SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE SLOW VEHICLES KEEP RIGHT

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68. 3x 2y 1500 67. x 2y 600

S L O P E O F A L I N E

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

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).

I n t h i s

s e c t i o n

● Slope

● Using Coordinates to Find Slope

● Parallel Lines

● Perpendicular Lines

● Applications of Slope

3.2

6 100

6%

GRADE

SLOW VEHICLES KEEP RIGHT

9 ft rise 12 ft

run

9–12 pitch

F I G U R E 3 . 1 0

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

Slope

Consider the line in Fig. 3.11(a) on the next page. In going from (0, 1) to (1, 3), there is a change of1 in the x-coordinate and a change of 2 in the y-coordinate,

rise run change in y-coordinate

change in x-coordinate

h e l p f u l h i n t

Since the amount of run is arbitrary, we can choose the run to be 1. In this case

slope ri 1 se rise.

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.

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or a run of 1 and a rise of 2. So the slope is²

1or 2. If we move from (1, 3) to (0, 1) as in Fig. 3.11(b) the rise is2 and the run is 1. So the slope is

2

1or 2. If we start at either point and move to the other point, we get the same slope.

E X A M P L E 1

Finding the slope from a graph

Find the slope of each line by going from point A to point B.

a) b) c)

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

slope 3 2.

b) In going from A(2, 1) to B(6, 3), we must rise 2 and run 4. So slope 2

4 1 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

slope

3

6 1

2.

■

y

1 x 2

– 2 – 3 5 4

– 2 – 1 – 3 – 4 – 5 – 6

1 A

B

3 y

1 2 3 4 5 6 x 2

– 2 – 3 5 4

– 1 1

B

A 3 y

1 3 x

2

– 2

4

– 3 5 4

– 1– 1 1

A

B 3

2

– 2 y

– 3 1 x

3

3 2

2 – 2

4 (0, 1)

+ 1 – 4

– 3 – 4 – 5 4 5

+ 2 (1, 3)

– 2 y

– 3 1 x

3

3 2

2 – 2

4 (0, 1)

– 1 – 4

– 3 – 4 – 5 4 5

– 2 (1, 3)

– 1 – 1

(a) (b)

F I G U R E 3 . 1 1

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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.

– 2 y

– 3 1 x

3

3 2

2 – 2 – 4

– 4 – 5 4 5

– 5

Rise

Run Hypotenuse

Rise

Run Hypotenuse – 1

F I G U R E 3 . 1 2

y

x (x₂, y₂)

(x₂, y₁) (x₁, y₁)

x₂ – x₁

y₂ – y₁ Rise

Run

F I G U R E 3 . 1 3

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 m , provided that x₂ x1 0.

E X A M P L E 2

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.

m

4

2 1 2 b) Let (x₁, y₁) (5, 1) and (x2, y₂) (2, 3):

m 4

3 3 (1)

2 (5) y₂ y1

x₂ x1

3 5

6 2 y₂ y1

x₂ x1

y₂ y1

x₂ x1

s t u d y t i p

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.”

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c) Let (x₁, y₁) (0, 0) and (x2, y₂) (6, 4):

m 2

3

■

Do not reverse the order of subtraction from numerator to denominator when finding the slope. If you divide y₂ y1by x₁ x2, you will get the wrong sign for the slope.

E X A M P L E 3

Slope for horizontal and vertical lines Find the slope of each line.

a) b)

Solution

a) Using (3, 2) and (4, 2) to find the slope of the horizontal line, we get m

0.

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 x1must be

nonzero, the slope is undefined for this line.

■

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.

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.

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

0

7 2 2

3 4

y

– 1 x – 2 – 2 3

1 2 4

– 1 3

4 5

(1, – 4) 2

5

– 3 – 4 – 5

(1, 2) y

– 1 x – 2 – 2 3

1 – 3 2

4

– 1 3

4 5 – 4 1

– 5

(– 3, 2) (4, 2)

4

6 4 0

6 0 C A U T I O N

C A U T I O N

h e l p f u l h i n t

Think about what slope means to skiers. No one skis on cliffs or even refers to them as slopes.

