3.2
67. x 2y 600
Slope of a Line
(3-11)
131
68. 3x 2y 1500
3.2
In this
section
●
Slope
●
Using Coordinates to Find
Slope
SLOPE OF A LINE
In Section 3.1 we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail and study the concept of slope
of a line.
Slope
●
Parallel Lines
●
Perpendicular Lines
●
Applications of Slope
If a highway has a 6% grade, then in 100 feet (measured horizontally) the road rises
6 feet (measured vertically). See Fig. 3.10. The ratio of 6 to 100 is 6%. If a roof rises
9 feet in a horizontal distance (or run) of 12 feet, then the roof has a 9–12 pitch. A
roof with a 9–12 pitch is steeper than a roof with a 6–12 pitch. The grade of a road
and the pitch of a roof are measurements of steepness. In each case the measurement is a ratio of rise (vertical change) to run (horizontal change).
6%
9 ft
rise
GRADE
6
100
SLOW VEHICLES
KEEP RIGHT
helpful
9–12 pitch
hint
Since the amount of run is arbitrary, we can choose the run
to be 1. In this case
rise
slope rise.
1
So the slope is the amount of
change in y for a change of 1
in the x-coordinate.This is why
rates like 50 miles per hour
(mph), 8 hours per day, and
two people per car are all
slopes.
12 ft
run
FIGURE 3.10
We measure the steepness of a line in the same way that we measure steepness
of a road or a roof. The slope of a line is the ratio of the change in y-coordinate, or
the rise, to the change in x-coordinate, or the run, between two points on the line.
Slope
rise
change in y-coordinate
Slope change in x-coordinate
run
Consider the line in Fig. 3.11(a) on the next page. In going from (0, 1) to (1, 3),
there is a change of 1 in the x-coordinate and a change of 2 in the y-coordinate,
132
(3-12)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
y
y
5
4
3
5
4
3
(1, 3)
2
(0, 1)
–4 –3 –2
–1
–2
–3
–4
–5
+2
+1
1
(1, 3)
2
(0, 1)
2
3
x
4
–4 –3 –2
–1
–2
–3
–4
–5
(a)
–2
–1
1
2
3
x
4
(b)
FIGURE 3.11
or a run of 1 and a rise of 2. So the slope is 2 or 2. If we move from (1, 3) to (0, 1)
1
2
as in Fig. 3.11(b) the rise is 2 and the run is 1. So the slope is or 2. If we start
1
at either point and move to the other point, we get the same slope.
E X A M P L E
a)
1
Finding the slope from a graph
Find the slope of each line by going from point A to point B.
b)
c)
y
y
5
4
3
2
1
–1
–1
–2
–3
5
4
A
B
1 2
3
4
x
–1
5
4
B
3
2
1
3
2
1 A
A
1
2
3
4
5
y
6
x
–2
–3
–6 –5 –4 –3 –2
B
–1
1
x
–2
–3
Solution
a) A is located at (0, 3) and B at (2, 0). In going from A to B, the change in y is 3
and the change in x is 2. So
3
slope .
2
b) In going from A(2, 1) to B(6, 3), we must rise 2 and run 4. So
2 1
slope .
4 2
c) In going from A(0, 0) to B(6, 3), we find that the rise is 3 and the run is
6. So
3 1
slope .
6 2
■
3.2
(3-13)
Slope of a Line
133
Note that in Example 1(c) we found the slope of the line of Example 1(b) by
using two different points. The slope is the ratio of the lengths of the two legs of a
right triangle whose hypotenuse is on the line. See Fig. 3.12. As long as one leg is
vertical and the other leg is horizontal, all such triangles for a given line have the
same shape: They are similar triangles. Because ratios of corresponding sides in
similar triangles are equal, the slope has the same value no matter which two points
of the line are used to find it.
y
Hypotenuse
5
4
y
3
(x 2, y2)
Rise
2
Rise
–5 –4 –3 –2
Hypotenuse
Run
–1
–2
Run
1 2
3
y2 – y1
x
x
(x1, y1)
Rise
–4
–5
FIGURE 3.12
x 2 – x1
(x 2, y1)
Run
FIGURE 3.13
Using Coordinates to Find Slope
We can obtain the rise and run from a graph, or we can get them without a graph by
subtracting the y-coordinates to get the rise and the x-coordinates to get the run for
two points on the line. See Fig. 3.13.
