The SlopeIntercept Form


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1 7.1 The SlopeIntercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph a line (, ) b In Chapter 6, we used two points to find the slope of a line. In this chapter we will use the slope to find the graph of an equation. First, we want to consider finding the equation of a line when its slope and intercept are known. Suppose that the intercept of a line is (0, b). Then the point at which the line crosses the ais has coordinates (0, b). Look at the sketch at left. Now, using an other point (, ) on the line and using our definition of slope, we can write (0, b) Change in. 0 m b 0 (1) or Change in. m b () Multipling both sides of equation () b, we have m b () Finall, adding b to both sides of equation () gives m b or m b (4) We can summarize the above discussion as follows: NOTE In this form, the equation is solved for. The coefficient of will give ou the slope of the line, and the constant term gives the intercept. Definitions: The SlopeIntercept Form for a Line An equation of the line with slope m and intercept (0, b) is m b Eample 1 Finding the Slope and Intercept Find the slope and intercept for the graph of the equation 5 m b The slope of the line is ; the intercept is (0, 5). 555
2 556 CHAPTER 7 GRAPHING AND INEQUALITIES CHECK YOURSELF 1 Find the slope and intercept for the graph of each of the following equations. (a) 7 (b) 4 5 As Eample illustrates, we ma have to solve for as the first step in determining the slope and the intercept for the graph of an equation. Eample Finding the Slope and Intercept NOTE If we write the equation as 6 it is more difficult to identif the slope and the intercept. Find the slope and intercept for the graph of the equation 6 First, we must solve the equation for. 6 6 Add ( ) to both sides. Divide each term b. The equation is now in slopeintercept form. The slope is, and the intercept is (0, ). CHECK YOURSELF Find the slope and intercept for the graph of the equation 5 10 As we mentioned earlier, knowing certain properties of a line (namel, its slope and intercept) will also allow us to write the equation of the line b using the slopeintercept form. Eample illustrates this approach. Eample Writing the Equation of a Line Write the equation of a line with slope and intercept (0, ). 4 We know that m and b. In this case, 4 m b 4 ( ) or 4 which is the desired equation.
3 THE SLOPEINTERCEPT FORM SECTION CHECK YOURSELF Write the equation of a line with the following: (a) slope and intercept (0, 7) (b) slope and intercept (0, ) We can also use the slope and intercept of a line in drawing its graph. Consider Eample 4. Eample 4 Graphing a Line NOTE m rise run The line rises from left to right because the slope is positive. Graph the line with slope and intercept (0, ). Because the intercept is (0, ), we begin b plotting the point (0, ). Because the horizontal change (or run) is, we move units to the right from that intercept. Then because the vertical change (or rise) is, we move units up to locate another point on the desired graph. Note that we will have located that second point at (, 4). The final step is to simpl draw a line through that point and the intercept. (, 4) Rise (0, ) Run The equation of this line is. CHECK YOURSELF 4 Graph the equation of a line with slope 5 and intercept (0, ). Step b Step: Step 1 Step Step Step 4 Step 5 Graphing b Using the SlopeIntercept Form Write the original equation of the line in slopeintercept form. Determine the slope m and the intercept (0, b). Plot the intercept at (0, b). Use m (the change in over the change in ) to determine a second point on the desired line. Draw a line through the two points determined above to complete the graph.
4 558 CHAPTER 7 GRAPHING AND INEQUALITIES You have now seen two methods for graphing lines: the slopeintercept method (Section 7.1) and the intercept method (Section 6.). When ou graph a linear equation, ou should first decide which is the appropriate method. Eample 5 Selecting an Appropriate Graphing Method Decide which of the two methods for graphing lines the intercept method or the slopeintercept method is more appropriate for graphing equations (a), (b), and (c). (a) 5 10 Because both intercepts are eas to find, ou should choose the intercept method to graph this equation. (b) 6 This equation can be quickl graphed b either method. As it is written, ou might choose the intercept method. It can, however, be rewritten as 6. In that case the slopeintercept method is more appropriate. (c) Because the equation is in slopeintercept form, that is the more appropriate method to choose. CHECK YOURSELF 5 Which would be more appropriate for graphing each equation, the intercept method or the slopeintercept method? (a) (b) 1 (c) 1 6 CHECK YOURSELF ANSWERS 1. (a) Slope is, intercept is (0, 7); (b) Slope is, intercept is (0, 5). 4 ; the slope is ; the intercept is (0, ) 5 5. (a) 7; (b) (a) Either; (b) intercept; (c) slopeintercept (0, ) (5, 1) Run 5 Rise 5
5 Name 7.1 Eercises Section Date Find the slope and intercept of the line represented b each of the following equations ANSWERS Write the equation of the line with given slope and intercept. Then graph each line, using the slope and intercept Slope: ; intercept: (0, 5) 16. Slope: ; intercept: (0, 4)
6 ANSWERS Slope: ; intercept: (0, 4) 18. Slope: 5; intercept: (0, ) Slope: ; intercept: (0, ) 0. Slope: ; intercept: (0, 8) 4 1. Slope: ; intercept: (0, 0). Slope: ; intercept: (0, ) 560
7 ANSWERS. Slope: ; intercept: (0, ) 4. Slope: ; intercept: (0, 0) In eercises 5 to, match the graph with one of the equations below. (a), (b) 1, (c), (d) 1, (e), (f), (g), (h)
