation to identify the interval(s)
where the function from Exercise 2 is increasing,
decreasing, or constant. Round to the nearest
hundredth.
4. Write the equation of the function that has the same
graph of f 1x2 ϭ 2x, shifted left 4 units and up 2
units.
5. For the graph given, (a) identify
the function family, (b) describe
or identify the end-behavior,
inflection point, and x- and
y-intercepts, (c) determine
the domain and range, and
(d) determine the value of k if
f 1k2 ϭ 2.5. Assume required
features have integer values.
Exercise 5
y
5
f(x)
Ϫ5
5 x
Ϫ5
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Reinforcing Basic Concepts
6. Use a graphing calculator to graph the given
functions in the same window and comment on
what you observe.
p1x2 ϭ 1x Ϫ 32 2
r1x2 ϭ Ϫ12 1x Ϫ 32 2
229
9. Solve the following absolute value inequalities.
Write solutions in interval notation.
a. 3.1Ϳd Ϫ 2Ϳ ϩ 1.1 Ն 7.3
Ϳ1 Ϫ yͿ
11
ϩ2 7
b.
3
2
c. Ϫ5Ϳk Ϫ 2Ϳ ϩ 3 6 4
q1x2 ϭ Ϫ1x Ϫ 32 2
7. Solve the following absolute value equations. Write
the solution in set notation.
2
11
a. Ϳd Ϫ 5Ϳ ϩ 1 ϭ 7
b. 5 Ϫ Ϳs ϩ 3Ϳ ϭ
3
2
10. Kiteboarding: With the correct sized kite, a person
can kiteboard when the wind is blowing at a speed
w (in mph) that satisfies the inequality Ϳw Ϫ 17Ϳ Յ 9.
Solve the inequality and determine if a person can
kiteboard with a windspeed of (a) 5 mph?
(b) 12 mph?
8. Solve the following absolute value inequalities.
Write solutions in interval notation.
x
a. 3Ϳq ϩ 4Ϳ Ϫ 2 6 10 b. ` ϩ 2 ` ϩ 5 Յ 5
3
REINFORCING BASIC CONCEPTS
Using Distance to Understand Absolute Value Equations and Inequalities
For any two numbers a and b on the number line, the distance between a and b can be written Ϳa Ϫ bͿ or Ϳb Ϫ aͿ. In exactly
the same way, the equation Ϳx Ϫ 3Ϳ ϭ 4 can be read, “the distance between 3 and an unknown number is equal to 4.”
The advantage of reading it in this way (instead of “the absolute value of x minus 3 is 4”), is that a much clearer
visualization is formed, giving a constant reminder there are two solutions. In diagram form we have Figure 2.51.
Distance between
3 and x is 4.
Ϫ5 Ϫ4 Ϫ3 Ϫ2
Figure 2.51
4 units
Ϫ1
0
1
4 units
2
3
4
5
Distance between
3 and x is 4.
6
7
8
9
From this we note the solutions are x ϭ Ϫ1 and x ϭ 7.
In the case of an inequality such as Ϳx ϩ 2Ϳ Յ 3, we rewrite the inequality as Ϳx Ϫ 1Ϫ22 Ϳ Յ 3 and read it, “the distance
between Ϫ2 and an unknown number is less than or equal to 3.” With some practice, visualizing this relationship
mentally enables a quick statement of the solution: x ʦ 3Ϫ5, 14 . In diagram form we have Figure 2.52.
Distance between Ϫ2
and x is less than or equal to 3.
Ϫ8 Ϫ7 Ϫ6
Figure 2.52
3 units
Distance between Ϫ2
and x is less than or equal to 3.
3 units
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
3
4
5
6
Equations and inequalities where the coefficient of x is not 1 still lend themselves to this form of conceptual understanding. For Ϳ2x Ϫ 1Ϳ Ն 3 we read, “the distance between 1 and twice an unknown number is greater than or equal to 3.”
On the number line (Figure 2.53), the number 3 units to the right of 1 is 4, and the number 3 units to the left of 1 is Ϫ2.
Distance between 1 and
2x is greater than or equal to 3.
Figure 2.53
3 units
Ϫ6 Ϫ5 Ϫ4 Ϫ3
؊2 Ϫ1
0
3 units
1
2
3
Distance between 1 and
2x is greater than or equal to 3
4
5
6
7
8
For 2x Յ Ϫ2, x Յ Ϫ1, and for 2x Ն 4, x Ն 2, and the solution set is x ʦ 1Ϫq, Ϫ1 4 ´ 32, q 2.
Attempt to solve the following equations and inequalities by visualizing a number line. Check all results algebraically.
Exercise 1: Ϳx Ϫ 2Ϳ ϭ 5
Exercise 2: Ϳx ϩ 1Ϳ Յ 4
Exercise 3: Ϳ2x Ϫ 3Ϳ Ն 5
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Basic Rational Functions and Power Functions;
More on the Domain
LEARNING OBJECTIVES
In Section 2.4 you will see
how we can:
A. Graph basic rational
functions, identify vertical
and horizontal
asymptotes, and describe
end-behavior
B. Use transformations to
graph basic rational
functions and write the
equation for a given
graph
C. Graph basic power
functions and state their
domains
D. Solve applications
involving basic rational
and power functions
In this section, we introduce two new kinds of relations, rational functions and power
functions. While we’ve already studied a variety of functions, we still lack the ability
to model a large number of important situations. For example, functions that model the
amount of medication remaining in the bloodstream over time, the relationship
between altitude and weightlessness, and the equations modeling planetary motion
come from these two families.