Slope Using Coordinates
The slope m of the line containing the points (x1, y1) and (x2, y2) is given by
y2 y1
m ,
provided that x2 x1 0.
x2 x1
E X A M P L E
study
2
tip
Don’t expect to understand a
new topic the first time that
you see it. Learning mathematics takes time, patience,
and repetition. Keep reading
the text, asking questions, and
working problems. Someone
once said, “All mathematics is
easy once you understand it.”
Finding slope from coordinates
Find the slope of each line.
a) The line through (2, 5) and (6, 3)
b) The line through (2, 3) and (5, 1)
c) The line through (6, 4) and the origin
Solution
a) Let (x1, y1) (2, 5) and (x2, y2) (6, 3). The assignment of (x1, y1) and (x2, y2) is
arbitrary.
y2 y1
3 5 2
1
m x2 x1
62
4
2
b) Let (x1, y1) (5, 1) and (x2, y2) (2, 3):
y2 y1
3 (1)
4
m x2 x1
2 (5) 3
134
(3-14)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
c) Let (x1, y1) (0, 0) and (x2, y2) (6, 4):
40
4
2
m 6 0
6
3
■
CAUTION
Do not reverse the order of subtraction from numerator to
denominator when finding the slope. If you divide y2 y1 by x1 x2, you will get
the wrong sign for the slope.
E X A M P L E
helpful
3
Slope for horizontal and vertical lines
Find the slope of each line.
a)
y
4
3
hint
Think about what slope
means to skiers. No one skis
on cliffs or even refers to them
as slopes.
b)
(– 3, 2)
(4, 2)
1
–5 –4 –3 –2 –1
–1
–2
1 2
3
4
5
x
y
5
4
3
2
1
–2 –1
–1
–2
–3
–4
–5
(1, 2)
2
3
4
5
x
(1, – 4)
Zero slope
Solution
a) Using (3, 2) and (4, 2) to find the slope of the horizontal line, we get
Small slope
22
m 3 4
0
0.
7
b) Using (1, 4) and (1, 2) to find the slope of the vertical line, we get x2 x1 0.
Because the definition of slope using coordinates says that x2 x1 must be
■
nonzero, the slope is undefined for this line.
Larger slope
Since the y-coordinates are equal for any two points on a horizontal line,
y2 y1 0 and the slope is 0. Since the x-coordinates are equal for any two points
on a vertical line, x2 x1 0 and the slope is undefined.
Horizontal and Vertical Lines
The slope of any horizontal line is 0.
Slope is undefined for any vertical line.
CAUTION
Do not say that a vertical line has no slope because “no slope”
could be confused with 0 slope, the slope of a horizontal line.
Undefined slope
As you move the tip of your pencil from left to right along a line with positive
slope, the y-coordinates are increasing. As you move the tip of your pencil from
3.2
(3-15)
Slope of a Line
135
left to right along a line with negative slope, the y-coordinates are decreasing. See
Fig. 3.14.
y
4
Increasing
3
y-coordinates
2
1
–4 –3 –2
–1
Positive slope
1
2
3
4
y
x
4
3
2
1
y
4
3
Decreasing
y-coordinates
–4 –3
–1
–2
Negative slope 1
–4 –3 –2 –1
–1
1
2
3
4
–4
–5
x
FIGURE 3.14
1
Slope —
3
1
2
3
4
x
1
Slope —
3
FIGURE 3.15
Parallel Lines
Consider the two lines shown in Fig. 3.15. Each of these lines has a slope of 1, and
3
these lines are parallel. In general, we have the following fact.
Parallel Lines
Nonvertical parallel lines have equal slopes.
Of course, any two vertical lines are parallel, but we cannot say that they have equal
slopes because slope is not defined for vertical lines.
E X A M P L E
4
Parallel lines
Line l goes through the origin and is parallel to the line through (2, 3) and (4, 5).
Find the slope of line l.
Solution
The line through (2, 3) and (4, 5) has slope
y
5 3
8
4
m .
4 (2)
6
3
(–1, 3)
4
1
–3 –2 –1
–1
–2
Slope 2
1
2
3
4
FIGURE 3.16
5
4
Because line l is parallel to a line with slope 3, the slope of line l is 3 also.
1
Slope – —
2
x
■
Perpendicular Lines
1
The lines shown in Fig. 3.16 have slopes 2 and 2. These two lines appear to be
perpendicular to each other. It can be shown that a line is perpendicular to another
line if its slope is the negative of the reciprocal of the slope of the other.
