1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 1 Chapter 1 Linear Equations and Linear Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 2 1.3 Slope of a Line

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 3 Comparing the Steepness of Two Objects Two ladders leaning against a building. Which is steeper? We compare the vertical distance from the base of the building to the ladder’s top with the horizontal distance from the ladder’s foot to the building.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 4 Comparing the Steepness of Two Objects Ratio of vertical distance to the horizontal distance: Ladder A: Ladder B: So, Ladder B is steeper.

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 5 Comparing the Steepness of Two Objects To compare the steepness of two objects such as two ramps, two roofs, or two ski slopes, compute the ratio for each object. The object with the larger ratio is the steeper object.

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 6 Example: Comparing the Steepness of Two Roads Road A climbs steadily for 135 feet over a horizontal distance of 3900 feet. Road B climbs steadily for 120 feet over a horizontal distance of 3175 feet. Which road is steeper? Explain.

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 7 Solution Sketches of the two roads are shown below. Note that the distances are not drawn to scale.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 8 Solution Calculate the approximate ratio of the vertical distance to the horizontal distance for each road: Road B is a little steeper because Road B’s ratio is greater than Road A’s.

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 9 Grade of a Road The grade of a road is the ratio of the vertical distance to the horizontal distance, written as a percentage.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 10 Slope of a nonvertical line Definition Let (x 1, y 1 ) and (x 2, y 2 ) be two distinct points of a nonvertical line. The slope of the line is In words, the slope of a nonvertical line is equal to the ratio of the rise to the run (in going from one point on the line to another point on the line).

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 11 Slope A formula is an equation that contains two or more variables. We refer to the equation as the slope formula. Here we list the directions associated with the signs of rises and runs:

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 12 Example: Finding the Slope of a Line Find the slope of the line that contains the points (1, 2) and (5, 4).

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 13 Solution Using the slope formula, where (x 1, y 1 ) = (1, 2) and (x 2, y 2 ) = (5, 4), we have

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 14 Solution By plotting the points, we find that if the run is 4, then the rise is 2. So, the slope is which is our result from using the slope formula.

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 15 Slope Warning It is a common error to substitute into the slope formula incorrectly.

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 16 Example: Finding the Slope of a Line Find the slope of the line that contains the points (2, 3) and (5, 1).

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 17 Solution By plotting the points, we find that if the run is 3, then the rise is –2. So, the slope is which is our result from using the slope formula.

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 18 Increasing and Decreasing Lines An increasing line has positive slope A decreasing line has negative slope

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 19 Example: Finding the Slope of a Line Find the slope of the line that contains the points (–9, –4) and (12, –8).

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 20 Solution Since the slope is negative, the line is decreasing.

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 21 Example: Comparing the Slopes of Two Lines Find the slopes of the two lines sketched at the right. Which line has the greater slope? Explain why this makes sense in terms of the steepness of a line.

22. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 22 Solution For line l 1, if the run is 1, the rise is 2. So,

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 23 Solution For line l 2, if the run is 1, the rise is 4. So,

24. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 24 Solution Note that the slope of line l 2 is greater than the slope of line l 1, which is what we would expect because line l 2 looks steeper than line l 1.

25. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 25 Example: Investigating the Slops of a Horizontal Line Find the slope of the line that contains the points (2, 3) and (6, 3).

26. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 26 Solution We plot the points (2, 3) and (6, 3) and sketch the line that contains the points. So, the slope of the horizontal line is zero, because such a line has “no steepness.”

27. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 27 Example: Investigating the Slops of a Vertical Line Find the slope of the line that contains the points (4, 2) and (4, 5).

28. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 28 Solution We plot the points (4, 2) and (4, 5) and sketch the line that contains the points. Since division by zero is undefined, the slope of the vertical line is undefined.

29. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 29 Slopes of Horizontal and Vertical Lines A horizontal line has slope equal to zero. A vertical line has undefined slope.

30. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 30 Parallel Lines Two lines are called parallel if they do not intersect.

31. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 31 Example: Finding Slopes of Parallel Lines Find the slopes of the parallel lines l 1 and l 2.

32. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 32 Solution For both lines, if the run is 3, the rise is 1. So, the slope of both lines is It makes sense that nonvertical parallel lines have equal slope, since parallel lines have the same steepness.

33. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 33 Slopes of Parallel Lines If lines l 1 and l 2 are nonvertical parallel lines on the same coordinate system, then the slopes of the lines are equal: m 1 = m 2 Also, if two distinct lines have equal slope, then the lines are parallel.

34. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 34 Perpendicular Lines Two lines are called perpendicular if they intersect at a 90° angle.

35. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 35 Example: Finding Slopes of Perpendicular Lines Find the slopes of the perpendicular lines l 1 and l 2.

36. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 36 Solution We see that the slope of line l 1 is and the slope of line l 2 is

37. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 37 Slopes of Perpendicular Lines If lines l 1 and l 2 are nonvertical perpendicular lines, then the slope of one line is the opposite of the reciprocal of the slope of the other line: Also, if the slope of one line is the opposite of the reciprocal of another line’s slope, then the lines are perpendicular.

38. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 38 Example: Finding Slopes of Parallel and Perpendicular Lines A line l 1 has slope 1. If line l 2 is parallel to line l 1, find the slope of line l 2. 2. If line l 3 is perpendicular to line l 1, find the slope of line l 3.

39. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.3, Slide 39 Solution 1. The slopes of lines l 2 and l 1 are equal, so line l 2 2. The slope of line l 3 is the opposite of the has slope reciprocal of or