1. 5.1 Rate of Change and Slope Rate of Change: The relationship between two changing quantities Slope: the ratio of the vertical change (rise) to the horizontal change (run). Rate of Change = Change in the dependent variable (y-axis) Change in the independent variable (x-axis) Slope = Vertical Change (y) = rise Horizontal Change (x) run

2. Real World:

3. Rate of Change can be presented in many forms such as: We can use the concept of change to solve the cable problem by using two sets of given data, for example: A band practices their march for the parade over time as follows:

4. Choosing the data from: Time and Distance 1min 260 ft. 2min 520 ft. We have the following:

5. Choosing the data from: Time and Distance 1min 260 ft. 3min 780 ft. We have the following:

6. Choosing the data from: Time and Distance 1min 260 ft. 4min 1040 ft. We have the following:

7. NOTE: When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change:

8. YOU TRY IT: Determine whether the following rate of change is constant in the miles per gallon of a car. GallonsMiles 128 384 5140 7196

9. Choosing the data from: Gallons and Miles 1 g 28 m 3g 84 m We have the following:

10. Choosing the data from: Gallons and Miles 1g 28 m. 5g 140 m. We have the following: THUS: the rate of change is CONSTANT.

11. Once Again: Real World

12. Remember: Rate of Change can be presented in many forms: We can use the concept of change to solve the cable problem by using two sets of given data: ( x, y ) A : Horizontal(x) = 20 Vertical(y) = 30  (20, 30) B : Horizontal(x) = 40 Vertical(y) = 35  (40, 35)

13. Using the data for A and B and the definition of rate of change we have: ( x, y ) A : Horizontal = 20 Vertical = 30  (20, 30) B : Horizontal = 40 Vertical = 35  (40, 35)

14. Using the data for B and C and the definition of rate of change we have: ( x, y ) B : Horizontal = 40 Vertical = 35  (40, 35) C : Horizontal = 60 Vertical = 60  (60, 60)

15. Using the data for C and D and the definition of rate of change we have: ( x, y ) C : Horizontal = 60 Vertical = 60  (60, 60) D : Horizontal = 100 Vertical = 70  (100, 70)

16. Comparing the slopes of the three: However, we must find all the combination that we can do. Try from A to C, from A to D and from B to C.

17. Finally: Finally we can conclude that the poles with the steepest path are poles B to C with slope of 5/4.

18. Class Work: Pages: 295-297 Problems: 1, 4, 8, 9,

19. Remember: When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change:

20. We further use the concept of CONSTANT slope when we are looking at the graph of a line:

21. We further use the concept of rise/run to find the slope: Make a right triangle to get from one point to another, that is your slope. rise run

22. CONSTANT rate of change: due to the fact that a line is has no curves, we use the following formula to find the SLOPE: A(x 1, y 1 ) B(x 2, y 2 ) y 2 -y 1 x 2 -x 1 A = (1, -1) B = (2, 1)

23. YOU TRY: Find the slope of the line:

24. YOU TRY (solution): -4 2 (0,4)(0,4) (2,0)(2,0)

25. YOU TRY IT: Provide the slope of the line that passes through the points A(1,3) and B(5,5):

26. YOU TRY IT: (Solution) Using the given data A(1,3) and B(5,5) and the definition of rate of change we have: A( 1, 3 )B(5, 5) (x 1, y 1 ) (x 2, y 2 )

27. YOU TRY: Find the slope of the line:

28. YOU TRY IT: (Solution) Choosing two points say: A(-5,3) and B(1,5) and the definition of rate of change (slope) we have: A( -2, 3 )B(1, 3) (x 1, y 1 ) (x 2, y 2 )

29. YOU TRY: Find the slope of the line:

30. YOU TRY IT: (Solution) Choosing two points say: A(-1,2) and B(-1,-1) and the definition of rate of change (slope) we have: We can never divide by Zero thus our slope = UNDEFINED. A( -1, 2 )B(-1, -1) (x 1, y 1 ) (x 2, y 2 )

31. THEREFORE: Horizontal ( ) lines have a slope of ZERO While vertical ( ) lines have an UNDEFINED slope.

32. VIDEOS: Graphs https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/slope-and- intercepts/v/slope-and-rate-of-change

33. Class Work: Pages: 295-297 Problems: As many as needed to master the concept