136
(3-16)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
Perpendicular Lines
Two lines with slopes m1 and m2 are perpendicular if and only if
1
m1 .
m2
Of course, any vertical line and any horizontal line are perpendicular, but we cannot
give a relationship between their slopes because slope is undefined for vertical lines.
E X A M P L E
5
Perpendicular lines
Line l contains the point (1, 6) and is perpendicular to the line through
(4, 1) and (3, 2). Find the slope of line l.
Solution
The line through (4, 1) and (3, 2) has slope
1 (2)
3
3
m .
4 3
7
7
3
Because line l is perpendicular to a line with slope 7, the slope of line l is 7.
3
■
Applications of Slope
When a geometric figure is located in a coordinate system, we can use slope to
determine whether it has any parallel or perpendicular sides.
E X A M P L E
6
Using slope with geometric figures
Determine whether (3, 2), (2, 1), (4, 1), and (3, 4) are the vertices of a
rectangle.
Solution
Figure 3.17 shows the quadrilateral determined by these points. If a parallelogram
has at least one right angle, then it is a rectangle. Calculate the slope of each side.
y
5
D (3, 4)
A (–3, 2)
1
–5 –3
B (– 2, –1)
–1
C (4, 1)
1
3
–3
FIGURE 3.17
5
x
2 (1)
mAB 3 (2)
3
3
1
14
mCD 43
3
3
1
1 1
mBC 2 4
2 1
6 3
24
mAD 3 3
2 1
6 3
Because the opposite sides have the same slope, they are parallel, and the figure is
a parallelogram. Because 1 is the opposite of the reciprocal of 3, the intersecting
3
■
sides are perpendicular. Therefore the figure is a rectangle.
The slope of a line is a rate. The slope tells us how much the dependent variable
changes for a change of 1 in the independent variable. For example, if the horizontal axis is hours and the vertical axis is miles, then the slope is miles per hour (mph).
3.2
Slope of a Line
(3-17)
137
If the horizontal axis is days and the vertical axis is dollars, then the slope is dollars
per day.
E X A M P L E
7
Slope as a rate
Worldwide carbon dioxide (CO2) emissions have increased from 14 billion tons in
1970 to 24 billion tons in 1995 (World Resources Institute, www.wri.org).
CO2 emission
(in billions of tons)
24
14
1970
1995
Year
FIGURE FOR EXAMPLE 7
study
tip
Finding out what happened in
class and attending class are
not the same. Attend every
class and be attentive. Don’t
just take notes and let your
mind wander. Use class time
as a learning time.
WARM-UPS
a) Find and interpret the slope of the line in the accompanying figure.
b) Predict the amount of worldwide CO2 emissions in 2005.
Solution
a) Find the slope of the line through (1970, 14) and (1995, 24):
24 14
m 0.4
1995 1970
The slope of the line is 0.4 billion tons per year.
b) If the (CO2) emissions keep increasing at 0.4 billion tons per year, then in 10 years
the level will go up 10(0.4) or 4 billion tons. So in 2005 CO2 emissions will be
■
28 billion tons.
True or false? Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
Slope is a measurement of the steepness of a line. True
Slope is run divided by rise. False
The line through (4, 5) and (3, 5) has undefined slope. False
The line through (2, 6) and (2, 5) has undefined slope. True
Slope cannot be negative. False
2
The slope of the line through (0, 2) and (5, 0) is 5. False
The line through (4, 4) and (5, 5) has slope 5. False
4
If a line contains points in quadrants I and III, then its slope is
positive. True
2
9. Lines with slope 2 and 3 are perpendicular to each other. False
3
10. Any two parallel lines have equal slopes. False
138
(3-18)
3. 2
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What does slope measure?
Slope measures the steepness of a line.
2. What is the rise and what is the run?
The rise is the change in y-coordinates and run is the
change in x-coordinates.
3. Why does a horizontal line have zero slope?
A horizontal line has zero slope because it has no rise.
4. Why is slope undefined for vertical lines?
Slope is undefined for vertical lines because the run is zero
and division by zero is undefined.
5. What is the relationship between the slopes of perpendicular lines?
If m1 and m2 are the slopes of perpendicular lines, then
.
m1 1
m
2
6. What is the relationship between the slopes of parallel lines?
If m1 and m2 are the slopes of parallel lines, then m1 m2.
Determine the slope of each line. See Example 1.
y
7.
