Preparation for Calculus 1 Copyright © Cengage Learning. All rights reserved.

Linear Models and Rates of Change Copyright © Cengage Learning. All rights reserved. 1.2

3 Objectives Find the slope of a line passing through two points. Write the equation of a line given a point and the slope. Interpret slope as a ratio or as a rate in a real- life application.

4 Objectives Sketch the graph of a linear equation in slope- intercept form. Write equations of lines that are parallel or perpendicular to a given line.

5 The Slope of a Line

6 The slope of a nonvertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right. Consider the two points (x 1, y 1 ) and (x 2, y 2 ) on the line in Figure The Slope of a Line  y = y 2 – y 1 = change in y  x = x 2 – x 1 = change in x Figure 1.12

7 As you move from left to right along this line, a vertical change of  y = y 2 – y 1 units corresponds to a horizontal change of  x = x 2 – x 1 units. (  is the Greek uppercase letter delta, and the symbols  y and  x are read “delta y” and “delta x.”) The Slope of a Line change in y change in x

8 The Slope of a Line

9 Figure 1.13 shows four lines: one has a positive slope, one has a slope of zero, one has a negative slope, and one has an “undefined” slope. The Slope of a Line If m is positive, then the line rises from left to right. If m is zero, then the line is horizontal. If m is negative, then the line falls from left to right. If m is undefined, then the line is vertical. Figure 1.13

10 In general, the greater the absolute value of the slope of a line, the steeper the line is. For instance, in Figure 1.13, the line with a slope of – 5 is steeper than the line with a slope of. The Slope of a Line

11 Equations of Lines

12 Any two points on a nonvertical line can be used to calculate its slope. This can be verified from the similar triangles shown in Figure (Recall that the ratios of corresponding sides of similar triangles are equal.) Equations of Lines Any two points on a nonvertical line can be used to determine its slope. Figure 1.14

13 You can write an equation of a nonvertical line if you know the slope of the line and the coordinates of one point on the line. Suppose the slope is m and the point is (x 1, y 1 ). If (x, y) is any other point on the line, then This equation, involving the two variables x and y can be rewritten in the form y – y 1 = m(x – x 1 ), which is called the point-slope form of the equation of a line. Equations of Lines

14 Equations of Lines

15 Example 1 – Finding an Equation of a Line Find an equation of the line that has a slope of 3 and passes through the point (1, –2). Solution: y – y 1 = m(x – x 1 ) y – (–2) = 3(x – 1) y + 2 = 3x – 3 y = 3x – 5 (See Figure 1.15.) The line with a slope of 3 passing through the point (1, –2) Figure 1.15 Point-slope form Substitute –2 for y 1, 1 for x 1, and 3 for m. Solve for y. Simplify.

16 Ratios and Rates of Change

17 The slope of a line can be interpreted as either a ratio or a rate. If the x- and y-axes have the same unit of measure, the slope has no units and is a ratio. If the x- and y-axes have different units of measure, the slope is a rate or rate of change. In your study of calculus, you will encounter applications involving both interpretations of slope. Ratios and Rates of Change

18 Example 2(a) – Population Growth and Engineering Design The population of Colorado was 3,827,000 in 1995 and 4,665,000 in Over this 10-year period, the average rate of change of the population was If Colorado’s population continues to increase at this same rate for the next 10 years, it will have a 2015 population of 5,503,000 (see Figure 1.16).(Source: U.S.Census Bureau) Population of Colorado Figure 1.16

19 Example 2(b) – Population Growth and Engineering Design In tournament water-ski jumping, the ramp rises to a height of 6 feet on a raft that is 21 feet long, as shown in Figure Dimensions of a water-ski ramp Figure 1.17 cont’d

20 Example 2(b) – Population Growth and Engineering Design The slope of the ski ramp is the ratio of its height (the rise) to the length of its base (the run). In this case, note that the slope is a ratio and has no units. Rise is vertical change, run is horizontal change. cont’d

21 The rate of change found in Example 2(a) is an average rate of change. An average rate of change is always calculated over an interval. In this case, the interval is [1995, 2005]. Ratios and Rates of Change

22 Graphing Linear Models

23 Many problems in analytic geometry can be classified in two basic categories: (1) Given a graph, what is its equation? and (2) Given an equation, what is its graph? The point-slope equation of a line can be used to solve problems in the first category. However, this form is not especially useful for solving problems in the second category. Graphing Linear Models

24 The form that is better suited to sketching the graph of a line is the slope-intercept form of the equation of a line. Graphing Linear Models

25 Example 3 – Sketching Lines in the Plane Sketch the graph of each equation. a. y = 2x + 1 b. y = 2 c. 3y + x – 6 = 0 Solution: a. Because b = 1, the y-intercept is (0, 1). Because the slope is m = 2, you know that the line rises two units for each unit it moves to the right, as shown in Figure 1.18(a). m = 2; line rises Figure 1.18(a)

26 b. Because b = 2 the y-intercept is (0, 2). Because the slope is m = 0, you know that the line is horizontal, as shown in Figure 1.18(b). Example 3 – Solution m = 0; line is horizontal Figure 1.18(b) cont’d

27 c. Begin by writing the equation in slope-intercept form. 3y + x – 6 = 0 3y = –x + 6 In this form, you can see that the y-intercept is (0, 2) and the slope is m =. Example 3 – Solution cont’d Write original equation. Isolate y-term on the left. Slope-intercept form

28 This means that the line falls one unit for every three units it moves to the right, as shown in Figure 1.18(c). Example 3 – Solution m = ; line falls Figure 1.18(c) cont’d

29 Because the slope of a vertical line is not defined, its equation cannot be written in the slope-intercept form. However, the equation of any line can be written in the general form where A and B are not both zero. For instance, the vertical line given by x = a can be represented by the general form x – a = 0. Graphing Linear Models General form of the equation of a line

30 Graphing Linear Models

31 Parallel and Perpendicular Lines

32 The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular, as shown in Figure Specifically, nonvertical lines with the same slope are parallel and nonvertical lines whose slopes are negative reciprocals are perpendicular. Parallel and Perpendicular Lines Parallel lines Perpendicular lines Figure 1.19

33 Parallel and Perpendicular Lines

34 Example 4 – Finding Parallel and Perpendicular Lines Find the general forms of the equations of the lines that pass through the point (2, –1) and are a. parallel to the line 2x – 3y = 5. b. perpendicular to the line 2x – 3y = 5. (See Figure 1.20.) Lines parallel and perpendicular to 2x – 3y = 5 Figure 1.20

35 By writing the linear equation 2x – 3y = 5 in slope-intercept form, you can see that the given line has a slope of m = a. The line through (2, –1) that is parallel to the given line also has a slope of y – y 1 = m(x – x 1 ) y – (–1) = (x – 2) Example 4 – Solution Point-slope form Substitute.

36 3(y + 1) = 2(x – 2) 2x – 3y – 7 = 0 Note the similarity to the original equation. b. Using the negative reciprocal of the slope of the given line, you can determine that the slope of a line perpendicular to the given line is Example 4 – Solution cont’d Simplify. General form

37 So, the line through the point (2, –1) that is perpendicular to the given line has the following equation. y – y 1 = m(x – x 1 ) y – (–1) = (x – 2) 2(y + 1) = –3(x – 2) 3x + 2y – 4 = 0 Example 4 – Solution cont’d Simplify. General form Point-slope form Substitute.