m81354 Understand Slope of a Line By the end of this section, you will be able to: Use geoboards to model slope Find the slope of a line from its graph Find the slope of horizontal and vertical lines Use the slope formula to find the slope of a line between two points Graph a line given a point and the slope Solve slope applications ea72caf9-e266-4a3a-870d-1a97e08b2881 Before you get started, take this readiness quiz. Simplify: 1−48−2. If you missed this problem, review . −12 Divide: 04,40. If you missed this problem, review . 0, undefined Simplify: 15−3,−153,−15−3. If you missed this problem, review . −5,−5,5 As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat?The steepness of the slant of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

In this section, we will explore the concepts of slope.Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. Doing the Manipulative Mathematics activity "Exploring Slope" will help you develop a better understanding of the slope of a line. We’ll start by stretching a rubber band between two pegs to make a line as shown in .

Does it look like a line?Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in . We carefully make a 90° angle around the third peg, so that one side is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in .

To help remember the terms, it may help to think of the images shown in .

On our geoboard, the rise is 2 units because the rubber band goes up 2 spaces on the vertical leg. See . What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across 3 spaces on the horizontal leg, so the run is 3 units.

The slope of a line is the ratio of the rise to the run. So the slope of our line is 23. In mathematics, the slope is always represented by the letter m. The slope of a line is m=riserun. The rise measures the vertical change and the run measures the horizontal change. What is the slope of the line on the geoboard in ? m=riserun m=23 The line has slope23. When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle. If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run. What is the slope of the line on the geoboard shown? Use the definition of slope. m=riserun Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg. Count the rise and the run as shown. The rise is3units.m=3runThe run is4units.m=34The slope is34. What is the slope of the line on the geoboard shown? 43 What is the slope of the line on the geoboard shown? 14 What is the slope of the line on the geoboard shown? Use the definition of slope. m=riserun Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative. The rise is−1.m=−1runThe run is3.m=−13m=−13The slope is−13. What is the slope of the line on the geoboard? −23 What is the slope of the line on the geoboard? −43 Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in .

As you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.

Use a geoboard to model a line with slope 12. To model a line with a specific slope on a geoboard, we need to know the rise and the run. Use the slope formula. m=riserun Replace m with 12. 12=riserun So, the rise is 1 unit and the run is 2 units. Start at a peg in the lower left of the geoboard. Stretch the rubber band up 1 unit, and then right 2 units. The hypotenuse of the right triangle formed by the rubber band represents a line with a slope of 12. Use a geoboard to model a line with the given slope: m=13. Use a geoboard to model a line with the given slope: m=32. Use a geoboard to model a line with slope −14, Use the slope formula. m=riserun Replace m with −14. −14=riserun So, the rise is −1 and the run is 4. Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down 1 unit, then to the right 4 units. The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is −14. Use a geoboard to model a line with the given slope: m=−21. Use a geoboard to model a line with the given slope: m=−13.

Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards. Doing the Manipulative Mathematics activity "Slope of Lines Between Two Points" will help you develop a better understanding of how to find the slope of a line from its graph.To find the slope, we must count out the rise and the run. But where do we start? We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run. Find the slope of the line shown: Locate two points on the graph, choosing points whose coordinates are integers. We will use (0,−3) and (5,1). Starting with the point on the left, (0,−3), sketch a right triangle, going from the first point to the second point, (5,1). Count the rise on the vertical leg of the triangle. The rise is 4 units. Count the run on the horizontal leg. The run is 5 units. Use the slope formula. m=riserun Substitute the values of the rise and run. m=45 The slope of the line is 45. Notice that the slope is positive since the line slants upward from left to right. Find the slope of the line: 25 Find the slope of the line: 34 Locate two points on the line whose coordinates are integers. Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. Count the rise and the run on the legs of the triangle. Take the ratio of rise to run to find the slope. m=riserun Find the slope of the line shown: Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use (0, 5) and (3, 3). Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. Count the rise – it is negative. The rise is −2. Count the run. The run is 3. Use the slope formula. m=riserun Substitute the values of the rise and run. m=−23 Simplify. m=−23 The slope of the line is −23. Notice that the slope is negative since the line slants downward from left to right. What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points (−3,7) and (6,1). Starting at (−3,7), sketch a right triangle to (6,1). Count the rise. The rise is −6. Count the run. The run is 9. Use the slope formula. m=riserun Substitute the values of the rise and run. m=−69 Simplify the fraction. m=−23 The slope of the line is −23.It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant! Find the slope of the line: −43 Find the slope of the line: −35 The lines in the previous examples had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points we used to find the slope. In the next example, the y-intercept is a fraction. The calculations are easier if we use two points with integer coordinates. Find the slope of the line shown: Locate two points on the graph whose coordinates are integers.(2,3) and (7,6) Which point is on the left? (2,3)Starting at (2,3), sketch a right angle to (7,6) as shown below. Count the rise. The rise is 3. Count the run. The run is 5. Use the slope formula. m=riserun Substitute the values of the rise and run. m=35 The slope of the line is 35. Find the slope of the line: 54 Find the slope of the line: 32

