2.2
Before
Now
Why?
Key Vocabulary
• slope
• parallel
• perpendicular
• rate of change
• reciprocal, p. 4
Find Slope and
Rate of Change
You graphed linear functions.
You will find slopes of lines and rates of change.
So you can model growth rates, as in Ex. 46.
For Your Notebook
KEY CONCEPT
Slope of a Line
Words
Algebra
Graph
y2 2 y1
rise
m5}
x2 2 x1 5 }
run
The slope m of a
nonvertical line is
the ratio of vertical
change (the rise) to
horizontal change
(the run).
run
x2 2 x1
y
rise
y2 2 y1
slope
(x2, y2)
(x1, y1)
x
EXAMPLE 1
Find slope in real life
SKATEBOARDING A skateboard ramp has a rise of
15 inches and a run of 54 inches. What is its slope?
rise 5 15 in.
run 5 54 in.
Solution
rise 5 15 5 5
slope 5 }
}
}
run
54
18
5.
c The slope of the ramp is }
18
★
EXAMPLE 2
Standardized Test Practice
What is the slope of the line passing through the points (21, 3) and (2, 21)?
3
4
4
3
3
C }
B 2}
A 2}
4
D }
4
3
Solution
AVOID ERRORS
When calculating slope,
be sure to subtract the
x- and y-coordinates in
a consistent order.
82
n2pe-0202.indd 82
y
Let (x1, y1) 5 (21, 3) and (x 2 , y 2 ) 5 (2, 21).
y 2y
21 2 3
2 2 (21)
(21, 3)
24 1
2
1
4
m5}
5 } 5 2}
x2 2 x1
3
1
(2, 21) x
3
c The correct answer is A. A B C D
Chapter 2 Linear Equations and Functions
10/20/05 9:02:17 AM
✓
GUIDED PRACTICE
for Examples 1 and 2
1. WHAT IF? In Example 1, suppose that the rise of the ramp is changed to
12 inches without changing the run. What is the slope of the ramp?
2. What is the slope of the line passing through the points (24, 9) and (28, 3)?
2
A 2}
1
B 2}
3
3
D }
2
C }
2
3
2
Find the slope of the line passing through the given points.
3. (0, 3), (4, 8)
4. (25, 1), (5, 24)
5. (23, 22), (6, 1)
6. (7, 3), (21, 7)
For Your Notebook
KEY CONCEPT
Classification of Lines by Slope
The slope of a line indicates whether the line rises from left to right, falls from
left to right, is horizontal, or is vertical.
y
y
y
y
READING
A vertical line has
“undefined slope”
because for any two
points, the slope
formula’s denominator
becomes 0, and division
by 0 is undefined.
x
x
x
x
Positive slope
Negative slope
Zero slope
Undefined slope
Rises from
left to right
Falls from
left to right
Horizontal
Vertical
EXAMPLE 3
Classify lines using slope
Without graphing, tell whether the line through the given points rises, falls,
is horizontal, or is vertical.
a. (25, 1), (3, 1)
b. (26, 0), (2, 24)
c. (21, 3), (5, 8)
d. (4, 6), (4, 21)
Solution
✓
121 5 0 50
a. m 5 }
}
8
3 2 (25)
Because m 5 0, the line is horizontal.
24 2 0 5 24 5 2 1
b. m 5 }
}
}
8
2
2 2 (26)
Because m < 0, the line falls.
823 5 5
c. m 5 }
}
6
5 2 (21)
Because m > 0, the line rises.
21 2 6 5 27
d. m 5 }
}
424
0
Because m is undefined, the line is vertical.
GUIDED PRACTICE
for Example 3
Without graphing, tell whether the line through the given points rises, falls,
is horizontal, or is vertical.
7. (24, 3), (2, 26)
8. (7, 1), (7, 21)
9. (3, 22), (5, 22)
10. (5, 6), (1, 24)
2.2 Find Slope and Rate of Change
n2pe-0202.indd 83
83
10/20/05 9:02:21 AM
PARALLEL AND PERPENDICULAR LINES Recall that two lines in a plane are
parallel if they do not intersect. Two lines in a plane are perpendicular if they
intersect to form a right angle.
