1. Slope of a Line Chapter 7 Section 3

2. Learning Objective Find the slope of a line Recognize positive and negative slopes Examine the slopes of horizontal and vertical lines Examine the slope of parallel and perpendicular lines Key Vocabulary: slope, rise, run, positive slope, negative slope, parallel, perpendicular, negative reciprocals

3. Find the Slope of a Line The slope of a line is a ratio of the vertical change to the horizontal change between any two selected points on the line. The slope of a line is a measure of the steepness of the line Road has a 8% grade, = Run 100 ft Rise 8 ft

4. Find the Slope of a Line Slope of the line goes through the points m is used for the slope and the Greek letter Delta (Δ) means “the change in” It makes no difference which two points are selected when finding the slope of the line. It makes no difference which points you labeled You read the slope of a graph from left to right just as you read a book Change in y Change in x

5. Find the Slope of a Line Example:(-2, 1) and (3, 4) (-2,1) (3,4) (0,0) Increase (up) 3 units on the y-axis, increase (over/right) 5 unit on the x-axis Positive Slope

6. Find the Slope of a Line Example:(1, 2) and (3, 6) (1,2) (3,6) (0,0) Increase (up) 2 units on the y-axis, increase (over/right) 1 unit on the x-axis Positive Slope

7. Find the Slope of a Line Example:(1, 0) and (3, 4) (1,0) (3,4) (0,0) Increase (up) 2 units on the y-axis, increase (over/right) 1 unit on the x-axis. This is a positive slope. Positive Slope

8. Find the Slope of a Line Example: Find the slope of the following line (4,2) (1,4) (0,0) Down (decrease) 2 units on the y-axis, increase (over/right) 3 unit on the x-axis Negative Slope

9. Find the Slope of a Line Example: Find the slope of the following line (20) (5, 15) (0,0) Down (decrease) 2 units on the y-axis, over (right) 3 unit on the x-axis (5) (10) (30, 5) (5) (5, 15) and (30, 10) (15) (10) (15) (25)

10. Slope of a Horizontal Line Every horizontal line has a slope of 0 Example: y = 2 (2,2) (0,0) (-2,2) There is no change in y. Slope is 0. Any two points would yield the same slope of 0. Horizontal line is parallel to the x-axis

11. Slope of a Vertical Line The slope of a vertical line is undefined Example: x = 2 (2,2) (0,0) (2,-2) There is no change in x. Slope is undefined. Any two points would yield the same results. Vertical line is parallel to the y-axis

12. Slope of Parallel Line Two nonvertical lines with the same slope and different y-intercepts are parallel lines. Any two vertical lines are parallel to each other. Example: draw two lines with a slope of 1; one through (1,3) and another through (1, -2) m = 1 (up 1 and right 1) (1,3) (0,0) (1,-2) These two lines are parallel. Parallel lines do not intersect. Two parallel lines will have the same slope, and a different y-intercept Line 1 Line 2

13. Slope of Perpendicular Line To determine if two lines are perpendicular multiply the slopes of the two lines together. If the product is -1 then the slopes are negative reciprocals, and the lines are perpendicular. m 1 represents line 1and m 2 represents line 2 m 1 = 2 m 2 = - ½

14. Slope of Perpendicular Line Two lines whose slopes are negative reciprocals of each other are perpendicular lines. Any two vertical lines is perpendicular to any horizontal line. Example: Draw a line with a slope of 2; through (1,-1) Up 2 right 1 Draw a line with a Slope of – ½ through (-2, 4) Down 1 and right 2 Perpemdicular The reciprocal of 2 is ½ The reciprocal of -½ is -2 The product of the slopes is -1. (2)(- ½) = -1 (1,-1) (0,0) (-2,4) Line 1 Line 2

15. Determining if lines are Parallel, Perpendicular, or neither

16. Positive Slopes Slopes are read from left to right, just like a book. Positive slope y increases as x increases Rises as it moves from left to right (0,0)

17. Negative Slopes Slopes are read from left to right, just like a book. Negative slope y decreases as x decreases Falls as it moves from left to right (0,0)

18. Remember The y goes in the numerator. The x goes in the denominator. Be careful when you subtract negatives. The signs become positive (+). Example: 3 – (-2) = 3 + 2 The term negative reciprocal may be thought of as opposite reciprocal.

19. HOMEWORK 7.3 Page 449-450: #11, 13, 15, 19, 25, 27, 33, 35