Section 4.4 The Slope Formula

Transcription

1 Section. The Slope Formula Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Find the slope of a line through Placing values into formulas (1.8) two points b counting rise and run. Plotting points in the --plane (.1) Create the equation of a graphed line. The -intercept point (.3) Find the slope of a line using the The slope of a line (.3) slope formula. Create the equation of a line using the slope formula. INTRODUCTION In Section.3, from a single point, we used the slope of the line to locate other points on the line. We begin this section b seeing points alread on a line and using the idea of rise and run to find the slope of the line. We must keep three things in mind when counting rise and run: 1. The rise: counting upward is a positive rise and downward is a negative rise.. The run: counting to the right is a positive run, and to the left is a negative run. 3. The slope is alwas rise over run, m = rise run, so count the rise first and then the run. THE SLOPE OF A LINE FROM A GRAPH If we know two points on the graph of a line, we can find the slope of the line b starting at one point and counting spaces toward the other. We first count the rise up or down and then the run left or right. For eample, if a line contains the points (1, ) and (5, ), then we can either start at (1, ) and count toward (5, ) or we can start at (5, ) and count toward (1, ). In either case the slope is the same. A. Start at (1, ) 1. The rise is up. The run is right m = rise run = up right = + + = 1 1 up right End (5, ) (1, ) 8 Start The Slope Formula Robert Prior, 010 page. - 1

2 B. Start at (5, ) 1. The rise is down. The run is left m = rise run = down left = - - = 1 End (1, ) Start (5, ) 1 down 8 left Eample 1: Procedure: Given the graph of the line, find the slope b counting the rise and the run. Start at either point. Count the rise first and then the run. Then write the slope as rise and simplif. run (-7, ) (-3, 1) Answer: (Two answers are shown for this eample, but onl one is necessar.) One option: A second option: Start at (-3, 1): Start at (-7, ): (-7, ) left (-3, 1) up (-7, ) down 3-8 right (-3, 1) - m = up 3 left = +3 - = - 3 m = down 3 right = -3 + = - 3 The Slope Formula Robert Prior, 010 page. -

3 YTI 1 Given the graph of the line, find the slope b counting the rise and the run. Use Eample 1 as a guide. a) b) (, 5) (-, 1) (-3, ) (7, -) GRAPHING A LINE FROM ITS INTERCEPTS If we know both the - and -intercept points, then we can plot these points and find the slope b counting the rise and run. We can then use the slope to find one or two more points on the line and graph the line. Eample : Given the - and -intercept points, plot them and identif the slope. Use the slope to find two more points on the line and graph the line. a) (3, 0) and (0, -) b) (-, 0) and (0, -3) Procedure: Plot the points and count the rise and run to identif the slope. a) Counting from the -intercept point, b) Counting from the -intercept point, the rise is up and the run is right 3. the rise is up 3 and the run is left. The slope is + +3, and we can use +3 The slope is -, and we can use m = 3 = up - right 3 and m = -3 = down left 3 m = 3 - = up 3 left and m = -3 = down 3 right to locate other points on the line. to locate other points on the line. The Slope Formula Robert Prior, 010 page. - 3

4 (3, 0) (, ) (0, -) (-3, -) - - (-8, 3) (-, 0) (0, -3) - (, -) - YTI Given the - and -intercept points, plot them and identif the slope. Use the slope to find two more points on the line and graph the line. Use Eample as a guide. a) (1, 0) and (0, 3) b) (-, 0) and (0, ) CREATING THE EQUATION FOR THE GRAPH For most lines, if we can see the graph of a line, then we can write the equation that goes with that line. The ke is identifing the -intercept (0, b) and the slope, m; then we use the form = m + b to write the equation. The Slope Formula Robert Prior, 010 page. -

5 Eample 3: Given the graph of the line, identif its slope and -intercept, and use them to write the equation of the line. b) (0, 3) (5, 1) (0, -) (-, -) - - Procedure: Identif the -intercept point, (0, b), giving us the value of b. Net, identif the slope b counting the rise and run, starting with the -intercept point; simplif m. Write the equation of the line using the slope-intercept form = m + b. a) The -intercept point is (0, -), so b = -. b) The -intercept point is (0, 3), so b = 3. Counting from (0, -) to (-, -), we see a rise of - and a run of -, so Counting from (0, 3) to (5, 1) we see a rise of - and a run of 5, so m = - - =. m = - 5 = - 5. The equation of the line is The equation of the line is Answer: = = The Slope Formula Robert Prior, 010 page. - 5