Slope can be used to determine whether two different nonvertical lines are
parallel or perpendicular.
For Your Notebook
KEY CONCEPT
Slopes of Parallel and Perpendicular Lines
Consider two different nonvertical lines l1 and l2 with slopes m1 and m2.
Parallel Lines The lines are parallel if and only if
they have the same slope.
l1
y
l2
m1 5 m2
x
Perpendicular Lines The lines are perpendicular if
and only if their slopes are negative reciprocals of
each other.
l1
l2
y
x
1 , or m m 5 21
m1 5 2}
1 2
m2
EXAMPLE 4
Classify parallel and perpendicular lines
Tell whether the lines are parallel, perpendicular, or neither.
a. Line 1: through (22, 2) and (0, 21)
Line 2: through (24, 21) and (2, 3)
b. Line 1: through (1, 2) and (4, 23)
Line 2: through (24, 3) and (21, 22)
Solution
a. Find the slopes of the two lines.
21 2 2 5 23 5 2 3
m1 5 }
}
}
2
0 2 (22)
3 2 (21)
2 2 (24)
2
(2, 3)
Line 2
2
(22, 2)
452
m2 5 } 5 }
}
6
y
(0, 21)
21
3
Line 1
3 p 2 5 21, m and m
c Because m1m2 5 2}
}
1
2
2
x
(24, 21)
3
are negative reciprocals of each other. So, the
lines are perpendicular.
b. Find the slopes of the two lines.
23 2 2
25 5 2 5
m1 5 } 5 }
}
421
22 2 3
21 2 (24)
3
y
(24, 3)
Line 2
3
1
25 5 2 5
m2 5 } 5 }
}
3
n2pe-0202.indd 84
Line 1
1
3
c Because m1 5 m2 (and the lines are different),
you can conclude that the lines are parallel.
84
(1, 2)
(21, 22)
x
(4, 23)
Chapter 2 Linear Equations and Functions
10/20/05 9:02:22 AM
✓
GUIDED PRACTICE
for Example 4
Tell whether the lines are parallel, perpendicular, or neither.
11. Line 1: through (22, 8) and (2, 24)
12. Line 1: through (24, 22) and (1, 7)
Line 2: through (25, 1) and (22, 2)
Line 2: through (21, 24) and (3, 5)
RATE OF CHANGE Slope can be used to represent an average rate of change, or
how much one quantity changes, on average, relative to the change in another
quantity. A slope that is a real-life rate of change involves units of measure such
as miles per hour or degrees per day.
REVIEW RATES
Remember that a
rate is a ratio of two
quantities that have
different units.
EXAMPLE 5
Solve a multi-step problem
FORESTRY Use the diagram, which illustrates the growth of a giant sequoia, to
find the average rate of change in the diameter of the sequoia over time. Then
predict the sequoia’s diameter in 2065.
IN
IN
Solution
STEP 1
Find the average rate of change.
Change in diameter
Change in time
Average rate of change 5 }
141 in. 2 137 in.
5}
2005 2 1965
4 in.
5}
40 years
5 0.1 inch per year
STEP 2 Predict the diameter of the sequoia in 2065.
Find the number of years from 2005 to 2065. Multiply this number by
the average rate of change to find the total increase in diameter during
the period 2005–2065.
Number of years 5 2065 2 2005 5 60
Increase in diameter 5 (60 years)(0.1 inch/year) 5 6 inches
c In 2065, the diameter of the sequoia will be about 141 1 6 5 147 inches.
✓
GUIDED PRACTICE
for Example 5
13. WHAT IF? In Example 5, suppose that the diameter of the sequoia is
248 inches in 1965 and 251 inches in 2005. Find the average rate of change
in the diameter, and use it to predict the diameter in 2105.
2.2 Find Slope and Rate of Change
n2pe-0202.indd 85
85
10/20/05 9:02:24 AM
2.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 9, 19, and 45
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 17, 35, 36, 44, 45, and 48
SKILL PRACTICE
1. VOCABULARY Copy and complete: The
? of a nonvertical line is the ratio
of vertical change to horizontal change.
