Precalculus Midterm Study Guide – Step-by-Step Guidance

Q4. Find the average rate of change of each function on the interval [−1, 2]

Background

Topic: Average Rate of Change

This question is testing your understanding of how to calculate the average rate of change of a function over a given interval. This is a foundational concept in precalculus and calculus, closely related to the concept of the slope of a secant line between two points on a graph.

Key Terms and Formula:

• Average Rate of Change: The change in the value of a function over a specified interval, divided by the change in the input (x-values).

The formula for the average rate of change of a function $f(x)$ from $x = a$ to $x = b$ is:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Step-by-Step Guidance

1. Identify the function and the interval. For each part (A, B, C), you will use the given representation (table, equation, or graph) to find the values of the function at $x = -1$ and $x = 2$.

2. For the table (A), locate the $y$-values that correspond to $x = -1$ and $x = 2$.

3. For the equation (B), substitute $x = -1$ and $x = 2$ into the function $g(x) = 8x - 3$ to find $g(-1)$ and $g(2)$.

4. For the graph (C), estimate or read off the $y$-values at $x = -1$ and $x = 2$ as accurately as possible.

5. Apply the average rate of change formula for each function using the values you found:

   $\text{Average Rate of Change} = \frac{f(2) - f(-1)}{2 - (-1)}$

Try solving on your own before revealing the answer!