4.2 the slope of a line - mcgraw hill higher … objectives 1. find the slope of a line 2. find the...

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207

The Slope of a Line4.2

4.2 OBJECTIVES

1. Find the slope of a line2. Find the slopes of parallel and perpendicular lines3. Find the slope of a line given an equation4. Find the slope given a graph5. Graph a linear equation using the slope and

a point

On the coordinate system below, plot a point, any point.

How many different lines can you draw through that point? Hundreds? Thousands?Millions? Actually, there is no limit to the number of different lines that pass through thatpoint.

Now, on the coordinate system above, plot a second point.How many different lines can you draw through those points? Only one! The two points

were enough to define the line.In Section 4.3, we will see how we can find the equation of a line if we are given two of

its points. The first part of finding that equation is finding the slope of the line, which is away of describing the steepness of a line.

The Definitions box contains a formula for slope. First, we choose any two distinctpoints on the line, say, P with coordinates (x1, y1) and Q with coordinates (x2, y2). As wemove along the line from P to Q, the x value, or coordinate, changes from x1 to x2. Thatchange in x, also called the horizontal change, is x2 � x1. Similarly, as we move from P toQ, the corresponding change in y, called the vertical change, is y2 � y1. The slope is thendefined as the ratio of the vertical change to the horizontal change. The letter m is used torepresent the slope, which we now define.

y

x2 4 6 8�2�4�6�8

2

4

6

8

�2

�4

�6

�8

208 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS

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The slope of a line through two distinct points P(x1, y1) and Q(x2, y2) is given by

when x1 � x2.

Change in yy2 � y1

Change in xx2 � x1

(x2, y1)

Q(x2, y2)

P(x1, y1)

y

x

L

m � change in ychange in x

� y2 � y1

x2 � x1

Definitions: Slope of a LineNOTE The difference, x2 � x1,is often called the run. Thedifference, y2 � y1, is the rise.So the slope can be thought ofas “rise over run.”

Note that x1 � x2 or x2 � x1 � 0 ensures that thedenominator is nonzero, so thatthe slope is defined. It alsomeans the line cannot bevertical.

Let’s look at some examples using the definition.

Example 1

Finding the Slope Through Two Points

Find the slope of the line through the points (�3, 2) and (3, 5).Let (x1, y1) � (�3, 2) and (x2, y2) � (3, 5). From the definition we have

Note that if the pairs are reversed, so that

(x1, y1) � (3, 5) and (x2, y2) � (�3, 2)

then we have

The slope in either case is the same.

m �2 � 5

�3 � 3�

�3

�6�

1

2

m �5 � 2

3 � (�3)�

3

6�

1

2

y

x

(3, 5)

(�3, 2)

NOTE The work here suggeststhat no matter which point ischosen as (x1, y1) or (x2, y2), theslope formula will give thesame result. Simply stay withyour choice once it is made, anduse the same order ofsubtraction in the numeratorand the denominator.

C H E C K Y O U R S E L F 1

Find the slope of the line through the points (�2, �1) and (1, 1).

The slope indicates both the direction of a line and its steepness. First, we will comparethe steepness of the lines in two examples.

THE SLOPE OF A LINE SECTION 4.2 209©

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Example 2

Finding the Slope

Find the slope of the line through (�2, �3) and (2, 5).Again, by equation (1),

Compare the lines in Examples 1 and 2. In Example 1 the line has slope . The slope here

is 2. Now look at the two lines. Do you see the idea of slope as measuring steepness? Thegreater the absolute value of the slope, the steeper the line.

1

2

m �5 � (�3)

2 � (�2)�

8

4� 2

The sign of the slope indicates in which direction the line tilts, as Example 3 illustrates.


Find the slope of the line through the points (�1, 2) and (2, 7). Draw a sketch of this line and the line in the Check Yourself 1 exercise on the same coordinate axes.Compare the lines and the two slopes.

y

x

(2, 5)

(�2, �3)

Example 3

Finding the Slope

Find the slope of the line through the points (�1, 2) and (4, �3).We see that

Now the slope is negative.Comparing this with our previous examples, we see that

1. In Examples 1 and 2, the lines were rising from left to right, and the slope waspositive.

m ��3 � 2

4 � (�1)�

�5

5� �1

y

x(�1, 2)

(4, �3)


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Example 4

Finding the Slope of a Horizontal Line

Find the slope of the line through (�2, 3) and (5, 3).

The slope of the line is 0. Note that the line is parallel to the x axis and y2 � y1 � 0. Theslope of any horizontal line will be 0.

m �3 � 3

5 � (�2)�

0

7� 0

y

x

(�2, 3) (5, 3)

y

x

(1, �3)

(1, 4)


Find the slope of the line through the points (�2, �4) and (3, �4).