8.
–5
y
4
3
4
3
1
2
1
–3 –2 –1
–1
–2
–3
–4
1 2
–5 –4 –3
x
3
1 2
3
–3 –2 –1
–1
–2
–3
–4
x
11.
–4 –3 –2 –1
–1
–2
–3
–4
1 2
3
4
x
1
2
3
4
–4 –3 –2 –1
–1
–2
–3
–4
x
1
y
4
3
2
1
–3 –2 –1
–1
–2
–3
x
2
3
4
x
1 2
3
4
x
1
1
y
14.
5
4
3
2
1
–4 –3 –2 –1
–1
–2
–3
–4
0
4
y
12.
4
3
2
1
1
3
1
Undefined
y
4
3
3
2
–1
–1
–3
–4
y
10.
13.
4
3
2
1
2
3
2
3
y
9.
y
15.
5
4
3
4
3
1
1
3
4
5
x
–5 –4 –3 –2
3
–1
–2
–3
–4
1 2
3
1
x
–4 –3 –2 –1
–1
–2
–3
1
3.2
42. Line l goes through the origin and is parallel to the line
through (3, 5) and (4, 1). 4
7
43. Line l is perpendicular to a line with slope 4. Both lines
5
contain the origin. 5
1
4
–4 –3 –2 –1
–1
–2
–3
1 2
3
x
4
1
2
Find the slope of the line that contains each of the following
pairs of points. See Examples 2 and 3.
5
17. (2, 6), (5, 1) 18. (3, 4), (6, 10) 2
3
4
19. (3, 1), (4, 3) 20. (2, 3), (1, 3) 2
7
11
21. (2, 2), (1, 7) 5
22. (3, 5), (1, 6) 4
1
5
23. (3, 5), (0, 0) 24. (0, 0), (2, 1) 3
2
3
25. (0, 3), (5, 0) 5
10
3
1
1 1
28. , 2, , 2
4 2
3
1 1
27. , 1 , , 4
2 2
29. (6, 212), (7, 209)
2
5
44. Line l is perpendicular to a line with slope 5. Both lines
contain the origin. 1
5
Solve each geometric figure problem. See Example 6.
45. If the opposite sides of a quadrilateral are parallel, then it is
a parallelogram. Use slope to determine whether the points
(6, 1), (2, 1), (0, 3), and (4, 1) are the vertices of a
parallelogram. Yes
46. Use slope to determine whether the points (7, 0), (1, 6),
(1, 2), and (6, 5) are the vertices of a parallelogram. See
Exercise 45. No
47. A trapezoid is a quadrilateral with one pair of parallel sides.
Use slope to determine whether the points (3, 2), (1, 1),
(3, 6), and (6, 4) are the vertices of a trapezoid. No
48. A parallelogram with at least one right angle is a rectangle.
Determine whether the points (4, 4), (1, 2), (0, 6), and
(3, 0) are the vertices of a rectangle. Yes
49. If a triangle has one right angle, then it is a right triangle.
Use slope to determine whether the points (3, 3), (1, 6),
and (0, 0) are the vertices of a right triangle. No
50. Use slope to determine whether the points (0, 1), (2, 5),
and (5, 4) are the vertices of a right triangle. See Exercise 49. Yes
Solve each problem. See Example 7.
51. Pricing the Crown Victoria. The list price of a new
Ford Crown Victoria four-door sedan was $20,115 in
1993 and $21,135 in 1998 (Edmund’s New Car Prices,
www.edmunds.com).
a) Find the slope of the line shown in the figure. 204
b) Use the graph to predict the price in 2005. $22,500
c) Use the slope to predict the price of a new Crown
Victoria in 2005. $22,563
6
3
30. (1988, 306), (1990, 315)
31.
32.
33.
34.
35.
139
7
5
4
3
26. (3, 0), (0, 10)
(3-19)
41. Line l goes through (2, 5) and is parallel to the line through
(3, 2) and (4, 1). 3
y
9
2
(4, 7), (12, 7) 0
(5, 3), (9, 3) 0
(2, 6), (2, 6) Undefined
(3, 2), (3, 0) Undefined
(24.3, 11.9), (3.57, 8.4) 0.169
36. (2.7, 19.3), (5.46, 3.28) 2.767
37. , 1 , , 0 1.273
4
2
38. , 1 , , 0 1.910
3
6
In each case, make a sketch and find the slope of line l. See
Examples 4 and 5.
