CH 24 SLOPE. rise = run. Ch 24 Slope. Introduction

Transcription

1 9 CH SLOPE Introduction A line has any attributes, or characteristics. Two of the ost iportant are its intercepts and its slope. The intercepts (previous chapter) tell us where the line crosses the x-axis and the y-axis; they are very good reference points. The slope of a line tells us how steep the line is -- it s kind of like the angle that a line akes, and is a concept used in econoics, cheistry, statistics, construction, and ountain clibing. Slope A trucker is keenly aware of the grade, or angle, of the road on which a truck travels -- it deterines the speed liit and the proper gear that the truck needs to be in. A roofer is concerned with the pitch, or steepness, of a roof. A construction worker needs to ake sure that a wheelchair rap has the correct angle with the street or sidewalk. All of these ideas are exaples of the concept steepness. We ll use the ter slope to represent steepness, and give it the letter (I don t know why -- aybe for ountain?). Our definition of slope in this course and all future ath courses (and cheistry, econoics, and nursing courses) is as follows: rise run Ch Slope

2 0 As we ll see shortly, a rise is a vertical (up/down) change, while a run is a horizontal (left/right) change. Slope is defined as the ratio of the rise to the run; we can also say that slope is the quotient of the rise and the run. EXAMPLE : Graph the line y x 5 and deterine its slope. Solution: Let s calculate a couple of points by choosing soe rando x-values. If we let x, then y, so the point (, ) is on the line. And if we let x, then y, giving us the point (, ). We could calculate ore points for our line, but let s cut to the chase and graph the line given the two points just coputed. (, ) Rise Run (, ) Notice that we ve constructed a right triangle using the line segent between the two given points as the hypotenuse. The rise and run are then just the lengths of the legs of the triangle. Counting squares fro left to right along the botto of the triangle, we see that the run is. Counting squares up the side of the triangle yields a rise of 6. Using the slope forula, we can calculate the slope of the line: rise 6 run Ch Slope

3 Note: The concept of slope is diensionless; that is, slope has no units. Here s why: Suppose that the units in the triangle are in feet. Then the slope is rise 6ft run ft 6 ft ft (since the feet cancel out) EXAMPLE : Find the slope of the line x + y. Solution: To graph this line, let s calculate the two intercepts (since they re generally the easiest points to calculate). Set x 0 to get (0) + y y y Thus, the y-intercept is x + (0) x (0, ). If we set y 0, we can solve for x: x, which iplies that the x-intercept is (, 0). Plotting these two intercepts gives us our line: Rise (0, ) Run (, 0) As we ove fro left to right, fro the y-intercept to the x-intercept, we notice that the rise is actually a drop -- this eans that the rise is negative. Since the height of the triangle is, we Ch Slope

4 conclude that the rise is. Since the run is fro left to right, the run is positive. Now we re ready for the calculation: rise run Note: Instead of oving fro left to right, fro the y-intercept to the x-intercept, we could also have oved fro right to left, fro the x-intercept to the y-intercept. In this case, the rise is positive because we re oving up, but the run is negative because we re oving to the left. This will still give us the sae answer, since now the calculation looks like: rise run Hoework. For each pair of points, plot the on a grid, find the rise and the run, and then use the forula for slope to calculate the slope of the line connecting the two points: a. (, ), (, 7) b. (, 0), (0, 6) c. (, ), (, 5) d. (, ), (7, 7) e. (, ), (0, 0) f. (, ), (, 5). Find the slope of the given line by graphing the line and using the rise and run. You ay, of course, use any two points on the line to calculate the rise and the run: a. y x + b. y x c. y x + d. y x + e. y x f. y x + g. x + y h. x y i. x y j. x + y 6 k. x + 5y 0 l. x y 8 Ch Slope

5 A New View of Slope Finding the slope, rise, of a line by plotting two run points and counting the squares to deterine the rise and the run works fine only when it s convenient to plot the points and you re in the ood to count squares. Indeed, consider the line connecting the points (, 000) and (, 5000). Certainly these points deterine a line, and that line has soe sort of slope, but plotting these points is not really feasible -- we need a sipler way to calculate slope. Recall Exaple fro this chapter, y x 5. We plotted the points (, ) and (, ) and then counted squares (as we oved fro left to right) to get a rise of 6 and a run of, giving us a slope of rise 6 run How can we get the nubers 6 and without referring to the points on the graph? Notice that if we subtract the y-coordinate of one point fro the y-coordinate of the other point, we get rise () + 6 Siilarly, if we subtract one x-coordinate fro the other, we get run (, ) Run (, ) Rise Now dividing the rise by the run gets us our slope of. We can now think of our rise forula as run change in y change in x The only issue we need to worry about is that we are consistent in the direction in which we do our subtractions. For instance, using the sae two points, (, ) and (, ), we can subtract in the reverse order fro above, as long as both subtractions are reversed. Ch Slope

