Copyright © Cengage Learning. All rights reserved. Graphs; Equations of Lines; Functions; Variation 3

Copyright © Cengage Learning. All rights reserved. Section 3.3 Slope of a Line

3 Objectives 1.Find the slope of a line given a graph. 2.Find the slope of a line passing through two specified points. 3.Find the slope of a line given an equation. 4.Identify the slope of a horizontal and vertical line

4 Objectives 5.Determine whether two lines are parallel, perpendicular, or neither parallel nor perpendicular. 5 5

5 Slope of a Line We have seen that two points can be used to graph a line. We can also graph a line if we know the coordinates of only one point and the slant (or steepness) of the line. A measure of this slant is called the slope of the line.

6 Find the slope of a line given a graph 1.

7 Finding the Slope of a Line A research service offered by an Internet company costs $2 per month plus $3 for each hour of connect time. The table shown in Figure 3-21(a) gives the cost y for different hours x of connect time. If we construct a graph from this data, we obtain the line shown in Figure 3-21(b). (a) (b) Figure 3-21

8 Finding the Slope of a Line From the graph, we can see that if x changes from 0 to 1, y changes from 2 to 5. As x changes from 1 to 2, y changes from 5 to 8, and so on. The ratio of the change in y divided by the change in x is the constant 3.

9 Finding the Slope of a Line The ratio of the change in y divided by the change in x between any two points on any line is always a constant. This constant rate of change is called the slope of the line and usually is denoted by the letter m. To distinguish between the coordinates of points P and Q in Figure 3-22, we use subscript notation. Figure 3-22

10 Finding the Slope of a Line Point P is denoted as P(x 1, y 1 ) and is read as “point P with coordinates of x sub 1 and y sub 1.” Point Q is denoted as Q(x 2, y 2 ) and is read as “point Q with coordinates of x sub 2 and y sub 2.” As a point on the line in Figure 3-22 moves from P to Q, its y-coordinate changes by the amount y 2 – y 1, and its x-coordinate changes by x 2 – x 1. The change in y is often called the rise of the line between points P and Q, and the change in x is often called the run.

11 Example Find the slope of the line shown in Figure 3-23(a). Figure 3-23(a)

12 In Figure 3-23(b), we choose two points on the line and call them A and B. Then we draw right triangle ABC, having a horizontal leg and a vertical leg. The longest side AB of the right triangle is called the hypotenuse. As we move from A to B, we move to the right, a run of 6, and then down, a rise of –3. Figure 3-23(b) Example – Solution

13 To find the slope of the line, we write a ratio. The slope of the line is cont’d Example – Solution

14 Find the slope of a line passing through two specified points 2.

15 Finding the Slope of a Line Once we know the coordinates of two points on a line, we can substitute those coordinates into the slope formula. Slope of a Nonvertical Line The slope of the nonvertical line passing through points P(x 1, y 1 ) and Q(x 2, y 2 ) is

16 Finding the Slope of a Line Comment You can use the coordinates of any two points on a line to compute the slope of the line and obtain the same result.

17 Example Use the two points shown in Figure 3-24 to find the slope of the line passing through the points P(–3, 2) and Q(2, –5). Figure 3-24

18 Example – Solution We can let P(x 1, y 1 ) = P(–3, 2) and Q(x 2, y 2 ) = Q(2, –5). Then x 1 = –3, y 1 = 2, x 2 = 2, and y 2 = –5. To find the slope, we substitute these values into the slope formula and simplify. Substitute –5 for y 2, 2 for y 1, 2 for x 2, and –3 for x 1.

19 Example – Solution The slope of the line is We would obtain the same result if we had let P(x 1, y 1 ) = P(2, –5) and Q(x 2, y 2 ) = Q(–3, 2). cont’d

20 Comment When calculating slope, always subtract the y-values and the x-values in the same order. However, the following are not true: Finding the Slope of a Line

21 Find the slope of a line given an equation 3.

22 Finding the Slope of a Line If we need to find the slope of a line from a given equation, we could graph the line and count squares to determine the rise and the run. Instead, we could find the x- and y-intercepts and then use the slope formula. In Example 3, we will find the slope of the line determined by the equation 3x – 4y = 12. To do so, we will find the x- and y-intercepts of the line and use the slope formula.