6 YTI 3 Given the graph of the line, identif its slope and -intercept, and use them to write the equation of the line. Use Eample 3 as a guide. b) (8, 5) (0, ) (0, -1) - (, -) c) d) (-, 1) (0, 3) (0, -) - (, -) - THE SLOPE FORMULA Recall from Section.3 the distinction between distance and difference. Both are found using subtraction, but distance is alwas a positive measure, whereas difference can be positive or negative. Consider this eample: Dana drove from her home in California to her college in Missouri. At the start of the trip, the car s odometer (mileage indicator) read and at the end of the trip it read How far did Dana drive on this trip? We calculate the distance b subtraction: end mileage start mileage = 31,877 30,53 = 1, miles. The Slope Formula Robert Prior, 010 page. -

7 This idea of subtracting the starting value from the ending value can be etended to the rise and run differences in the slope of a line. The rise is the difference of -values, end start, and the run is the difference of -values, end start. This means that we can write the slope as m = rise run = end start end start As these two diagrams demonstrate, either point can be the starting point. end Rise: start Start end Run: start End End end Run: start Start end Rise: start However, instead of subscripts start and end, we usuall use 1 and to represent the first point ( 1, 1 ) and the second point (, ). This leads us to the slope formula: The Slope Formula: The slope of the line that passes through ( 1, 1 ) and (, ) is m = 1 1 For eample, if we want to find the slope of the line that passes through the points (3, ) and (7, 8), then we can designate either one to be the first point: We ll designate which is the first point and which is the second b placing ( 1, 1 ) over one point and (, ) over the other point. Choosing ( 1, 1 ) (, ) ( 3, ) and ( 7, 8) the slope formula is m = 1 1 = = = 3. Instead, choosing ( 1, 1 ) (, ) ( 7, 8) and ( 3, ) the slope formula is m = 1 1 = = - - = 3. Notice that it doesn t matter which point we choose as the first point, the slope is the same. The Slope Formula Robert Prior, 010 page. - 7

8 Eample : Find the slope of the line that passes through the given points and simplif. a) (-5, ) and (-1, ) b) (1, -5) and (-, ) Procedure: Use the slope formula. It doesn t matter which point is chosen as the first point and which as the second, either choice will lead to the same slope. a) Let s choose (-5, ) as the first point b) Let s choose (-, ) as the first point and (-1, ) as the second point: and (1, -5) as the second point: m = -1 (-5) = = = 1 m = -5 1 (-) = = -9 3 = -3 YTI Find the slope of the line that passes through the given points and simplif. Use Eample as a guide. a) (1, ) and (5, 10) b) (, -) and (-, 8) c) (-, 3) and (0, -7) d) (-, 0) and (0, ) The Slope Formula Robert Prior, 010 page. - 8

9 Think about it 1 Given two points, (, -10) and (-0, 15) decide whether the slope formula is set up correctl or not. If not, state what the error is. a) m = (-0) b) m = (-0) c) m = d) m = 15 (-10) (-0) -10 We generall use the slope formula when we can t count to get the slope, because either 1. the points are not alread plotted in the --plane, or. one or both points have coordinates that are outside of the grid region that we have been using. Caution: It can be eas to misplace some of the numbers within the slope formula. One strateg that might be helpful is to align the ordered pairs, one above the other, as shown in the diagram. Then place m = below this alignment and prepare to correctl place the numbers in the formula. The preparation (, ) (7, 8) (3, ) Placing the values correctl (, ) (7, 8) (3, ) m = m = 8! 7! 3 The Slope Formula Robert Prior, 010 page. - 9

10 YTI 5 Find the slope of the line that passes through the given points and simplif. Use Eample as a guide. a) (0, 0) and (-3, ) b) (-1, 3) and (0, -5) c) (-11, -10) and (3, ) d) (-13, -1) and (-17, -11) CREATING THE EQUATION OF A LINE USING THE SLOPE FORMULA As ou saw earlier in this section, if we have the right information, we can easil write the equation of the line. In particular, if we know the -intercept point, (0, b), of a line, and if we can find the slope, m, then we can use the slope-intercept form of a line, = m + b, to write the equation. Eample 5: Find the equation of the line that passes through the given points. a) (0, 0) and (1, -5) b) (3, 1) and (0, -3) Procedure: Notice that the -intercept point, (0, b), is one of the given points in the pair. This means that we have the value of b. We can use the slope formula to find m, and place these values into = m + b. a) The -intercept point is (0, 0), so b = 0. b) The -intercept point is (0, -3), so b = -3. Find the slope: m = = -5 1 Find the slope: = -5 m = = - -3 = 3 The equation of the line is The equation of the line is Answer: = , or = -5 = 3 3 The Slope Formula Robert Prior, 010 page. - 10