Example 5

Finding the Slope of a Vertical Line

Find the slope of the line through the points (1, �3) and (1, 4).

Here the line is parallel to the y axis, and x2 � x1 (the denominator of the slope formula) is0. Because division by 0 is undefined, we say that the slope is undefined, as will be the casefor any vertical line.

m �4 � (�3)

1 � 1�

7

0, which is undefined.


Find the slope of the line through the points (�2, 5) and (4, �1).

Let’s continue by looking at the slopes of lines in two particular cases.

2. In this example, the line is falling from left to right, and the slope is negative.


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Find the slope of the line through the points (2, �3) and (2, 7).

Here is a summary of our work in the previous examples.

1. If the slope of a line is positive, the line is rising from left to right.

2. If the slope of a line is negative, the line is falling from left to right.

3. If the slope of a line is 0, the line is horizontal.

y

L3

x

m is zero

y

L2

x

m isnegative

y

L1

x

m ispositive

Be very careful not to confuse a slope of 0 (in the case of a horizontal line) with anundefined slope (in the case of a vertical line).


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Are lines L1 through (�2, �1) and (1, 4) and L2 through (�3, 4) and (0, 8) parallel, or do they intersect?

For nonvertical lines L1 and L2, if line L1 has slope m1 and line L2 has slope m2,then

L1 is parallel to L2 if and only if m1 � m2

Note: All vertical lines are parallel to each other.

Definitions: Slopes of Parallel LinesNOTE This means that if thelines are parallel, then theirslopes are equal. Conversely, ifthe slopes are equal, then thelines are parallel.

Mathematicians use thesymbol ⇔ to represent “if andonly if.”

Example 6

Parallel Lines

Are lines L1 through (2, 3) and (4, 6) and L2 through (�4, 2) and (0, 8) parallel, or do theyintersect?

Because the slopes of the lines are equal, the lines are parallel. They do not intersect.

m2 �8 � 2

0 � (�4)�

6

4�

3

2

m1 �6 � 3

4 � 2�

3

2

NOTE Unless, of course, L1 andL2 are actually the same line. Inthis case, a quick sketch willshow that the lines are distinct.

4. If the slope of a line is undefined, the line is vertical.

There are two more important results regarding the slope. Recall from geometry thattwo distinct lines in the plane either intersect at a point or never intersect. Two lines in theplane that do not intersect are called parallel lines. It can be shown that two distinct paral-lel lines will always have the same slope, and we can state the following.

y

L4

x

m isundefined


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Are lines L1 through points (1, 3) and (4, 1) and L2 through points (�2, 4) and (2, 10)perpendicular?

For nonvertical lines L1 and L2, if line L1 has slope m1 and line L2 has slope m2,then

L1 is perpendicular to L2 if and only if m1 � �

or, equivalently,

Note: Horizontal lines are perpendicular to vertical lines.

m1 � m2 � �1

1m2

Definitions: Slopes of Perpendicular Lines

Example 7

Perpendicular Lines

Are lines L1 through points (�2, 3) and (1, 7) and L2 through points (2, 4) and (6, 1)perpendicular?

Because the slopes are negative reciprocals of each other, the lines are perpendicular.

L1

L2

x

(�2, 3) (2, 4)

(6, 1)

(1, 7)

ym1 �

43

m2 � � 34

m2 �1 � 4

6 � 2� �

3

4

m1 �7 � 3

1 � (�2)�

4

3

Two lines are perpendicular if they intersect at right angles. Also, if two lines (which arenot vertical or horizontal) are perpendicular, their slopes are the negative reciprocals ofeach other. We can then state the following result for perpendicular lines.

NOTE

�43��

34� � �1


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Example 8

Finding the Slope from an Equation

Find the slope of the line with equation 3x � 2y � 6.First, find any two points on the line. In this case, (2, 0) and (0, 3), the x and y intercepts,

will work and are easy to find. From the slope formula,

The slope of the line with equation 3x � 2y � 6 is �3

2.

m �0 � 3

2 � 0�

�3

2� �

3

2


Find the slope of the line with equation 3x � 4y � 12.

We can find the slope of a graphed line by identifying two points on the line. We will usethat technique in Example 9.

NOTE Let’s try solving theoriginal equation for y:

3x � 2y � 6

2y � �3x � 6

Consider the coefficient of x.What do you observe?

y � �32

x � 3

Example 9

Finding the Slope from a Graph

Determine the slope of the line from its graph.

We can choose any two points to determine the slope of the line. Here we pick (3, 6) and(�2, �4). We find the slope by the usual method.

The slope of the line is 2.

m �6 � (�4)

3 � (�2)�

10

5� 2

y

x

(3, 6)

(�2, �4)

Given the equation of a line, we can also find its slope, as Example 8 illustrates.