Zero slope

Small slope

Larger slope

Undefined slope

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left to right along a line with negative slope, the y-coordinates are decreasing. See Fig. 3.14.

1 – 2

y

– 1 2 x

– 1 3

– 2 3 1

– 3 4

4 y

2 x – 1 1

3

3 1

– 3 4

4

2

– 4 Increasing y-coordinates

– 4

Decreasing y-coordinates

Positive slope

Negative slope

F I G U R E 3 . 1 4

y

3 x 1

4

– 1 3

1 2

– 2

– 4 – 5 – 3

– 4 2 4

Slope^—¹₃

Slope—¹ 3

F I G U R E 3 . 1 5

Parallel Lines

Consider the two lines shown in Fig. 3.15. Each of these lines has a slope of¹

3, and 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

m

6

8 4 3.

Because line l is parallel to a line with slope ⁴₃, the slope of line l is⁴₃also.

■

Perpendicular Lines

The lines shown in Fig. 3.16 have slopes 2 and¹₂. 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.

5 3

4 (2)

– 2 y

1 3 x

1 2 – 2 – 1 4 (– 1, 3)

Slope 2 – 1

– 3 5

Slope –^—¹₂

F I G U R E 3 . 1 6

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Perpendicular Lines

Two lines with slopes m₁and m₂are perpendicular if and only if m₁

m 1

2

.

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

m .

Because line l is perpendicular to a line with slope³₇, the slope of line l is⁷

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.

m_AB m_BC

3

31

2

6 1 3

m_CD m_AD

1

33

2

6 1 3

Because the opposite sides have the same slope, they are parallel, and the figure is a parallelogram. Because¹

3is the opposite of the reciprocal of3, the intersecting 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).

2 4

3 3 1 4

4 3

1 1

2 4 2 (1)

3 (2)

3 7

3

7 1 (2)

4 3

y

1 3 5 x

1 5

– 3 – 5 – 3 – 1 A (– 3, 2)

C (4, 1) D (3, 4)

B (– 2, – 1)

F I G U R E 3 . 1 7

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a) Find and interpret the slope of the line in the accompanying figure.

b) Predict the amount of worldwide CO₂emissions in 2005.

Solution

a) Find the slope of the line through (1970, 14) and (1995, 24):

m 0.4

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 CO₂emissions will be

28 billion tons.

■

True or false? Explain your answer.

1. Slope is a measurement of the steepness of a line. True 2. Slope is run divided by rise. False

3. The line through (4, 5) and (3, 5) has undefined slope. False 4. The line through (2, 6) and (2, 5) has undefined slope. True 5. Slope cannot be negative. False

6. The slope of the line through (0, 2) and (5, 0) is 2

5. False 7. The line through (4, 4) and (5, 5) has slope ⁵

4. False

8. If a line contains points in quadrants I and III, then its slope is positive. True

9. Lines with slope²

3and²₃are perpendicular to each other. False 10. Any two parallel lines have equal slopes. False

24 14

1995 1970

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).

W A R M - U P S

CO2 emission (in billions of tons) 24

14

1970 1995

Year

F I G U R E F O R E X A M P L E 7

s t u d y t i p

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.

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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.

E X E R C I S E S

3 . 2

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 perpendicu- lar lines?

If m₁and m₂are the slopes of perpendicular lines, then m₁ ^.

6. What is the relationship between the slopes of parallel lines?

If m₁and m₂are the slopes of parallel lines, then m₁ m2.

1m₂

Determine the slope of each line. See Example 1.

7. 8. 9.

2

3 2

3 Undefined

10. 11. 12.

0 1 1

13. 14. 15.

