Introduction The rate of change is a ratio that describes how much one quantity changes with respect to the change in another quantity. For a function, the rate of change is the amount by which function values change for each change in domain values. When the rate of change of a function increases by the same amount for every consecutive domain value, the rate of change is said to be constant. Linear functions exhibit this trait; however, not all functions have a constant rate of change : Calculating Average Rates of Change

Key Concepts The rate of change of a single function is dependent on the change that occurs between two points (ordered pairs) on the graph of that function. Recall that this is the same as the slope when the function is a line : Calculating Average Rates of Change

Key Concepts, continued The rate of change of a function is the difference between the values of the function at two different domain values; mathematically, this can be represented by the formula, in which f 1 (x) is the function value at the domain value x 1 and f 2 (x) is the function value at the domain value x : Calculating Average Rates of Change

Key Concepts, continued The uppercase Greek letter delta ( ) is commonly used to represent the “change” in a value; for example, f (x) can be read as “change in f of x.” Therefore, f (x) and x are more concise ways of representing the numerator and denominator, respectively, in the rate of change formula. Note that the order in which the function values are compared must be the same as the order in which the domain values are compared. Mixing up the order of the values in the numerator with the order of the values in the denominator will result in an incorrect rate calculation : Calculating Average Rates of Change

Key Concepts, continued The average rate of change is the average of all of the different rates of change of a function value over a restricted domain. Formally, it is the ratio of the difference of output values to the difference of the corresponding input values: The average rate of change of a function or a specific variable is useful in those mathematical or real-world problems in which the average rate of change of different function values varies over equal restricted domain intervals : Calculating Average Rates of Change

Key Concepts, continued The average rate of change is often calculated by substituting the function values that correspond to the endpoints or bounds of a restricted domain interval into the formula for calculating the rate of change. The rate of change of a function or variable in a real- world situation can be positive, negative, 0, undefined, or a constant, depending on the situation. In particular, the rate of change of a variable can be negative or 0 even if the variable values are always positive, such as in problems involving area, mass, time, volume, and any number of similar quantities : Calculating Average Rates of Change

Key Concepts, continued A special case of the constant rate of change can be found with linear functions. For a linear function ax + by = c, the rate of change of the function values y is a constant given by the quantity This quantity is the slope of the straight-line graph that represents the linear function. In the slope-intercept form of the linear function f(x) = mx + b, the constant m is the rate of change of the function values : Calculating Average Rates of Change

Key Concepts, continued The average rate of change of a quantity can have special meaning in situations in which the restricted domain over which it is computed is very small or approaches 0. A graphing calculator can be helpful in approximating or visualizing the rate of change in such applications : Calculating Average Rates of Change

Common Errors/Misconceptions reversing the order of the function values and domain values in a rate of change calculation switching the order of two function values in the numerator of a rate of change calculation from that of the two domain values in the denominator neglecting to check a rate of change calculation, especially in real-world problems, for accuracy or reasonableness in terms of the problem assumptions, conditions, and restrictions : Calculating Average Rates of Change

Guided Practice Example 2 Two school clubs are selling plants to raise money for a trip. The juggling club started out with 250 plants and had 30 left after selling them for 7 days. The mock trial club started out with 450 plants and had 50 left after selling them for 8 days. Compare the rates at which the two clubs are selling their plants. If the two clubs started selling plants at the same time, which club will sell out of plants first? Explain : Calculating Average Rates of Change

Guided Practice: Example 2, continued 1.Determine how many plants the juggling club has sold. The juggling club has sold 250 – 30 or 220 plants : Calculating Average Rates of Change

Guided Practice: Example 2, continued 2.Calculate the average number of plants the juggling club sold per day. The juggling club sold 220 plants over 7 days. Divide to determine the average. The juggling club sold an average of approximately 31 plants each day : Calculating Average Rates of Change

Guided Practice: Example 2, continued 3.Determine how many plants the mock trial club has sold. The mock trial club has sold 450 – 50 or 400 plants : Calculating Average Rates of Change

Guided Practice: Example 2, continued 4.Calculate the average number of plants the mock trial club sold per day. The mock trial club sold 400 plants over 8 days. Divide to determine the average. The mock trial club sold an average of 50 plants each day : Calculating Average Rates of Change