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable. horizontal line y=b; all the y-coordinates are the same. vertical line x=a; all the x-coordinates are the same. So how do we find the slope of the horizontal line y=4? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens in . We’ll use the two points (0,4) and (3,4) to count the rise and run.

What is the rise? The rise is 0. What is the run? The run is 3. What is the slope? m=riserun m=03 m=0 The slope of the horizontal line y=4 is 0. All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0. The slope of a horizontal line, y=b, is 0. Now we’ll consider a vertical line, such as the line x=3, shown in . We’ll use the two points (3,0) and (3,2) to count the rise and run.

What is the rise? The rise is 2. What is the run? The run is 0. What is the slope? m=riserun m=20 But we can’t divide by 0. Division by 0 is undefined. So we say that the slope of the vertical line x=3 is undefined. The slope of all vertical lines is undefined, because the run is 0. The slope of a vertical line, x=a, is undefined. Find the slope of each line: ⓐ x=8ⓑ y=−5 ⓐ x=8This is a vertical line, so its slope is undefined. ⓑ y=−5This is a horizontal line, so its slope is 0. Find the slope of the line: x=−4. undefined Find the slope of the line: y=7. 0

Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing. Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair (x,y) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol (x,y) be used to represent two different points? Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable. (x1,y1)readxsub1,ysub1 (x2,y2)readxsub2,ysub2 We will use (x1,y1) to identify the first point and (x2,y2) to identify the second point. If we had more than two points, we could use (x3,y3),(x4,y4), and so on. To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points (2,3) and (7,6) in .

Since we have two points, we will use subscript notation. (2,3)x1,y1(7,6)x2,y2On the graph, we counted the rise of 3. The rise can also be found by subtracting the y-coordinates of the points. y2−y16−33 We counted a run of 5. The run can also be found by subtracting the x-coordinates. x2−x17−25 We know m=riserun So m=35 We rewrite the rise and run by putting in the coordinates. m=6−37−2 But 6 is the y-coordinate of the second point, y2 and 3 is the y-coordinate of the first point y1. So we can rewrite the rise using subscript notation. m=y2−y17−2 Also 7 is the x-coordinate of the second point, x2 and 2 is the x-coordinate of the first point x2. So we rewrite the run using subscript notation. m=y2−y1x2−x1 We’ve shown that m=y2−y1x2−x1 is really another version of m=riserun. We can use this formula to find the slope of a line when we have two points on the line. The slope of the line between two points (x1,y1) and (x2,y2) is m=y2−y1x2−x1 Say the formula to yourself to help you remember it: Slope isyof the second point minusyof the first point over xof the second point minusxof the first point. Doing the Manipulative Mathematics activity “Slope of Lines Between Two Points” will help you develop a better understanding of how to find the slope of a line between two points. Find the slope of the line between the points (1,2) and (4,5). We’ll call (1,2) point #1 and (4,5)point #2. (1,2)x1,y1and(4,5)x2,y2 Use the slope formula. m=y2−y1x2−x1 Substitute the values in the slope formula: y of the second point minus y of the first point m=5−2x2−x1 x of the second point minus x of the first point m=5−24−1 Simplify the numerator and the denominator. m=33 m=1 Let’s confirm this by counting out the slope on the graph. The rise is 3 and the run is 3, so m=riserunm=33m=1 Find the slope of the line through the given points: (8,5) and (6,3). 1 Find the slope of the line through the given points: (1,5) and (5,9). 1 How do we know which point to call #1 and which to call #2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative. We’ll call (4,5) point #1 and (1,2) point #2. (4,5)x1,y1and(1,2)x2,y2 Use the slope formula. m=y2−y1x2−x1 Substitute the values in the slope formula: y of the second point minus y of the first point m=2−5x2−x1 x of the second point minus x of the first point m=2−51−4 Simplify the numerator and the denominator. m=−3−3 m=1 The slope is the same no matter which order we use the points. Find the slope of the line through the points (−2,−3) and (−7,4). We’ll call (−2,−3) point #1 and (−7,4) point #2. (−2,−3)x1,y1and(−7,4)x2,y2 Use the slope formula. m=y2−y1x2−x1 Substitute the values y of the second point minus y of the first point m=4−(−3)x2−x1 x of the second point minus x of the first point m=4−(−3)−7−(−2) Simplify. m=7−5 m=−75 Let’s confirm this on the graph shown. m=riserunm=−75m=−75 Find the slope of the line through the pair of points: (−3,4) and (2,−1). −1 Find the slope of the line through the pair of points: (−2,6) and (−3,−4). 10