2. ★ WRITING How can you use slope to decide whether two nonvertical lines
are parallel? whether two nonvertical lines are perpendicular?
EXAMPLES
2 and 3
on pp. 82–83
for Exs. 3–17
FINDING SLOPE Find the slope of the line passing through the given points.
Then tell whether the line rises, falls, is horizontal, or is vertical.
3. (2, 24), (4, 21)
4. (8, 9), (24, 3)
5. (5, 1), (8, 24)
6. (23, 22), (3, 22)
7. (21, 4), (1, 24)
8. (26, 5), (26, 25)
9. (25, 24), (21, 3)
10. (23, 6), (27, 3)
11. (4, 4), (4, 9)
13. (0, 23), (4, 23)
14. (1, 21), (21, 24)
12. (5, 5), (7, 3)
"MHFCSB
at classzone.com
ERROR ANALYSIS Describe and correct the error in finding the slope of the line
passing through the given points.
15.
16.
(24, 23), (2, 21)
21 2 (23)
24 2 2
(21, 4), (5, 1)
1
m 5 } 5 2}
3
5 2 (21)
124
m 5 } 5 22
17. ★ MULTIPLE CHOICE What is true about the line through (2, 24) and (5, 1)?
A It rises from left to right.
B It falls from left to right.
C It is horizontal.
D It is vertical.
EXAMPLE 4
CLASSIFYING LINES Tell whether the lines are parallel, perpendicular, or
on p. 84
for Exs. 18–23
neither.
18. Line 1: through (3, 21) and (6, 24)
19. Line 1: through (1, 5) and (3, 22)
Line 2: through (24, 5) and (22, 7)
Line 2: through (23, 2) and (4, 0)
20. Line 1: through (21, 4) and (2, 5)
Line 2: through (26, 2) and (0, 4)
22. Line 1: through (23, 2) and (5, 0)
Line 2: through (21, 24) and (3, 23)
EXAMPLE 5
on p. 85
for Exs. 24–27
86
n2pe-0202.indd 86
21. Line 1: through (5, 8) and (7, 2)
Line 2: through (27, 22) and (24, 21)
23. Line 1: through (1, 24) and (4, 22)
Line 2: through (8, 1) and (14, 5)
AVERAGE RATE OF CHANGE Find the average rate of change in y relative to x for
the ordered pairs. Include units of measure in your answer.
24. (2, 12), (5, 30)
x is measured in hours and y is measured in dollars
25. (0, 11), (3, 50)
x is measured in gallons and y is measured in miles
26. (3, 10), (5, 18)
x is measured in seconds and y is measured in feet
27. (1, 8), (7, 20)
x is measured in seconds and y is measured in meters
Chapter 2 Linear Equations and Functions
10/20/05 9:02:38 AM
28. REASONING The Key Concept box on page 84 states that lines l1 and l2 must
be nonvertical. Explain why this condition is necessary.
FINDING SLOPE Find the slope of the line passing through the given points.
3 , 0, 7
29. 21, }
}
2
2
1
21
3 , 22 , 5 , 23
30. 2}
}
4
4
1
2
32. (24.2, 0.1), (23.2, 0.1)
21
1, 5 , 5, 3
31. 2}
} }
2 2
2
2
1
33. (20.3, 2.2), (1.7, 20.8)
21
34. (3.5, 22), (4.5, 0.5)
35. ★ SHORT RESPONSE Does it make a difference
which two points on a line you choose when finding
the slope? Does it make a difference which point is
(x1, y1) and which point is (x2, y 2) in the formula for
slope? Support your answers using three different
pairs of points on the line shown.
2
P
2
y
Œ
R
4
x
S
T
36. ★ OPEN-ENDED MATH Find two additional points on the line that passes
through (0, 3) and has a slope of 24.
CHALLENGE Find the value of k so that the line through the given points has the
given slope. Check your solution.
