 # 5 1 Rate of Change and Slope Rate

• Slides: 33 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). Slope = Vertical Change (y) Horizontal Change (x) = rise run Real World: 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: Choosing the data from: Time and Distance 1 min 260 ft. 2 min 520 ft. We have the following: Choosing the data from: Time and Distance 1 min 260 ft. 3 min 780 ft. We have the following: Choosing the data from: Time and Distance 1 min 260 ft. 4 min 1040 ft. We have the following: NOTE: When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change: YOU TRY IT: Determine whether the following rate of change is constant in the miles per gallon of a car. Gallons Miles 1 28 3 84 5 140 7 196 Choosing the data from: Gallons and Miles 1 g 28 m 3 g 84 m We have the following: Choosing the data from: Gallons and Miles 1 g 28 m. 5 g 140 m. We have the following: THUS: the rate of change is CONSTANT. Once Again: Real World X Y 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) 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) 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) 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) 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. Finally: Finally we can conclude that the poles with the steepest path are poles B to C with slope of 5/4. Class Work: Pages: 295 -297 Problems: 1, 4, 8, 9, Remember: When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change: 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: We further use the concept of rise/run to find the slope: run rise Make a right triangle to get from one point to another, that is your slope. CONSTANT rate of change: due to the fact that a line is has no curves, we use the following formula to find the SLOPE: y 2 -y 1 x 2 -x 1 B(x 2, y 2) A(x 1, y 1) A = (1, -1) B = (2, 1) YOU TRY: Find the slope of the line: YOU TRY (solution): -4 (0, 4) (2, 0) 2 YOU TRY IT: Provide the slope of the line that passes through the points A(1, 3) and B(5, 5): 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) YOU TRY: Find the slope of the line: 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) YOU TRY: Find the slope of the line: 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) (x 1, y 1) (x 2, y 2) We can never divide by Zero thus our slope = UNDEFINED. THEREFORE: Horizontal ( slope of ZERO ) lines have a While vertical ( ) lines have an UNDEFINED slope. VIDEOS: Graphs https: //www. khanacademy. org/math/algebra/line ar-equations-and-inequalitie/slope-andintercepts/v/slope-and-rate-of-change Class Work: Pages: 295 -297 Problems: As many as needed to master the concept