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line. Graph the line passing through the point (1,−1) whose slope is m=34. Plot the given point, (1,−1). Use the slope formula m=riserun to identify the rise and the run. m=34riserun=34rise=3run=4 Starting at the point we plotted, count out the rise and run to mark the second point. We count 3 units up and 4 units right. Then we connect the points with a line and draw arrows at the ends to show it continues. We can check our line by starting at any point and counting up 3 and to the right 4. We should get to another point on the line. Graph the line passing through the point with the given slope: (2,−2),m=43 Graph the line passing through the point with the given slope: (−2,3),m=14 Plot the given point. Use the slope formula to identify the rise and the run. Starting at the given point, count out the rise and run to mark the second point. Connect the points with a line. Graph the line with y-intercept (0,2) and slope m=−23. Plot the given point, the y-intercept (0,2). Use the slope formula m=riserun to identify the rise and the run. m=−23riserun=−23rise=–2run=3 Starting at (0,2), count the rise and the run and mark the second point. Connect the points with a line. Graph the line with the given intercept and slope: y-intercept 4,m=−52 Graph the line with the given intercept and slope: x-intercept −3,m=−34 Graph the line passing through the point (−1,−3) whose slope is m=4. Plot the given point. Identify the rise and the run. m=4 Write 4 as a fraction. riserun=41 rise=4run=1 Count the rise and run. Mark the second point. Connect the two points with a line. Graph the line passing through the point (−2,1) and with slope m=3. Graph the line passing through the point (4,−2) and with slope m=−2.

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now. The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown? Use the slope formula. m=riserun Substitute the values for rise and run. m=9 ft18 ft Simplify. m=12 The slope of the roof is 12. Find the slope given rise and run: A roof with a rise =14 and run =24. 712 Find the slope given rise and run: A roof with a rise =15 and run =36. 512 Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house. Sewage pipes must slope down 14 inch per foot in order to drain properly. What is the required slope? Use the slope formula. m=riserun m=−14in.1ft m=−14in.1ft Convert 1 foot to 12 inches. m=−14in.12in. Simplify. m=−148 The slope of the pipe is −148. Find the slope of the pipe: The pipe slopes down 13 inch per foot. −136 Find the slope of the pipe: The pipe slopes down 34 inch per yard. −148 Determine Positive slope from a Graph Determine Negative slope from a Graph Determine Slope from Two Points

Find the slope from a graphLocate two points on the line whose coordinates are integers. Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.Count the rise and the run on the legs of the triangle. Take the ratio of rise to run to find the slope, m=riserun Slope of a Horizontal LineThe slope of a horizontal line, y=b, is 0. Slope of a Vertical LineThe slope of a vertical line, x=a, is undefined.Slope FormulaThe slope of the line between two points (x1,y1) and (x2,y2) is m=y2-y1x2-x1 Graph a line given a point and a slope.Plot the given point. Use the slope formula to identify the rise and the run. Starting at the given point, count out the rise and run to mark the second point. Connect the points with a line.

