6.4
The Slope of a Line
6.4
OBJECTIVES
1. Find the slope of a line through two given points
2. Find the slope of a line from its graph
We saw in Section 6.3 that the graph of an equation such as
y 2x 3
is a straight line. In this section we want to develop an important idea related to the equation of a line and its graph, called the slope of a line. Finding the slope of a line gives us a
numerical measure of the “steepness” or inclination of that line.
y
Q(x2, y2)
P(x1, y1)
(x2, y1)
NOTE Recall that an equation
Change in x
x2 x1
such as y 2x 3 is a linear
equation in two variables. Its
graph is always a straight line.
NOTE x1 is read “x sub 1,” x2 is
read “x sub 2,” and so on. The 1
in x1 and the 2 in x2 are called
subscripts.
Change
in y
y2 y1
x
To find the slope of a line, we first let P(x1, y1) and Q(x2, y2) be any two distinct points
on that line. The horizontal change (or the change in x) between the points is x2 x1. The
vertical change (or the change in y) between the points is y2 y1.
We call the ratio of the vertical change, y2 y1, to the horizontal change, x2 x1, the
slope of the line as we move along the line from P to Q. That ratio is usually denoted by the
letter m, and so we have the following formula:
Definitions: The Slope of a Line
© 2001 McGraw-Hill Companies
NOTE The difference x2 x1 is
sometimes called the run
between points P and Q. The
difference y2 y1 is called the
rise. So the slope may be
thought of as “rise over run.”
If P(x1, y1) and Q(x2, y2) are any two points on a line, then m, the slope of the
line, is given by
m
vertical change
y y1
2
horizontal change
x2 x1
when x2 Z x1
This definition provides exactly the numerical measure of “steepness” that we want. If a
line “rises” as we move from left to right, the slope will be positive—the steeper the line,
the larger the numerical value of the slope. If the line “falls” from left to right, the slope will
be negative.
Let’s proceed to some examples.
Example 1
Finding the Slope
Find the slope of the line containing points with coordinates (1, 2) and (5, 4).
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CHAPTER 6
AN INTRODUCTION TO GRAPHING
Let P(x1, y1) (1, 2) and Q(x2, y2) (5, 4). By the definition of slope, we have
m
y2 y1
42
2
1
x2 x1
51
4
2
y
(5, 4)
422
(5, 2)
x
(1, 2)
514
Note: We would have found the same slope if we had reversed P and Q and subtracted in
the other order. In that case, P(x1, y1) (5, 4) and Q(x2, y2) (1, 2), so
m
24
2
1
15
4
2
It makes no difference which point is labeled (x1, y1) and which is (x2, y2). The resulting
slope will be the same. You must simply stay with your choice once it is made and not
reverse the order of the subtraction in your calculations.
CHECK YOURSELF 1
Find the slope of the line containing points with coordinates (2, 3) and (5, 5).
By now you should be comfortable subtracting negative numbers. Let’s apply that skill
to finding a slope.
Example 2
Finding the Slope
Find the slope of the line containing points with the coordinates (1, 2) and (3, 6).
Again, applying the definition, we have
m
6 (2)
62
8
2
3 (1)
31
4
y
(3, 6)
6 (2) 8
x
(1, 2)
(3, 2)
3 (1) 4
© 2001 McGraw-Hill Companies
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THE SLOPE OF A LINE
SECTION 6.4
521
The figure below compares the slopes found in the two previous examples. Line l1, from
1
Example 1, had slope . Line l2, from Example 2, had slope 2. Do you see the idea of slope
2
measuring steepness? The greater the slope, the more steeply the line is inclined upward.
y
l2
m2
l1
1
m2
x
CHECK YOURSELF 2
Find the slope of the line containing points with coordinates (1, 2) and (2, 7). Draw
a sketch of this line and the line of Check Yourself 1. Compare the lines and the two
slopes.
Let’s look at lines with a negative slope.
Example 3
Finding the Slope
Find the slope of the line containing points with coordinates (2, 3) and (1, 3).
By the definition,
m
3 3
6
2
1 (2)
3
y
(2, 3)
© 2001 McGraw-Hill Companies
m 2
x
(1, 3)
This line has a negative slope. The line falls as we move from left to right.
CHECK YOURSELF 3
Find the slope of the line containing points with coordinates (1, 3) and (1, 3).
CHAPTER 6
AN INTRODUCTION TO GRAPHING
We have seen that lines with positive slope rise from left to right and lines with negative
slope fall from left to right. What about lines with a slope of zero? A line with a slope of 0
is especially important in mathematics.
Example 4
Finding the Slope
Find the slope of the line containing points with coordinates (5, 2) and (3, 2).