A. Rational Functions and Asymptotes
Just as a rational number is the ratio of two integers, a rational function is the ratio of
two polynomials. In general,
Rational Functions
A rational function V(x) is one of the form
V1x2 ϭ
p1x2
d1x2
,
where p and d are polynomials and d1x2 0.
The domain of V(x) is all real numbers, except the zeroes of d.
The simplest rational functions are the reciprocal function y ϭ 1x and the reciprocal square function y ϭ x12, as both have a constant numerator and a single term in
the denominator. Since division by zero is undefined, the domain of both excludes
x ϭ 0. A preliminary study of these two functions will provide a strong foundation for
our study of general rational functions in Chapter 4.
The Reciprocal Function: y ؍
1
x
The reciprocal function takes any input (other than zero) and gives its reciprocal as the
output. This means large inputs produce small outputs and vice versa. A table of
values (Table 2.1) and the resulting graph (Figure 2.54) are shown.
Table 2.1
230
Figure 2.54
x
y
x
y
Ϫ1000
Ϫ1/1000
1/1000
1000
Ϫ5
Ϫ1/5
1/3
3
Ϫ4
Ϫ1/4
1/2
2
Ϫ3
Ϫ1/3
1
1
Ϫ2
Ϫ1/2
2
1/2
Ϫ1
Ϫ1
3
1/3
Ϫ1/2
Ϫ2
4
1/4
Ϫ1/3
Ϫ3
5
1/5
Ϫ1/1000
Ϫ1000
1000
1/1000
0
undefined
y
3
a, 3
yϭ
1
x
2
(1, 1)
Ϫ3, Ϫ a
1
Ϫ5
3, a
5
Ϫ5, Ϫ Q
(Ϫ1, Ϫ1)
Ϫ a, Ϫ3
5, Q
x
Ϫ1
Ϫ2
Ϫ3
2–44
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Section 2.4 Basic Rational Functions and Power Functions; More on the Domain
WORTHY OF NOTE
The notation used for graphical
behavior always begins by
describing what is happening to the
x-values, and the resulting effect on
the y-values. Using Figure 2.55,
visualize that for a point (x, y) on
the graph of y ϭ 1x , as x gets larger,
y must become smaller, particularly
since their product must always be
1 1y ϭ 1x 1 xy ϭ 12 .
Table 2.1 and Figure 2.54 reveal some interesting features. First, the graph passes
the vertical line test, verifying y ϭ 1x is indeed a function. Second, since division by
zero is undefined, there can be no corresponding point on the graph, creating a break
at x ϭ 0. In line with our definition of rational functions, the domain is
x ʦ 1Ϫq, 02 ´ 10, q 2 . Third, this is an odd function, with a “branch” of the graph in
the first quadrant and one in the third quadrant, as the reciprocal of any input maintains
its sign. Finally, we note in QI that as x becomes an infinitely large positive number, y
becomes infinitely small and closer to zero. It seems convenient to symbolize this endbehavior using the following notation:
Figure 2.55
as x S q,
y
yS0
as x becomes an infinitely
large positive number
Graphically, the curve becomes very close to, or approaches the x-axis.
We also note that as x approaches zero from the right, y becomes an infinitely large
positive number: as x S 0 ϩ , y S q . Note a superscript ϩ or Ϫ sign is used to indicate the direction of the approach, meaning from the positive side (right) or from the
negative side (left).
y
x
x
EXAMPLE 1
y approaches 0
ᮣ
Describing the End-Behavior of Rational Functions
For y ϭ 1x in QIII (Figure 2.54),
a. Describe the end-behavior of the graph.
b. Describe what happens as x approaches zero.
Solution
ᮣ
Similar to the graph’s behavior in QI, we have
a. In words: As x becomes an infinitely large negative number, y approaches zero.
In notation: As x S Ϫq , y S 0.
b. In words: As x approaches zero from the left, y becomes an infinitely large
negative number. In notation: As x S 0 Ϫ , y S Ϫq .
Now try Exercises 7 and 8
The Reciprocal Square Function: y ؍
ᮣ
1
x2
From our previous work, we anticipate this graph will also have a break at x ϭ 0. But
since the square of any negative number is positive, the branches of the reciprocal
square function are both above the x-axis. Note the result is the graph of an even function. See Table 2.2 and Figure 2.56.
Table 2.2
Figure 2.56
x
y
x
y
Ϫ1000
1/1,000,000
1/1000
1,000,000
Ϫ5
1/25
1/3
9
Ϫ4
1/16
1/2
4
Ϫ3
1/9
1
1
Ϫ2
1/4
2
1/4
Ϫ1
1
3
1/9
Ϫ1/2
4
4
1/16
Ϫ1/3
9
5
1/25
Ϫ1/1000
1,000,000
1000
1/1,000,000
0
undefined
y ϭ x12
y
3
(Ϫ1, 1)
Ϫ5,
1
1
25
Ϫ5
2
Ϫ3, 19
(1, 1)
3, 19
5,
5
Ϫ1
Ϫ2
Ϫ3
1
25
x