Use Geoboards to Model Slope In the following exercises, find the slope modeled on each geoboard. 14 −32 In the following exercises, model each slope. Draw a picture to show your results. 23 34 14 43 −12 −34 −23 −32 Find the Slope of a Line from its Graph In the following exercises, find the slope of each line shown. 25 54 −13 −34 34 −52 −23 14 Find the Slope of Horizontal and Vertical Lines In the following exercises, find the slope of each line. y=3 0 y=1 x=4 undefined x=2 y=−2 0 y=−3 x=−5 undefined x=−4 Use the Slope Formula to find the Slope of a Line between Two Points In the following exercises, use the slope formula to find the slope of the line between each pair of points. (1,4),(3,9) 52 (2,3),(5,7) (0,3),(4,6) 34 (0,1),(5,4) (2,5),(4,0) −52 (3,6),(8,0) (−3,3),(2,−5) −85 (−2,4),(3,−1) (−1,−2),(2,5) 73 (−2,−1),(6,5) (4,−5),(1,−2) −1 (3,−6),(2,−2) Graph a Line Given a Point and the Slope In the following exercises, graph the line given a point and the slope. (1,−2);m=34 (1,−1);m=12 (2,5);m=−13 (1,4);m=−12 (−3,4);m=−32 (−2,5);m=−54 (−1,−4);m=43 (−3,−5);m=32 (0,3);m=−25 (0,5);m=−43 (−2,0);m=34 (−1,0);m=15 (−3,3);m=2 (−4,2);m=4 (1,5);m=−3 (2,3);m=−1 Solve Slope Applications In the following exercises, solve these slope applications. Slope of a roof A fairly easy way to determine the slope is to take a 12-inch level and set it on one end on the roof surface. Then take a tape measure or ruler, and measure from the other end of the level down to the roof surface. You can use these measurements to calculate the slope of the roof. What is the slope of the roof in this picture? 13 What is the slope of the roof shown? Road grade A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. ⓐ Find the slope of the road as a fraction and then simplify the fraction. ⓑ What rise and run would reflect this slope or grade? ⓐ 350ⓑ rise=3;run=50 Highway grade A local road rises 2 feet for every 50 feet of highway. ⓐ What is the slope of the highway? ⓑ The grade of a highway is its slope expressed as a percent. What is the grade of this highway?

Wheelchair ramp The rules for wheelchair ramps require a maximum 1 inch rise for a 12 inch run. ⓐ What run must the ramp have to accommodate a 24-inch rise to the door? ⓑ Draw a model of this ramp. ⓐ 288 inches (24 feet)ⓑ Models will vary. Wheelchair ramp A 1-inch rise for a 16-inch run makes it easier for the wheelchair rider to ascend the ramp. ⓐ What run must the ramp have to easily accommodate a 24-inch rise to the door? ⓑ Draw a model of this ramp.

What does the sign of the slope tell you about a line? Answers will vary. How does the graph of a line with slope m=12 differ from the graph of a line with slope m=2? Why is the slope of a vertical line undefined? Answers will vary. Explain how you can graph a line given a point and its slope.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Plot Points in a Rectangular Coordinate SystemIn the following exercises, plot each point in a rectangular coordinate system. (1,3),(3,1) (2,5),(5,2) In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. ⓐ (−1,−5) ⓑ (−3,4) ⓒ (2,−3) ⓓ (1,52) ⓐ IIIⓑ IIⓒ IVⓐ I ⓐ (3,−2) ⓑ (−4,−1) ⓒ (−5,4) ⓓ (2,103) Identify Points on a Graph In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system. ⓐ (5,3)ⓑ (2,−1)ⓒ (−3,−2)ⓓ (−1,4) ⓐ (2,0)ⓑ (0,−5)ⓒ (−4,0)ⓓ (0,3) Verify Solutions to an Equation in Two Variables In the following exercises, find the ordered pairs that are solutions to the given equation. 5x+y=10 ⓐ (5,1) ⓑ (2,0) ⓒ (4,−10) ⓑ (2, 0),ⓒ (4, –10) y=6x−2 ⓐ (1,4) ⓑ (13,0) ⓒ (6,−2) Complete a Table of Solutions to a Linear Equation in Two Variables In the following exercises, complete the table to find solutions to each linear equation. y=4x−1 x y (x,y) 0 1 −2 x y (x,y) 0 −1 (0,−1) 1 3 (1,3) −2 −9 (−2,−9) y=−12x+3 x y (x,y) 0 1 −2 x+2y=5 x y (x,y) 0 1 −1 x y (x,y) 5 0 (5,0) 1 2 (1,2) −1 3 (−1,3) 3x−2y=6 x y (x,y) 0 0 −2 Find Solutions to a Linear Equation in Two Variables In the following exercises, find three solutions to each linear equation. x+y=3 Answers will vary. x+y=−4 y=3x+1 Answers will vary. y=−x−1