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The slope of a line can also be useful in graphing a line. In Example 10, the slope of aline is used in sketching its graph.

Example 10

Graphing a Line with a Given Slope

Suppose a line has slope and passes through the point (5, 2). Graph the line.

First, locate the point (5, 2) in the coordinate system. Now, because the slope, is the

ratio of the change in y to the change in x, move 2 units to the right in the x direction andthen 3 units up in the y direction. This determines a second point, here (7, 5), and we candraw our graph.

x

y

2 units right

3 units up

(7, 5)

(5, 2)

3

2,

3

2

C H E C K Y O U R S E L F 1 0

Graph the line with slope that passes through the point (2, 3). Hint: Consider

the x change as 4 units and the y change as �3 units (down).

�34

Because, given a point on a line and its slope, we can graph the line, we also should beable to write its equation. That is, in fact, the case, as we will see in Section 4.3.


Determine the slope of the line from its graph.

y

x


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C H E C K Y O U R S E L F A N S W E R S

1. 2. . This line is steeper than the line of check yourself 1.

3. m � �1 4. m � 0 5. Undefined

6. The lines intersect. 7. The lines are perpendicular. 8.

9. m � 10.

3 units down

4 units right

(2, 3)

y

x(6, 0)

1

2

m �3

4

m �5

3m �

2

3

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Exercises

In exercises 1 to 12, find the slope (if it exists) of the line determined by the followingpairs of points. Sketch each line so that you can compare the slopes.

1.(2, 3) and (4, 7) 2. (�1, 2) and (5, 3) 3. (2, �3) and (�2, �5)

4. (0, 0) and (5, 7) 5. (2, 5) and (�3, 5) 6. (�2, �4) and (5, 3)

7.(�1, 4) and (�1, 7) 8. (4, 2) and (�2, 5) 9. (8, �3) and (�2, �5)

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

4.2

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

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10. (4, �3) and (�2, 7) 11. (�4, �3) and (2, �7) 12. (3, 6) and (3, �4)

In exercises 13 to 20, find the slope of the line determined by each equation.

13. 14.

15. 16.

17. 2x � 3y � 6 18. x � 4y � 4

19. 3x � 4y � 12 20. x � 3y � 9

In exercises 21 to 26, are the pairs of lines parallel, perpendicular, or neither?

21. L1 through (�2, �3) and (4, 3); L2 through (3, 5) and (5, 7)

22. L1 through (�2, 4) and (1, 8); L2 through (�1, �1) and (�5, 2)

23. L1 through (8, 5) and (3, �2); L2 through (�2, 4) and (4, �1)

24. L1 through (�2, �3) and (3, �1); L2 through (�3, 1) and (7, 5)

25. L1 with equation x � 3y � 6; L2 with equation 3x � y � 3

26. L1 with equation x � 2y � 4; L2 with equation 2x � 4y � 5

2y � 3x � 5 � 0y �1

2x � 2

y �1

4x � 3y � �3x �

1

2

y

x

y

x

y

x

ANSWERS

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

218

27. Find the slope of any line parallel to the line through points (�2, 3) and (4, 5).

28. Find the slope of any line perpendicular to the line through points (0, 5) and(�3, �4).

29. A line passing through (�1, 2) and (4, y) is parallel to a line with slope 2. What is thevalue of y?

30. A line passing through (2, 3) and (5, y) is perpendicular to a line with slope . Whatis the value of y?

If points P, Q, and R are collinear (lie on the same line), the slope of the line through Pand Q must equal the slope of the line through Q and R. In exercises 31 to 36, use theslope concept to determine whether the sets of points are collinear.

31. P(�2, �3), Q(3, 2), and R(4, 3) 32. P(�5, 1), Q(�2, 4), and R(4, 9)

33. P(0, 0), Q(2, 4), and R(�3, 6) 34. P(�2, 5), Q(�5, 2), and R(1, 12)

35. P(2, 4), Q(�3, �6), and R(�4, 8) 36. P(�1, 5), Q(2, �4), and R(�2, 8)

In exercises 37 to 44, graph the lines through each of the specified points having the givenslope.

37. (0, 1), m � 3 38. (0, �2), m � �2 39. (3, �1), m � 2

y

x

y

x

y

x

3

4

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27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

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40. (2, �3), m � �3 41. (2, 3), 42. (�2, 1),

43. (4, 2), m � 0 44. (3, 0), m is undefined

45. On the same graph, sketch lines with slope 2 through each of the following points:(�1, 0), (2, 0), and (5, 0).

46. On the same graph, sketch one line with slope and one line with slope �3, havingboth pass through point (2, 3).

y

x

1

3

y

x

y

x

y

x

y

x

y

x

y

x

m � �3

4m �

2

3

ANSWERS

40.

41.

42.

43.

44.

45.

46.