By the definition,
22
0
0
3 (5)
8
m
y
m0
(3, 2)
(5, 2)
x
The slope of the line is 0. In fact, that will be the case for any horizontal line. Because any
two points on the line have the same y coordinate, the vertical change y2 y1 must always
be 0, and so the resulting slope is 0.
CHECK YOURSELF 4
Find the slope of the line containing points with coordinates (2, 4) and (3, 4).
Because division by 0 is undefined, it is possible to have a line with an undefined slope.
Example 5
Finding the Slope
Find the slope of the line containing points with coordinates (2, 5) and (2, 5).
By the definition,
m
5 (5)
10
22
0
Remember that division by zero is undefined.
y
(2, 5)
An undefined
slope
x
(2, 5)
© 2001 McGraw-Hill Companies
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THE SLOPE OF A LINE
SECTION 6.4
523
We say that the vertical line has an undefined slope. On a vertical line, any two points have
the same x coordinate. This means that the horizontal change x2 x1 must always be 0 and
because division by 0 is undefined, the slope of a vertical line will always be undefined.
CHECK YOURSELF 5
Find the slope of the line containing points with the coordinates (3, 5) and
(3, 2).
Given the graph of a line, we can find the slope of that line. Example 6 illustrates this.
Example 6
Finding the Slope from the Graph
Find the slope of the line graphed below.
y
x
We can find the slope by identifying any two points. It is almost always easiest to use the x
and y intercepts. In this case, those intercepts are (3, 0) and (0, 4).
Using the definition of slope, we find
m
0 (4)
4
30
3
4
The slope of the line is .
3
CHECK YOURSELF 6
© 2001 McGraw-Hill Companies
Find the slope of the line graphed below.
y
x
CHAPTER 6
AN INTRODUCTION TO GRAPHING
In Section 6.3, we saw that a line could be drawn from two ordered pairs. Given equations
of the form y kx, it is fairly easy to find two ordered pairs. In the next example, we will
use those ordered pairs to find the graph of the equation.
Example 7
Graphing an Equation of the Form y kx
(a) Find the graph of the equation y 2x.
From the table to the right, we know that the ordered pairs (0, 0) and (1, 2) are solutions
to the equation.
x
y
0
1
0
2
The graph is displayed below
y
x
Note that the slope of the line that passes through the points (0, 0) and (1, 2) is
m
0 (2)
2
2
01
1
1
x.
3
From the table to the right, we know that the ordered pairs (0, 0) and (3, 1) are solutions to
the equation.
(b) Find the graph of the equation y x
y
0
3
0
1
© 2001 McGraw-Hill Companies
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THE SLOPE OF A LINE
SECTION 6.4
525
The graph is displayed below
y
x
Note that the slope of the line that passes through the points (0, 0) and (3, 1) is
m
01
1
03
3
CHECK YOURSELF 7
1
Find the graph of the equation y x.
2
In Example 7, we noted that the slope of the line for the equation y 2x is 2, and the
1
1
slope of the line for the equation y x is . This leads us to the following observation.
3
3
The slope of a line for an equation of the form y kx will always be k. Because k is the
slope, we generally write the form as
y mx
Note that (0, 0) will be a solution for any equation of this form. As a result, the line for an
equation of the form y mx will always pass through the origin.
The following sketch summarizes the results of our previous examples.
y
The slope is undefined.
m is positive.
NOTE As the slope gets closer
x
© 2001 McGraw-Hill Companies
to 0, the line gets “flatter.”
m is 0.
m is negative.
Four lines are illustrated in the figure. Note that
1.
2.
3.
4.
The slope of a line that rises from left to right is positive.
The slope of a line that falls from left to right is negative.
The slope of a horizontal line is 0.
A vertical line has an undefined slope.
CHAPTER 6
AN INTRODUCTION TO GRAPHING
CHECK YOURSELF ANSWERS
1. m 2
3
2. m 5
3
y
(2, 7)
(5, 5)
(1, 2)
(2, 3)
x
3. m 3
7.
4. m 0
5. m is undefined
6. m 2
5
y
(Note: Your second point could have been
(2, 1) or (2, 1).)
x
© 2001 McGraw-Hill Companies
526
Name
6.4 Exercises
Section
Date
Find the slope of the line through the following pairs of points.
1. (5, 7) and (9, 11)
ANSWERS
2. (4, 9) and (8, 17)
1.
3. (2, 5) and (2, 15)
2.
4. (3, 2) and (0, 17)
3.
4.
6. (3, 4) and (3, 2)
5. (2, 3) and (3, 7)
5.