Guided Practice: Example 2, continued 5.Describe the rates at which the two clubs sold their plants. Compare the kinds of numbers that represent days, plants, and rates. The variables “days” and “plants” are positive whole numbers, whereas “rate” can be a negative number. In this situation, the rates for the two clubs are negative, because the rates reflect a reduction in the number of plants each club has in stock : Calculating Average Rates of Change

Guided Practice: Example 2, continued The rate for the juggling club is approximately –31 plants per day because the quantity of plants is being reduced by an average of about 31 per day; similarly, the rate for the mock trial club is –50 plants per day : Calculating Average Rates of Change

Guided Practice: Example 2, continued 6.Calculate how long it will take the juggling club to sell its remaining plants. The juggling club has 30 plants left and is selling them at a rate of about –31 plants per day, so the juggling club’s plants will sell out in 1 day : Calculating Average Rates of Change

Guided Practice: Example 2, continued 7.Calculate how long it will take the mock trial club to sell its remaining plants. The mock trial club has 50 plants left and is selling them at a rate of –50 plants per day, so the mock trial club’s plants will sell out in 1 day : Calculating Average Rates of Change

Guided Practice: Example 2, continued 8.Explain which club will sell out of plants first. Both clubs will sell off their remaining stock in 1 day. However, assuming each club started selling the plants on the same day, the juggling club will sell out first because it took 1 day less to sell off its initial stock of plants. (Recall that the juggling club sold plants for 7 days and the mock trial club sold plants for 8 days.) : Calculating Average Rates of Change ✔

Guided Practice: Example 2, continued : Calculating Average Rates of Change

Guided Practice Example 3 Find the average rate of change of the function f(x) = x 3 + 2x 2 – x – 2 over the restricted intervals [–2, –1] and [–1, 1]. Determine if the rate of change over both intervals is positive, negative, 0, undefined, and/or constant. Use the graph of f(x) to explain how the average rate of change relates to the behavior of the graph across its whole domain : Calculating Average Rates of Change

Guided Practice: Example 3, continued 1.Calculate the average rate of change of f(x) over the interval [–2, –1]. First find the values of f(–2) and f(–1). Calculate f(–2) : Calculating Average Rates of Change Original function Substitute –2 for x. Simplify.

Guided Practice: Example 3, continued Calculate f(–1) : Calculating Average Rates of Change Original function Substitute –1 for x. Simplify.

Guided Practice: Example 3, continued Substitute the values for f(–2) and f(–1) into the rate of change formula and solve. The rate of change of f(x) over the interval [–2, –1] is : Calculating Average Rates of Change Rate of change formula Substitute 0 for f 1 (x), 0 for f 2 (x), –2 for x 1, and –1 for x 2. Simplify.

Guided Practice: Example 3, continued 2.Calculate the average rate of change of f(x) over the interval [–1, 1]. The value of f(–1) is 0, as found in the previous step. Find the value of f(1). Calculate f(1) : Calculating Average Rates of Change Original function Substitute 1 for x. Simplify.

Guided Practice: Example 3, continued Substitute the values for f(–1) and f(1) in the rate of change formula. The rate of change of f(x) over the interval [–1, 1] is : Calculating Average Rates of Change Rate of change formula Substitute 0 for f 1 (x), 0 for f 2 (x), –1 for x 1, and 1 for x 2. Simplify.

Guided Practice: Example 3, continued 3.Determine if the rate of change over both intervals is positive, negative, 0, undefined, and/or constant. The rate of change that occurred over each of the two intervals is 0 and constant : Calculating Average Rates of Change

Guided Practice: Example 3, continued 4.Graph f(x). Use a graphing calculator to view the graph of f(x) : Calculating Average Rates of Change

Guided Practice: Example 3, continued 5.Describe what happens to the function values over the restricted domains [–2, –1] and [–1, 1]. The function value goes from 0 to a maximum value and back to 0 over the restricted domain [–2, –1]. The function value goes from 0 to a minimum value and back to 0 over the restricted domain [–1, –1]. For the domain interval (−∞, −2), the function value range is given by the interval (−∞, 0); for the domain interval (1,  ∞), the function value range is given by the interval (0,  ∞) : Calculating Average Rates of Change ✔

Guided Practice: Example 3, continued : Calculating Average Rates of Change