A function is given. Determine the average rate of change of the function between the given values of the variable.

To determine the **average rate of change** of a function between two given values of the variable, you can use the following formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] Here: - \( f(x) \) is the function. - \( a \) and \( b \) are the two values of the variable between which you're calculating the average rate of change. - \( f(a) \) is the value of the function at \( x = a \). - \( f(b) \) is the value of the function at \( x = b \). ### **Step-by-Step Process** 1. **Identify the Function and Interval:** - Determine the function \( f(x) \). - Identify the two points \( a \) and \( b \) between which you want to find the average rate of change. 2. **Calculate \( f(a) \) and \( f(b) \):** - Substitute \( a \) into the function to find \( f(a) \). - Substitute \( b \) into the function to find \( f(b) \). 3. **Apply the Formula:** - Plug the values of \( f(a) \), \( f(b) \), \( a \), and \( b \) into the average rate of change formula. 4. **Simplify:** - Perform the subtraction and division to find the average rate of change. ### **Example** Let's go through an example for clarity. **Function:** \( f(x) = x^2 + 3x + 2 \) **Interval:** Between \( x = 1 \) and \( x = 4 \) **Step 1:** Identify \( a = 1 \) and \( b = 4 \). **Step 2:** Calculate \( f(1) \) and \( f(4) \). - \( f(1) = (1)^2 + 3(1) + 2 = 1 + 3 + 2 = 6 \) - \( f(4) = (4)^2 + 3(4) + 2 = 16 + 12 + 2 = 30 \) **Step 3:** Apply the formula. \[ \text{Average Rate of Change} = \frac{f(4) - f(1)}{4 - 1} = \frac{30 - 6}{3} = \frac{24}{3} = 8 \] **Conclusion:** The average rate of change of the function \( f(x) = x^2 + 3x + 2 \) between \( x = 1 \) and \( x = 4 \) is **8**. ### **Interpretation** The average rate of change represents the average slope of the function over the interval \([a, b]\). In the example above, it means that, on average, the function increases by 8 units in function value for each unit increase in \( x \) between \( x = 1 \) and \( x = 4 \). ### **Additional Notes** - The average rate of change is closely related to the concept of the **derivative**, which represents the instantaneous rate of change at a specific point. - For linear functions, the average rate of change is constant and is equal to the slope of the line. - For non-linear functions, the average rate of change gives a sense of the overall trend between two points but doesn't capture variations within the interval. If you have a specific function and interval you'd like to analyze, feel free to provide them, and I can walk you through the calculation!