1. Find Slope and Rate of Change Chapter 2.2

2. How Fast is He Walking? 2

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5. Rate of Change In this second animation, the man’s speed is itself changing after every second passes The first animation is an example of a constant rate of change In a motor vehicle, you experience a constant rate of change (change of position over time) when the value on your speedometer is not changing The second animation is an example of a variable rate of change In a motor vehicle, you experience a variable rate of change occurs the value on a speedometer changes as time passes 5

6. Rate of Change The graph of two related quantities (like distance and time when something is moving) will be a line if the rate of change is constant This rate of change is commonly called the slope of the line It turns out that the graph of two related quantities, like distance and time, is curved when the rate of change is not constant Such a graph also has a kind of slope; finding this “slope” is the object of one of the two major branches of calculus 6

7. Rate of Change 7

8. Slope of a Line 8

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11. Example 11

12. Example 12

13. Guided Practice 13

14. Guided Practice 14

15. Classifying Lines by Slope You should be able to tell by looking at a line whether its slope is positive, negative, zero, or undefined Vertical lines have undefined slopes because, using the slope formula, the denominator yields zero and division by zero is not defined Horizontal lines have zero slopes because, using the slope formula, the numerator yields zero and every non-zero number divided by zero is zero For the last two cases, imagine that you are walking on a line from left to right in the coordinate plane 15

16. Negative Slope 16

17. Negative Slope 17

18. Positive Slope 18

19. Positive Slope 19

20. Classification of Line by Slope A vertical line has an undefined slope A horizontal line has a slope of zero A line that falls from left to right has a negative slope A line that rises from left to right has a positive slope 20

21. Example 21

22. Guided Practice 22

23. Guided Practice 23

24. Parallel & Perpendicular Lines Recall from geometry that Two lines are parallel if they never intersect Two lines are perpendicular if they intersect at right angles (90˚) It is possible to show how the slopes of lines that are parallel or lines that are perpendicular are related, but this is lengthy so we will just remember the relationship 24

25. Parallel & Perpendicular Lines Two non-vertical lines are parallel if and only if their slopes are equal The phrase “if and only if” means two things: Two lines are parallel if their slopes are equal Two lines with equal slopes are parallel 25

26. Parallel & Perpendicular Lines 26

27. Example 27

28. Example 28

29. Example 29

30. Guided Practice 30

31. Guided Practice 31

32. Guided Practice 32

33. Rate of Change At the beginning of this presentation, we considered slope as the ratio of the change in the distance a man walked compared to the time that passed The slope of a line is always the ratio of the change in one quantity compared to the change in another That is, slope is an average rate of change 33

34. Example 34

35. Guided Practice A Giant Sequoia tree has a diameter of 248 inches in 1965 and a diameter of 251 inches in 2005. Find the average rate of change in the diameter. Include units in your answer. 35

36. Guided Practice 36

37. Exercise 2.1 Handout 37