Lesson 3.4 Constant Rate of Change (linear functions) 3.3.2: Proving Average Rate of Change

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-values to the change in x-values. Formula: m = Linear functions have a constant rate of change, meaning values increase or decrease at the same rate over a period of time. 3.3.2: Proving Average Rate of Change

Constant Rate of Change (slope) Recall….. The rate of change between any two points of a linear function will be equal. Calculating Constant Rate of Change (slope) from a Table Choose two points from the table. Assign one point to be (x1, y1) and the other point to be (x2, y2). Substitute the values into the slope formula: 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑦 2 The result is the rate of change for the interval between the two points chosen. 3.3.2: Proving Average Rate of Change

Number of Customers (f(x)) 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 $6 105 $8 90 $10 75 3.3.2: Proving Average Rate of Change

Number of Customers (f(x)) Guided Practice: Example 1, continued Choose two points from the table. (4, 120) and (10, 75) 2. Assign one point to be (x1, y1) and the other to be (x2, y2). It doesn’t matter which is which. Let (4, 120) be (x1, y1) and (10,76) be (x2, y2). Carwash Price (x) Number of Customers (f(x)) $4 120 $6 106 $8 92 $10 78 3.3.2: Proving Average Rate of Change

Guided Practice: Example 1, continued 3. Substitute (0, 1.5) and (155, 0) into the slope formula to calculate the rate of change. Slope formula Substitute (4, 120) and (10, 78) for (x1, y1) and (x2, y2). Simplify as needed. = 78−120 10−4 = −42 6 = -7 The rate of change for this function is -7 customers per dollar. For every dollar the carwash price increases, 7 customers are lost. 3.3.3: Recognizing Average Rate of Change

Constant Rate of Change (slope) Recall…. The rate of change between any two points of a linear function will be equal. Estimating Constant Rate of Change (slope) from a Graph Pick two points from the graph. Identify (x1, y1) as one point and (x2, y2) as the other point. Substitute (x1, y1) and (x2, y2) into the slope formula to calculate the rate of change. The result is the estimated constant rate of change (slope) for the graph. m = 3.3.2: Proving Average Rate of Change

Calculate the constant rate of change (slope) for these tables. You Try Calculate the constant rate of change (slope) for these tables. 1) 2) x f(x) 1 -6 2 -11 3 -16 4 -21 x f(x) -3 -5 -4 3 6 -2 3.3.2: Proving Average Rate of Change

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? 3.3.3: Recognizing Average Rate of Change

Guided Practice: Example 2, continued 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 (x1, y1) as one point and (x2, y2) as the other point. It doesn’t matter which is which. Let’s have (0, 1.5) be (x1, y1) and (155,0) be (x2, y2) 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. Slope formula Substitute (0,1.5) and (155, 0) for (x1, y1) and (x2, y2). Simplify as needed. = 0−1.5 155−0 = −1.5 155 ≈ -0.01 3.3.3: Recognizing Average Rate of Change

Calculate the constant rate of change (slope) for these graphs. You Try Calculate the constant rate of change (slope) for these graphs. 1) 2) 3.3.2: Proving Average Rate of Change