Recognize the Relation Between the Solutions of an Equation and its Graph In each of the following exercises, an equation and its graph is shown. For each ordered pair, decideⓐ if the ordered pair is a solution to the equation.ⓑ if the point is on the line. y=−x+4 (0,4) (−1,3) (2,2) (−2,6) ⓐ yes ⓑ yesⓐ no ⓑ noⓐ yes ⓑ yesⓐ yes ⓑ yes y=23x−1 (0,−1) (3,1) (−3,−3) (6,4) Graph a Linear Equation by Plotting Points In the following exercises, graph by plotting points. y=4x−3 y=−3x 2x+y=7 Graph Vertical and Horizontal lines In the following exercises, graph the vertical or horizontal lines. y=−2 x=3

Identify the Intercepts on a Graph In the following exercises, find the x- and y-intercepts. (0,3) (3,0) Find the Intercepts from an Equation of a Line In the following exercises, find the intercepts. x+y=5 x−y=−1 (−1,0) (0,1) y=34x−12 y=3x (0,0) Graph a Line Using the Intercepts In the following exercises, graph using the intercepts. −x+3y=3 x+y=−2 Choose the Most Convenient Method to Graph a Line In the following exercises, identify the most convenient method to graph each line. x=5 y=−3 horizontal line 2x+y=5 x−y=2 intercepts y=12x+2 y=34x−1 plotting points

Use Geoboards to Model Slope In the following exercises, find the slope modeled on each geoboard. 43 −23 In the following exercises, model each slope. Draw a picture to show your results. 13 32 −23 −12 Find the Slope of a Line from its Graph In the following exercises, find the slope of each line shown. 1 −12 Find the Slope of Horizontal and Vertical Lines In the following exercises, find the slope of each line. y=2 x=5 undefined x=−3 y=−1 0 Use the Slope Formula to find the Slope of a Line between Two Points In the following exercises, use the slope formula to find the slope of the line between each pair of points. (2,1),(4,5) (−1,−1),(0,−5) −4 (3,5),(4,−1) (−5,−2),(3,2) 12 Graph a Line Given a Point and the Slope In the following exercises, graph the line given a point and the slope. (2,−2);m=52 (−3,4);m=−13 Solve Slope Applications In the following exercise, solve the slope application. A roof has rise 10 feet and run 15 feet. What is its slope?

Plot and label these points: ⓐ (2,5) ⓑ (−1,−3) ⓒ (−4,0) ⓓ (3,−5) ⓔ (−2,1) Name the ordered pair for each point shown. Find the x-intercept and y-intercept on the line shown. (4,0), (0,−2) Find the x-intercept and y-intercept of the equation 3x−y=6. Is (1,3) a solution to the equation x+4y=12? How do you know? no; 1 + 4 · 3 ≠ 12 Complete the table to find four solutions to the equation y=−x+1. x y (x,y) 0 1 3 −2 Complete the table to find three solutions to the equation 4x+y=8 x y (x,y) 0 0 3 x y (x,y) 0 8 (0,8) 2 0 (2,0) 3 −4 (3,−4) In the following exercises, find three solutions to each equation and then graph each line. y=−3x 2x+3y=−6 In the following exercises, find the slope of each line. −52 Use the slope formula to find the slope of the line between (0,−4) and (5,2). Find the slope of the line y=2. 0 Graph the line passing through (1,1) with slope m=32. A bicycle route climbs 20 feet for 1,000 feet of horizontal distance. What is the slope of the route? 150

slope of a line The slope of a line is m=riserun. The rise measures the vertical change and the run measures the horizontal change.