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
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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
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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)
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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)
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THE SLOPE OF A LINE
SECTION 6.4
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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
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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
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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
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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
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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.
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18. 19.
20.
y
y
19. 20. x
x
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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
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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 40 530
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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]
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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