8 ANSWERS In which quadrant(s) are there no solutions for each line?
9 ANSWERS 41. Reccling. The equation describes the award mone in a reccling contest. What are the slope and the intercept for this equation? Fundraising. The equation describes the amount of mone a high school class might earn from a paper drive. What are the slope and intercept for this equation? 4. Science. On a certain Februar da in Philadelphia, the temperature at 6:00 A.M. was 10 F. B :00 P.M. the temperature was up to 6 F. What was the hourl rate of temperature change? Slope of a roof. A roof rises 8.75 feet (ft) in a horizontal distance of ft. Find the slope of the roof to the nearest hundredth. 45. Slope of airplane descent. An airplane covered 15 miles (mi) of its route while decreasing its altitude b 4,000 ft. Find the slope of the line of descent that was followed. (1 mi 580 ft.) Round to the nearest hundredth Slope of road descent. Driving down a mountain, Tom finds that he has descended 1800 ft in elevation b the time he is.5 mi horizontall awa from the top of the mountain. Find the slope of his descent to the nearest hundredth. 47. Complete the following statement: The difference between undefined slope and zero slope is Complete the following: The slope of a line tells ou In a stud on nutrition conducted in 1984, 18 normal adults aged to 61 ears old were measured for bod fat, which is given as percentage of weight. The mean (average) bod fat percentage for women 40 ears old was 8.6 percent, and for women 5 ears old was 8.4 percent. Work with a partner to decide how to show this information on a scatterplot. Tr to find a linear equation that will tell ou percentage of bod fat based on a woman s age. What does our equation give for 0 ears of age? For 60? Do ou think a linear model works well for predicting bod fat percentage in women as the age? 56
10 ANSWERS On two occasions last month, Sam Johnson rented a car on a business trip. Both times it was the same model from the same compan, and both times it was in San Francisco. Sam now has to fill out an epense account form and needs to know how much he was charged per mile and the base rate. On both occasions he dropped the car at the airport booth and just got the total charge, not the details. All Sam knows is that he was charged $10 for 65 miles on the first occasion and $1.50 for 70 miles on the second trip. Sam has called accounting to ask for help. Plot these two points on a graph, and draw the line that goes through them. What question does the slope of the line answer for Sam? How does the intercept help? Write a memo to Sam eplaining the answers to his question and how a knowledge of algebra and graphing has helped ou find the answers On the same graph, sketch the following lines: 1 and What do ou observe about these graphs? Will the lines intersect? Repeat eercise 51 using 4 and 1 5. On the same graph, sketch the following lines: 5. and What do ou observe concerning these graphs? Find the product of the slopes of these two lines Repeat eercise 5 using 4 and 4 564
11 ANSWERS 55. Based on eercises 5 and 54, write the equation of a line that is perpendicular to a. b. c. Getting Read for Section 7. [Section 6.4] Find the slope of the line connecting the given points. (a) ( 4, 6) and (, 0) (b) (, 8) and ( 6, 8) (c) (5, 7) and ( 5, ) (d) (, 8) and (, 5) (e) (6, 9) and (, 9) (f ) (4, 6) and ( 4, ) d. e. f. Answers 1. Slope, intercept (0, 5). Slope, intercept (0, 5) 5. Slope, intercept (0, 1) 7. Slope, intercept (0, 0) 4 9. Slope 4, intercept (0, 4) 11. Slope 0, intercept (0, 9) 1. Slope, intercept (0, 4)
12 g 7. e 9. h 1. c. IV 5. III 7. I 9. III and IV 41. Slope 0.10; intercept (0, 00) 4. /hr Parallel lines; no 5. Perpendicular lines; a. b. c. 1 d. Undefined e. 0 f
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