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5.1 Rate of Change and Slope

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

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5.1 Rate of Change and Slope

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

  2. Real World:

  3. Rate of Changecan 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.

  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) Rate of Change = Rate of Change = Rate of Change = Rate of Change from A to B =

  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) Rate of Change = Rate of Change = Rate of Change = Rate of Change from B to C =

  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) Rate of Change = Rate of Change = Rate of Change = Rate of Change from B to C =

  16. Comparing the slopes of the three: Rate of Change from A to B = Rate of Change from B to C = Rate of Change from C to D = As we can see right now the pole from B to C is the one with the biggest change of rate(steepest) = 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: A to B = B to C = A to C = C to D = A to D = B to D = 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. When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change: 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/runto find the slope: rise Make a right triangleto get from one point to another, that is your slope. run = SLOPE=

  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: Slope = x2-x1 B(x2, y2) Slope = y2-y1 A = (1, -1) B = (2, 1) A(x1, y1) Slope = =

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

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

  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) (x1, y1) (x2, y2) Slope = Slope = Slope = Rate of Change from A to B is =

  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) (x1, y1) (x2, y2) Slope = Slope = Slope = Rate of Change (slope) from A to B is =

  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: A( -1 , 2 ) B(-1 , -1) (x1, y1) (x2, y2) Slope = Slope = Slope = We can never divide by Zero thus our slope = UNDEFINED.

  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/linear-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

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