11 YTI Find the equation of the line that passes through the given points. Use Eample 5 as a guide. a) (0, -5) and (, 3) b) (0, 8) and (1, 0) c) (-, ) and (0, ) d) (0, 0) and (-, -) Answers: You Tr It and Think About It YTI 1: a) m = = 3 b) m = - 10 = - 5 YTI : Some points shown ma be different from ours. a) m = - 3 b) m = 1 1 (-1, ) (0, 3) (, ) (1, 0) (-, 0) (0, ) (, -3) (-8, -) YTI 3: a) = -3 + b) = 3 1 c) = d) = - 1 The Slope Formula Robert Prior, 010 page. - 11

12 YTI : a) m = 3 b) m = -1 c) m = d) m = 3 YTI 5: a) m = - b) m = - 1 c) m = 7 d) m = - 1 YTI : a) = 5 b) = c) = d) = 3 Think About It: 1. a) is not correct; the difference in -values is in the numerator. b) is correct. c) is correct. d) is not correct; both the numerator and denominator is a difference between an -value and a -value. Section. Eercises Think Again. Given two points, (-8, -9) and (13, -18) decide whether the slope formula is set up correctl or not. If not, state what the error is. (Refer to Think About It 1) 1. m = -9 (-18) m = (-8) 3. m = -9 (-18) 13 (-8). m = 13 (-8) -18 (-9) Focus Eercises. Given the graph of each line, find the slope b counting the rise and the run. 5. Line A. Line B 7. Line C 8. Line D (-, 3) A (, ) (, ) (-, -3) - - B (-3, 5) C (7, -) (-7, -5) - - (1, -) D The Slope Formula Robert Prior, 010 page. - 1

13 9. Line E 10. Line F 11. Line G 1. Line H F (1, ) H (, ) E (-8, 1) (8, ) (, -5) G (-7, -) (7, -3) (3, -) Given the - and -intercept points, plot them and identif the slope. Use the slope to find two more points on the line and graph the line. 13. (-3, 0) and (0, ) 1. (, 0) and (0, -) 15. (0, ) and (, 0) 1. (0, -3) and (-1, 0) 17. (-5, 0) and (0, 5) 18. (0, -) and (, 0) Given the graph of the line, identif its slope and -intercept, and use them to write the equation of the line. 19. Line A 0. Line B 1. Line C. Line D A (0, 1) (-3, -) - - (, -) (0, -) B C (0, ) (5, ) (-7, ) (0, -3) - - D The Slope Formula Robert Prior, 010 page. - 13

14 3. Line E. Line F 5. Line G. Line H E F (0, 5) H (-, ) (-, ) (8, 1) 8 (0, -) G (-8, 1) (0, 3) 8 (0, -5) Use the slope formula to find the slope of the line that passes through the given points. Simplif, if possible. 7. (1, ) and (, 5) 8. (9, 8) and (, ) 9. (, -1) and (, 3) 30. (5, -) and (, 0) 31. (5, 3) and (9, -5) 3. (-, 7) and (7, 10) 33. (0, -) and (-, 0) 3. (-, 1) and (-7, -1) 35. (-3, -) and (0, 0) 3. (9, -) and (-3, -) 37. (, -5) and (, ) 38. (-1, -3) and (-5, -9) 39. (-, 0) and (, 3) 0. (-10, 5) and (, -7) 1. (5, 7) and (0, ). (-5, 9) and (, 0) 3. (-1, 5) and (, 5). (3, -) and (1, -) 5. (, -7) and (, 3). (7, -) and (7, -) The Slope Formula Robert Prior, 010 page. - 1

15 Find the equation of the line that passes through the given points. 7. (0, 7) and (3, 1) 8. (0, -8) and (-, -18) 9. (0, 11) and (1, ) 50. (0, -) and (15, ) 51. (-9, 10) and (0, -8) 5. (-, -10) and (0, -1) 53. (0, 0) and (9, -15) 5. (-1, 8) and (0, 0) 55. (0, 10) and (5, 0) 5. (, 0) and (0, 1) 57. (-0, 0) and (0, -15) 58. (0, -18) and (-1, 0) Think Outside the Bo: For each pair of points: a) Plot the points in the --plane. b) Find the slope of the line. c) Use the slope to locate the -intercept. d) Graph the line that passes through these two points e) Write the equation of the line. 59. (, 1) and (3, ) 0. (8, ) and (, 1) 1. (-3, -) and (-, -). (-8, 3) and (-7, ) The Slope Formula Robert Prior, 010 page. - 15