7. (3, 2) and (2, 8)
8. (6, 1) and (2, 7)
6.
7.
9. (3, 3) and (5, 0)
8.
10. (2, 4) and (3, 1)
9.
11. (5, 4) and (5, 2)
12. (5, 4) and (2, 4)
10.
11.
13. (4, 2) and (3, 3)
14. (5, 3) and (5, 2)
12.
13.
15. (3, 4) and (2, 4)
16. (5, 7) and (2, 2)
14.
15.
18. (4, 2) and (6, 4)
17. (1, 7) and (2, 3)
16.
In exercises 19 to 24, two points are shown. Find the slope of the line through the given
points.
17.
© 2001 McGraw-Hill Companies
18.
19.
20.
y
y
19.
20.
x
x
527
ANSWERS
21.
21.
22.
y
y
22.
23.
x
24.
x
25.
26.
27.
23.
24.
y
y
28.
x
x
In exercises 25 to 30, find the slope of the lines graphed.
26.
y
y
x
27.
28.
y
x
528
x
y
x
© 2001 McGraw-Hill Companies
25.
ANSWERS
29.
30.
y
y
29.
30.
31.
x
x
32.
33.
34.
35.
Find the graph of the following equations.
31. y 4x
36.
32. y 3x
y
y
x
33. y 2
x
3
x
3
4
34. y x
y
y
x
© 2001 McGraw-Hill Companies
35. y 5
x
4
x
4
5
36. y x
y
y
x
x
529
ANSWERS
37. (a)
(b)
37. Consider the equation y 2x 5.
(a) Complete the following table:
y
3
4
(c)
38. (a)
(b)
x
(b) Use the ordered pairs found in part (a) to calculate the slope of the line.
(c) What do you observe concerning the slope found in part (b) and the given
equation?
(c)
39. (a)
38. Repeat exercise 37 for y 3
x 5 and
2
(b)
(c)
40. (a)
x
2
4
1
3
39. Repeat exercise 37 for y x 2 and
x
40. Repeat exercise 37 for y 4x 6 and
(c)
x
y
1
3
41. (a)
(b)
y
3
6
(b)
(c)
y
41. Consider the equation: y 2x 3
(a) Complete the following table of values, and plot the resulting points.
(d)
(b)
(c)
(d)
43. (a)
(b)
(c)
(d)
Point
x
A
B
C
D
E
5
6
7
8
9
y
(b) As the x coordinate changes by 1 (for example, as you move from point A to
point B), how much do the corresponding y coordinates change?
(c) Is your answer to part (b) the same if you move from B to C? from C to D? from
D to E?
(d) Describe the “growth rate” of the line using these observations. Complete the
following statement: When the x value grows by 1 unit, the y value __________.
44. (a)
(b)
42. Repeat exercise 41 using: y 2x 5
(c)
(d)
43. Repeat exercise 41 using: y 4x 50
44. Repeat exercise 41 using: y 4x 40
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© 2001 McGraw-Hill Companies
42. (a)
ANSWERS
In the following exercises, (a) plot the given point; (b) using the given slope, move from
the point plotted in (a) to plot a new point; (c) draw the line that passes through the points
plotted in (a) and (b).
45.
46.
47.
45. (3, 1), m 2
46. (1, 4), m 2
y
48.
a.
y
b.
c.
x
x
d.
e.
f.
47. (2, 1), m 4
48. (3, 5), m 2
y
y
x
x
Getting Ready for Section 6.5 [Section 2.2]
© 2001 McGraw-Hill Companies
Solve each equation for x.
(a) 25 5x
(d) 14 3x
(b) 36 12x
(e) 24 9x
(c) 49 7x
(f ) 72 24x
Answers
1. 1
15. 0
3. 5
17. 5.
4
3
4
5
7. 2
19. 2
9. 21. 2
3
2
11. Undefined
23. 0
25. 4
13.
5
7
27. 5
29.
1
3
531
33. y 31. y 4x
2
x
3
y
y
x
35. y 5
x
4
x
37. (a) (3, 1), (4, 3); (b) 2; (c) slope equals coefficient of x
1
3
39. (a) (3, 1), (6, 0); (b) ; (c) slope equals
y
coefficient of x
41. (a) (5, 13), (6, 15), (7, 17), (8, 19), (9, 21);
(b) 2; (c) Yes; (d) increases by 2
x
43. (a) (5, 30), (6, 26), (7, 22), (8, 18), (9, 14);
(b) 4; (c) Yes; (d) decreases by 4
45.
47.
y
y
x
x
b. 3
c. 7
d.
14
3
e. 8
3
f. 3
© 2001 McGraw-Hill Companies
a. 5
532