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

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