8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 This is a derivation of the Pythagorean Theorem and can be used to find the lengths of segments between two points. Distance Formula Example 1: Find the distance between (-5, 2) and (-1,6). Solution:

8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 2 Slope: The slope of a line tells you the “steepness” of a line (or the rate of which a line is increasing or decreasing). The line is increasing if the slope is positive and decreasing if the slope is negative. The variable ‘m’ will be used to represent the slope. The slope of the line in figure a. is positive since it is increasing (from left to right). The slope of the line in figure b. is negative since it is decreasing (from left to right). We will begin by considering the concept of slope. Consider the graphs below. Figure a.Figure b. Positive slope (m>0) Negative slope (m<0) The slope of a line is the ratio of the vertical change to the horizontal change when moving from one point to another on a line. Sometimes referred to as rise divided by run. Next Slide (m is used as an abbreviation for the slope) Slope

8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 3 Step 1. Plot the points and graph the line. Step 2. Find the slope. Example 2: Graph the line determined by the two points (3,1) and (-2,-2) and then find the slope of the line. Note: either point may be designated as P 1 or P 2. The result is equal. Next Slide

8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 4 Your Turn Problem #2 Graph the line determined by the two points (-3,3) and (2,3) and find the slope of the line. x axis y axis Answer:

8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 5 Finding coordinates on a line given the slope and a point on the line. The slope is defined as the ratio of vertical change to horizontal change (change in y divided by change in x). Procedure: To find coordinates on a line given the slope and a point on the line. Step 2. To find another point on the same line, repeat the same process. However, use the coordinate obtained from Step 1 as the starting point. Step 1. Use the point given as the starting point. Using the slope, add the numerator to the y value of the starting point and add the denominator to the x value of the starting point. This will be another coordinate on the same line. Example 3: Given one point on a line and the slope, find coordinates of three other points on the line. and Step 1. (3, –2) is the starting point. Using the slope, add 2 to the y coordinate and add 5 to the x coordinate. Change in y Change in x (8, 0) Step 2. To find another point continue the process. (13, 2) (18, 4) Next Slide

8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 6 There are many correct answers to the last example. The slope could have been written differently. Instead of adding 2 to the y coordinate and 5 to the x coordinate, we could have added –2 to the y coordinate and –5 to the x coordinate. (–2,–4) (–7, –6) Your Turn Problem #3 Given the slope and one point on a line, find the coordinates of three other points on the line. Note: There are many other answers.

8.4 Distance and Slope BobsMathClass.Com Copyright © 2010 All Rights Reserved. 7 Example 4: Given one point on a line and the slope, find the coordinates of three other points on the line. and Step 1. (4, –1) is the starting point. Using the slope, add –3 to the y coordinate and add 4 to the x coordinate. Change in y Change in x Step 2. To find more points, continue the process. (8, –4) (12, –7) (16, –10) The slope in this example is negative. The negative was used with the numerator as –3. The negative could have instead been used with the denominator. Change in y Change in x In this case, add 3 to the y coordinate and add –4 to the x coordinate. (0, 2) (–4, 5) (–8, 8) Your Turn Problem #4 Note: Answers may vary. The End. B.R