Lesson 3.3 Constant Rate of Change (linear functions) Concept: Characteristics of a function EQ: How do we analyze a real world scenario to interpret a constant rate of change? (F.IF.7) Vocabulary: Rate of change 1
Introduction A rate of change is a ratio that describes how much one quantity changes with respect to the change in another quantity of the function. With linear functions the rate of change is called the slope. The slope of a line is the ratio of the change in y y-values to the change in x-values. Formula: m = 2 - y 1 x 2 - x 1 Linear functions have a constant rate of change, meaning values increase or decrease at the same rate over a period of time. 2
Recall.. Calculating Constant Rate of Change (slope) from a Table 1. Choose two points from the table (highest and lowest points). 2. Assign one point to be (x 1, y 1 ) and the other point to be (x 2, y 2 ). 3. Substitute the values into the slope formula m = y 2 y 1 x 2 y 2 4. The result is the rate of change for the interval between the two points chosen. * The rate of change between any two points of a linear function will be equal. 5. Interpret your answer for the context of the problem. 3
Starter Activity Tommy's Amazing Car Wash Technology! http://youtu.be/q-x6qeirrva
Guided Practice Example 1 To raise money, students plan to hold a car wash. They ask some adults how much they would pay for a car wash. The table on the right shows the results of their research. What is the rate of change for their results? Carwash Price (x) Number of Customers (f(x)) $4 120 $6 106 $8 92 $10 78 5
Guided Practice: Example 1, continued 1. Choose two points from the table. (4, 120) and (10, 78) Carwash Price (x) Number of Customers (f(x)) $4 120 $6 106 $8 92 2. Assign one point to be (x 1, y 1 ) and the other $10 78 to be (x 2, y 2 ). It doesn t matter which is which. Let (4, 120) be (x 1, y 1 ) and (10,78) be (x 2, y 2 ). 6
Guided Practice: Example 1, continued 3. Substitute (4, 120) and (10, 78) into the slope formula to calculate the rate of change. y 2 - y 1 x 2 - x 1 Slope formula = 78 120 10 4 = 42 6 Substitute (4, 120) and (10, 78) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. = -7 The rate of change for this function is -7 customers per dollar. For every dollar the carwash price increases, 7 customers are lost. 7 3.3.3: Recognizing Average Rate of Change
Recall.. Estimating Constant Rate of Change (slope) from a Graph 1. Pick two points from the graph. 2. Identify (x 1, y 1 ) as one point and (x 2, y 2 ) as the other point. 3. Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula to calculate the rate of change. m = y 2 y 1 x 2 y 2 4. The result is the estimated rate of change (slope) for the graph. *The rate of change between any two points of a linear function will be equal. 5. Interpret your answer for the context of the problem. 8
Guided Practice Example 2 The graph to the right compares the distance a small motor scooter can travel in miles to the amount of fuel used in gallons. What is the rate of change for this scenario? 9 3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 2, continued 1. Pick two points from the graph. The function is linear, so the rate of change will be constant for any interval (continuous portion) of the function. Choose points on the graph with coordinates that are easy to estimate. For example, (0, 1.5) and (155,0) 2. Identify (x 1, y 1 ) as one point and (x 2, y 2 ) as the other point. It doesn t matter which is which. Let s have (0, 1.5) be (x 1, y 1 ) and (155,0) be (x 2, y 2 ) 10 3.3.3: Recognizing Average Rate of Change
Guided Practice: Example 2, continued 3. Substitute (0, 1.5) and (155, 0) into the slope formula to calculate the rate of change. y 2 - y 1 x 2 - x 1 Slope formula = 0 1.5 155 0 = 1.5 155 Substitute (0,1.5) and (155, 0) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. -0.01 11 3.3.3: Recognizing Average Rate of Change
You Try 1 Calculate the constant rate of change(slope) from the table of values. You are a cashier at McDonald s and you are paid per hour. After two hours of work, you earn $15. Then after 6 hours of work, you have earned $45. How do you find your hourly rate? Hours worked (x) Money earned (f(x)) 2 15 4 30 6 45 8 60 12
You Try 2 Calculate the constant rate of change (slope) for the graph. Tim is walking to his grandmother s house and is using a smartphone app to calculate his speed and the distance he ran. The app gives him the graph to the right. What is his speed(rate of change)? 13
$2.00 Summary Create a sentence consisting of 20 words which summarizes what you learned about constant rate of change.