3

2 3 1

y

3 x 1

4

– 1 3

1

– 3 – 1

– 3– 2 4

– 2 – 4

5

2 y

3 x 1

3 4

– 1 1 – 2

– 4

– 5 2

– 3 – 2 – 3 – 4 y

3 x 1

4

– 1 3

– 1 1 2

– 3– 2 4 5

– 2 – 3

y

3 x 4

3

– 4 – 3

– 3– 2 4

– 2 – 4 2

2 1

1 – 1 – 1 y

3 x 4

– 1 3

– 4 – 3 – 1

– 3– 2 4

– 2 – 4 2

2 1

1 y

3 x 1

4

– 1 3

1

– 4 – 3 – 1

– 3– 2 4

– 2 – 4 2

y

3 x 1

4

– 1 3

1

– 4 – 3 – 1

2

– 3– 2 4 5

– 2 y

3 x 1

4

– 1 3

1

– 4

– 5 2

– 3

– 4 – 1

2

– 3 y

3 x 1

4

– 1 3

1 – 2

– 4

– 5 2

– 3 – 2 – 1 – 3

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16.

1 2

Find the slope of the line that contains each of the following pairs of points. See Examples 2 and 3.

17. (2, 6), (5, 1) 18. (3, 4), (6, 10) 2 19. (3, 1), (4, 3) 4

7 20. (2, 3), (1, 3) 2 21. (2, 2), (1, 7) 5 22. (3, 5), (1, 6) 23. (3, 5), (0, 0) 24. (0, 0), (2, 1) 25. (0, 3), (5, 0)

26. (3, 0), (0, 10) 27.

³⁴^{, 1}

^,

¹²^,¹²

28.

¹²^{, 2}

^,

⁴¹^,¹²

⁶

29. (6, 212), (7, 209) 3 30. (1988, 306), (1990, 315) 31. (4, 7), (12, 7) 0 32. (5, 3), (9, 3) 0 33. (2, 6), (2, 6) Undefined 34. (3, 2), (3, 0) Undefined 35. (24.3, 11.9), (3.57, 8.4) 0.169 36. (2.7, 19.3), (5.46, 3.28) 2.767 37.

⁴^{, 1}

^,

²^{, 0}

^1.273

38.

³^{, 1}

^,

⁶^{, 0}

^1.910

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

9 2 2 5 10

3 3 5

1 2 5

3

11 4 5

3 y

3 x 1

4

– 1 3

1

– 3 – 1 – 3 – 2

– 2

– 4 2

5

4

41. Line l goes through (2, 5) and is parallel to the line through (3, 2) and (4, 1). 3

7

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

5. Both lines contain the origin. ⁵₄

44. Line l is perpendicular to a line with slope5. 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 Exer- cise 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

93 94 95 96 97 98 99 00 20

21 22

Year

List price (in thousands of dollars)

(1993, 20,115)

(1998, 21,135)

F I G U R E F O R E X E R C I S E 5 1

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).

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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

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 know- ing 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.

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

2, perpendicular

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, ¹₂, perpendicular

G R A P H I N G C A LC U L ATO R E X E RC I S E S

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 decreas- ing 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 are1, 2, 3, and 4.

I n t h i s

s e c t i o n

● Point-Slope Form

● Slope-Intercept Form

● Standard Form

● Using Slope-Intercept Form for Graphing

● Linear Functions

T H R E E F O R M S F O R T H E

E Q U A T I O N O F A L I N E

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 slope²

3 and contains the point (3, 5). In Sec- tion 3.2 you learned that the slope is the same no matter which two points of the line

3.3

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

10 15

Age (in years)

Selling price (in thousands of dollars)

(3, 11,095) (1, 13,595)

F I G U R E F O R E X E R C I S E 5 2 M I S C E L L A N E O U S

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 slope²

3, then what is the value of y on this line when x 8, x 11, and x 12?

4, 6, 6

55. Find k so that the line through (2, k) and (3, 5) has slope¹

2. ⁵₂

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

G E T T I N G M O R E I N VO LV E D

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.

2 3