39. Line l contains the point (3, 4) and is perpendicular to the
line through (5, 1) and (3, 2). 8
3
40. Line l goes through (3, 5) and is perpendicular to the
line through (2, 6) and (5, 3). 7
3
List price (in thousands of dollars)
16.
Slope of a Line
22
(1998, 21,135)
21
20
(1993, 20,115)
93
94
95
96 97
Year
98
99
00
FIGURE FOR EXERCISE 51
52. Depreciating Monte Carlo. In 1998 the average retail
price of a one-year-old Chevrolet Monte Carlo was
$13,595, whereas the average retail price of a 3-year-old
Monte Carlo was $11,095 (Edmund’s Used Car Prices).
140
(3-20)
Chapter 3
Graphs and Functions in the Cartesian Coordinate System
Selling price (in thousands of dollars)
a) Use the graph on the next page to estimate the average
retail price of a 2-year-old car in 1998. $12,000
b) Find the slope of the line shown in the figure. 1250
c) Use the slope to predict the price of a 2-year-old car.
$12,345
15
(1, 13,595)
(3, 11,095)
10
5
0
0.5
1.0
1.5
2.0 2.5 3.0
Age (in years)
3.5
4.0
4.5
FIGURE FOR EXERCISE 52
53. The points (3, ) and ( ,7) are on the line that passes
through (2, 1) and has slope 4. Find the missing coordinates
of the points. (3, 5), (0, 7)
54. If a line passes through (5, 2) and has slope 2, then what is
3
the value of y on this line when x 8, x 11, and x 12?
4, 6, 6 23
55. Find k so that the line through (2, k) and (3, 5) has
slope 1. 5
2
56. Find k so that the line through (k, 3) and (2, 0) has slope 3.
3 or 1
57. What is the slope of a line that is perpendicular to a line
with slope 0.247?
4.049
58. What is the slope of a line that is perpendicular to the line
through (3.27, 1.46) and (5.48, 3.61)?
1.726
GET TING MORE INVOLVED
59. Writing. What is the difference between zero slope and
undefined slope?
A horizontal line has a zero slope and a vertical line has undefined slope.
3.3
In this
section
●
Point-Slope Form
●
Slope-Intercept Form
●
Standard Form
●
Using Slope-Intercept Form
for Graphing
●
Linear Functions
61. Exploration. A rhombus is a quadrilateral with four equal
sides. Draw a rhombus with vertices (3, 1), (0, 3),
(2, 1), and (5, 3). Find the slopes of the diagonals of the
rhombus. What can you conclude about the diagonals of
this rhombus?
2, 1, perpendicular
2
MISCELL ANEOUS
2
60. Writing. Is it possible for a line to be in only one quadrant?
Two quadrants? Write a rule for determining whether a line
has positive, negative, zero, or undefined slope from knowing in which quadrants the line is found.
Every line goes through at least two quadrants. A nonhorizontal, nonvertical line that misses quadrant II or IV or
both has a positive slope. A nonhorizontal, nonvertical
line that misses quadrant I or III or both has a negative
slope.
62. Exploration. Draw a square with vertices (5, 3),
(3, 3), (1, 5), and (3, 1). Find the slopes of the diagonals
of the square. What can you conclude about the diagonals of
this square?
2, 1, perpendicular
2
GR APHING C ALCUL ATOR
EXERCISES
63. Graph y 1x, y 2x, y 3x, and y 4x together in
the standard viewing window. These equations are all of
the form y mx. What effect does increasing m have on
the graph of the equation? What are the slopes of these four
lines?
Increasing m makes the graph increase faster. The slopes of
these lines are 1, 2, 3, and 4.
64. Graph y 1x, y 2x, y 3x, and y 4x
together in the standard viewing window. These equations
are all of the form y mx. What effect does decreasing m have on the graph of the equation? What are the
slopes of these four lines?
Decreasing m makes the graph decrease faster. The slopes
of these lines are 1, 2, 3, and 4.
THREE FORMS FOR THE
EQUATION OF A LINE
In Section 3.1 you learned how to graph a straight line corresponding to a linear
equation. The line contains all of the points that satisfy the equation. In this section
we start with a line or a description of a line and write an equation corresponding to
the line.
Point-Slope Form
Figure 3.18 shows the line that has slope 2 and contains the point (3, 5). In Sec3
tion 3.2 you learned that the slope is the same no matter which two points of the line