6 change in y 6 change in x the sae value of slope calculated before. EXAMPLE : Solution: Find the slope of the line connecting the points (7, ) and (, 0). Then calculate the slope again by subtracting in the reverse direction. Subtracting in one direction coputes the slope as: change in y ( 0) 0 change in x Reversing the direction in which we subtract the coordinates: change in y 0 ( ) 0 change in x ( 7) 7 9 Either way, we get the sae slope; thus, the order in which you subtract is entirely up to you, as long as each subtraction (top and botto) is done in the sae direction. New Notation We re just about ready to find the slope of a line using the points entioned at the beginning of this section: (, 000) and (, 5000). But first we introduce soe new notation. The natural world is filled with changes. In slope, we ve seen changes in x and y in the notions of rise and run. In cheistry, there are changes in the volue and pressure of a gas. In nursing, there are changes in teperature and blood pressure, and in econoics there are changes in supply and deand. This concept occurs so often that there s a special notation for a change in soething. We use the Greek capital letter delta,, to represent a change in soething. A change in volue ight be denoted by V and a change in tie by t. Ch Slope

7 5 And so now we can redefine slope as y x Slope is the ratio of the change in y to the change in x. which is, of course, just fancy notation for what we already know. EXAMPLE : Solution: Find the slope of the line connecting the points (, 000) and (, 5000). A siple ratio calculation will give us the slope: y 000 ( 5000) x 500 In the last step of this calculation we used the fact that a positive nuber divided by a negative nuber is negative. Also, we could obtain an approxiate answer by dividing 500 by. -- then attaching the negative sign -- to get about,.65. Notice that there s no need to plot points and count squares on a grid. We ve turned the geoetric concept of slope into an arithetic proble. Try reversing the order of the subtractions above to ake sure you get the sae slope. change Why use delta,, to represent a change in soething? Because delta begins with a d, and d is the first letter of the word difference, and difference eans subtract, and subtract is what you do when you want to calculate the in soething. Ch Slope

8 6 Hoework. Use the forula y x to find the slope of the line connecting the given pair of points: a. (, ) and (, 7) b. (, 0) and (0, 6) c. (, ) and (, 5) d. (, ) and (7, 7) e. (, ) and (0, 0) f. (, ) and (, 5) g. (, ) and (, ) h. (, ) and (0, 0) i. (, ) and (, ) j. (, ) and (, ) k. (, 5) and (0, 0) l. (, ) and (, ) The Slopes of Increasing and Decreasing Lines Looking back at Exaple of this chapter, let s ake a quick sketch of the line. We can call this an increasing line, because as we ove fro left to right, the line is rising, or increasing, since the y-values are getting bigger. Now notice that the slope of this line, as calculated before, was, a positive nuber. Referring now to Exaple, we find that its graph, unlike the previous one, is falling as we ove fro left to right -- that is, we have a decreasing line. And this is due to the fact that the y-values are getting saller. Next we note that the slope was calculated to be the negative nuber. Ch Slope

9 7 This connection between the increasing/decreasing of a line and the sign of its slope is always true. Our conclusion is the following: An increasing line has a positive slope, while a decreasing line has a negative slope. Review Probles. Find the slope of the line connecting the given pair of points. Use the slope to deterine whether the graph of the line is increasing or decreasing. a. (0, 7) and (, 8) b. (, 0) and (8, 5) c. (, ) and (, 0) d. (, ) and (0, 5) e. (8, 0) and (, 8) f. (9, ) and (0, ) g. (, ) and (, 5) h. (6, ) and (, ) i. (, 6) and (9, 5) j. (, ) and (, 6) Ch Slope

10 8 Solutions. a. b. c. 8 d. e. f.. a. b. c. d. e. f. g. h. i. j. k. l. 5. a. b. c. g. h. i. 8 d. e. f. j. k. 5 l. 5. If the slope is positive, the line is increasing; if the slope is negative, the line is decreasing. But what about part h. of this proble? a. 5 b. 5 c. 7 d. 5 f. 0 g. h. 0 i. 9 e. 5 j. 0 Huan history becoes ore and ore a race between education and catastrophe. H.G. Wells (866-96) Ch Slope