220

A four-sided figure (quadrilateral) is a parallelogram if the opposite sides have the sameslope. If the adjacent sides are perpendicular, the figure is a rectangle. In exercises 47 to50, for each quadrilateral ABCD, determine whether it is a parallelogram; then determinewhether it is a rectangle.

47. A(0, 0), B(2, 0), C(2, 3), D(0, 3)

48. A(�3, 2), B(1, �7), C(3, �4), D(�1, 5)

49. A(0, 0), B(4, 0), C(5, 2), D(1, 2)

50. A(�3,�5),B(2,1),C(�4, 6),D(�9,0)

In exercises 51 to 54, solve each equation for y, then use your graphing utility to grapheach equation.

51. 2x � 5y � 10 52. 5x � 3y � 12

53. x � 7y � 14 54. �2x � 3y � 9

4

2

�2

�4

�2�4 2 4

4

2

�2

�4

�2 2 4 6

4

2

�2

�4

�2�4 2 4

4

2

�2

�4

�2�4 2 4

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47.

48.

49.

50.

51.

52.

53.

54.

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In exercises 55 to 62, use the graph to determine the slope of the line.

55. 56.

57. 58.

59. 60.

61. 62. y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

ANSWERS

55.

56.

57.

58.

59.

60.

61.

62.

222

63. Consider the equation y � 2x � 3.

(a) Complete the following table of values, and plot the resulting points.

(b) As the x coordinates change by 1 (for example, as you move from point A topoint B), by how much does the corresponding y coordinate change?

(c) Is your answer to part (b) the same if you move from B to C? from C to D? fromD to E?

(d) Describe the “growth rate” of the line using these observations. Complete thefollowing statement: When the x value grows by 1 unit, the y value ________________________.

64. Repeat exercise 63 using y � 2x � 5.



67. Repeat exercise 63 using y � �4x � 50.

68. Repeat exercise 63 using y � �4x � 40.

69. Summarize the results of exercises 63 to 68. In particular, how does the concept of“growth rate” connect to the concept of slope?

70. Consumer affairs. In 1995, the cost of a soft drink was 75¢. By 1999, the cost ofthe same soft drink had risen to $1.25. During this period, what was the rate ofchange of the cost of the soft drink? (Hint: Assume a linear growth rate for the priceas the years go by. Find the slope of the line.)

Point x y

A 5B 6C 7D 8E 9

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63. (a)

(b)

(c)(d)

64. (a)

(b)(c)(d)

65. (a)

(b)(c)(d)

66. (a)

(b)(c)(d)

67. (a)

(b)(c)(d)

68. (a)

(b)(c)(d)

69.

70.

223

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71. Science. On a certain February day in Philadelphia, the temperature at 6:00 A.M. was10°F. By 2:00 P.M. the temperature was up to 26°F. What was the average hourly rateof temperature change? (Hint: Assume a linear growth rate for the temperature as thehours go by. Find the slope.)

72. Construction. The rise-to-run ratio used to describe the steepness of the roof on acertain house is 4 to 12. Determine the maximum height of the attic if the house is32 ft wide.

1101009080706050403020100

–10–20

1101009080706050403020100

–10–20

ANSWERS

71.

72.

224

Answers

1. 2 3.

5. 0 7. Undefined

9. 11.

13. �3 15. � 17. 19. 21. Parallel

23. Neither 25. Perpendicular 27. 29. 12

31. Collinear 33. Not collinear 35. Not collinear

1

3

�3

4

2

3

1

2

(4, 7)

y

x

(2, �7)

(�4, �3)

(4, 7)

y

x

(8, �3)

(�2, �5)

�2

3

1

5

(4, 7)

y

x

(�1, 7)

(�1, 4)

(4, 7)

y

x

(2, 5)(�3, 5)

(4, 7)

y

x

(2, �3)(�2, �5)

(4, 7)

y

x

(2, 3)

(4, 7)

1

2

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225

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37. 39.

41. 43.

45. 47. Parallelogram, rectangle49. Parallelogram, not a rectangle

51. 53.

55. 2 57. �3 59. 3 61. �363. (a) y values: 13, 15, 17, 19, 21; (b) y increases by 2; (c) yes; (d) grows by 2 units 65. (a) y values: 13, 16, 19, 22, 25; (b) y increases by 3; (c) yes; (d) grows by 3 units 67. (a) y values: 30, 26, 22, 18, 14; (b) y decreases by 4; (c) yes; (d) drops by 4 units 69. 71. 2°/hour

4

�

�

��2 2 4 6

4

�

�

�2 2 4

y � �1

7x � 2y � �

2

5x � 2

(4, 7)

y

x

(4, 7)

y

x

(4, 7)

y

x

y

xx

y

226

4.2 the slope of a line - mcgraw hill higher … objectives 1. find the slope of a line 2. find the...

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