23 Example Find the slope of the line determined by 3x – 4y = 12. Solution: We first find the coordinates of the intercepts of the line. If x = 0, then y = –3, and the point (0, –3) is on the line. If y = 0, then x = 4, and the point (4, 0) is on the line.

24 Example – Solution We then refer to Figure 3-25 and use the slope formula to find the slope of the line passing through (0, –3) and (4, 0). The slope of the line is Figure 3-25 Substitute 0 for y 2, –3 for y 1, 4 for x 2, and 0 for x 1. cont’d Simplify.

25 Identify the slope of a horizontal and vertical line 4.

26 Slope of Vertical and Horizontal Lines If P(x 1, y 1 ) and Q(x 2, y 2 ) are points on the horizontal line shown in Figure 3-26(a), then y 1 = y 2, and the numerator of the slope formula is 0. Thus, the value of the fraction is 0, and the slope of the horizontal line is 0. On a horizontal line, x 2  x 1. Figure 3-26(a)

27 Slope of Vertical and Horizontal Lines If P(x 1, y 1 ) and Q(x 2, y 2 ) are two points on the vertical line shown in Figure 3-26(b), then x 1 = x 2, and the denominator of the slope formula is 0. Since the denominator cannot be 0, a vertical line has an undefined slope. On a vertical line, y 2  y 1. Figure 3-26(b)

28 Slope of Vertical and Horizontal Lines Slopes of Horizontal and Vertical Lines All horizontal lines (lines with equations of the form y = b) have a slope of 0. Vertical lines (lines with equations of the form x = a) have no defined slope. Comment It is also true that if a line has a slope of 0, it is a horizontal line with an equation of the form y = b. Also, if a line has an undefined slope, it is a vertical line with an equation of the form x = a.

29 Slope of Vertical and Horizontal Lines If a line rises as we follow it from left to right as in Figure 3-27(a), the line is said to be increasing and its slope is positive. If a line drops as we follow it from left to right, as in Figure 3-27(b), it is said to be decreasing and its slope is negative. (a) Positive slope (b) Negative slope Figure 3-27

30 Slope of Vertical and Horizontal Lines If a line is horizontal, as in Figure 3-27(c), it is said to be constant and its slope is 0. If a line is vertical, as in Figure 3-27(d), it has an undefined slope. (c) Zero slope (d) Undefined slope Figure 3-27

31 Determine whether two lines are parallel, perpendicular, or neither parallel nor perpendicular 5.

32 Determining Parallel Lines To see a relationship between parallel lines and their slopes, we refer to the parallel lines l 1 and l 2 shown in Figure 3-28, with slopes of m 1 and m 2, respectively. Figure 3-28

33 Determining Parallel Lines Because right triangles ABC and DEF are similar, it follows that This illustrates that if two nonvertical lines are parallel, they have the same slope. It is also true that when two distinct nonvertical lines have the same slope, they are parallel lines. Read as “the change in y.” Read as “the change in x.”

34 Determining Parallel Lines Slopes of Parallel Lines Nonvertical parallel lines have the same slope. Two distinct nonvertical lines having the same slope are parallel. Since vertical lines are parallel, two distinct lines each with an undefined slope are parallel. Comment The triangle is an uppercase delta in the Greek alphabet.

35 Example The lines in Figure 3-29 are parallel. Find the slope of line l 2. Figure 3-29

36 Example – Solution From the information in the figure, we can find the slope of line l 1. Since the lines are parallel, they will have equal slopes. Therefore, the slope of line l 2 will be equal to the slope of line l 1.

37 Example – Solution The slope of line l 1 is Because the lines are parallel, the slope of line l 2 is also equal to cont’d

38 Determining Parallel Lines Two real numbers a and b are called negative reciprocals or opposite reciprocals if a  b = –1. For example, are negative reciprocals, because

39 Determining Parallel Lines The following relates perpendicular lines and their slopes. Slopes of Perpendicular Lines If two nonvertical lines are perpendicular, their slopes are negative reciprocals. If the slopes of two lines are negative reciprocals, the lines are perpendicular. Because a horizontal line is perpendicular to a vertical line, a line with a slope of 0 is perpendicular to a line with an undefined slope.

40 Two Perpendicular Lines b b a a l1l1 l2l2 m 1 = b/a m 2 = -a/b m 1 · m 2 = (b/a)(-a/b) = -1