37. (2, 23) and (k, 7); m 5 22
38. (0, k) and (3, 4); m 5 1
39. (24, 2k) and (k, 25); m 5 21
40. (22, k) and (2k, 2); m 5 20.25
PROBLEM SOLVING
EXAMPLE 1
on p. 82
for Exs. 41–44
41. ESCALATORS An escalator in an airport rises 28 feet over a horizontal
distance of 48 feet. What is the slope of the escalator?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
42. INCLINE RAILWAY The Duquesne Incline, a cable car railway, rises
400 feet over a horizontal distance of 685 feet on its ascent to an overlook
of Pittsburgh, Pennsylvania. What is the slope of the incline?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
43. ROAD GRADE A road’s grade is its slope expressed as a percent. A road rises
195 feet over a horizontal distance of 3000 feet. What is the grade of the road?
44. ★ SHORT RESPONSE The diagram shows a
three-section ramp to a bridge. Each section
has the same slope. Compare this slope with
the slope that a single-section ramp would
have if it rose directly to the bridge from the
same starting point. Explain the benefits of
a three-section ramp in this situation.
EXAMPLE 5
on p. 85
for Exs. 45–46
45. ★ MULTIPLE CHOICE Over a 30 day period, the amount of propane in a
tank that stores propane for heating a home decreases from 400 gallons to
214 gallons. What is the average rate of change in the amount of propane?
A 26.2 gallons per day
B 26 gallons per day
C 20.16 gallon per day
D 6 gallons per day
2.2 Find Slope and Rate of Change
n2pe-0202.indd 87
87
10/20/05 9:02:40 AM
46. BIOLOGY A red sea urchin grows its
Growth of Red Sea Urchin
entire life, which can last 200 years.
The diagram gives information about
the growth in the diameter d of one
red sea urchin. What is the average
growth rate of this urchin over the
given period?
Age 30
Age 110
d = 11.9 cm
d = 15.5 cm
47. MULTI-STEP PROBLEM A building code requires the minimum slope, or
pitch, of an asphalt-shingle roof to be a rise of 3 feet for each 12 feet of run.
The asphalt-shingle roof of an apartment building has the dimensions
shown.
a. Calculate What is the slope of the roof?
15 ft
b. Interpret Does the roof satisfy the building code?
80 ft
c. Reasoning If you answered “no” to part (b), by
how much must the rise be increased to satisfy the
code? If you answered “yes,” by how much does the
rise exceed the code minimum?
48. ★ EXTENDED RESPONSE Plans for a new water slide in an amusement park
call for the slide to descend from a platform 80 feet tall. The slide will drop
1 foot for every 3 feet of horizontal distance.
a. What horizontal distance do you cover when descending the slide?
b. Use the Pythagorean theorem to find the length of the slide.
c. Engineers decide to shorten the slide horizontally by 5 feet to allow for
a wider walkway at the slide’s base. The plans for the platform remain
unchanged. How will this affect the slope of the slide? Explain.
49. CHALLENGE A car travels 36 miles per gallon of gasoline in highway driving
and 24 miles per gallon in city driving. If you drive the car equal distances
on the highway and in the city, how many miles per gallon can you expect
to average? (Hint: The average fuel efficiency for all the driving is the total
distance traveled divided by the total amount of gasoline used.)
MIXED REVIEW
Identify the property that the statement illustrates. (p. 2)
PREVIEW
Prepare for
Lesson 2.3
in Exs. 56–61.
50. 5(8 1 12) 5 5(8) 1 5(12)
51. (7 1 9) 1 13 5 7 1 (9 1 13)
52. 4 1 (24) 5 0
53. 5 p 10 5 10 p 5
1 51
54. 15 p }
15
2
2 p15}
55. }
3
3
Solve the equation for y. (p. 26)
56. 3x 1 y 5 7
57. 2x 2 y 5 3
58. y 2 4x 5 26
59. 2x 1 3y 5 212
60. 7x 2 4y 5 10
61. 2x 1 2y 5 9
Solve the equation or inequality. (p. 51)
88
n2pe-0202.indd 88
62. 5 1 2x 5 7
63. 4x 2 9 5 5
64. 6 2 5x 5 9
65. 3 2 7x < 10
66. 3x 1 1 > 25
67. 3 2 4x ≥ 7
PRACTICE
for Lesson 2.2, p. 1011
Chapter 2 EXTRA
Linear Equations
and Functions
ONLINE QUIZ at classzone.com
10/20/05 9:02:41 AM