https://file.aiquickdraw.com/mathbot/file/1729952565486kh4oz8ll.jpg

To find the average rate of change of the function \( f(x) = 2x^2 - x - 4 \) over the interval \( -4 \leq x \leq 2 \), we use the formula for the average rate of change:

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

where \( a = -4 \) and \( b = 2 \).

### Step-by-Step Solution:

1. **Calculate \( f(-4) \):**
   \[
   f(-4) = 2(-4)^2 - (-4) - 4 = 2 \times 16 + 4 - 4 = 32 + 4 - 4 = 32
   \]

2. **Calculate \( f(2) \):**
   \[
   f(2) = 2(2)^2 - (2) - 4 = 2 \times 4 - 2 - 4 = 8 - 2 - 4 = 2
   \]

3. **Calculate the Average Rate of Change:**
   \[
   \text{Average Rate of Change} = \frac{f(2) - f(-4)}{2 - (-4)} = \frac{2 - 32}{2 + 4} = \frac{-30}{6} = -5
   \]

### Answer:
The average rate of change is \(-5\), so the correct choice is **B (-5)**.

Would you like more details or have any questions?

### Related Questions:
1. How do you interpret the average rate of change of a function geometrically?
2. What is the significance of a negative average rate of change?
3. Can the average rate of change be different for nonlinear functions?
4. How is average rate of change related to the concept of derivatives?
5. How do you compute the average rate of change for a function over a different interval?

### Tip:
Remember, the average rate of change of a function over an interval can be thought of as the slope of the secant line connecting the points at the endpoints of that interval.

What is the average rate of change of f(x) = 2x^2 - x - 4 over the interval -4 ≤ x ≤ 2?

Find the average rate of change of the function f(x) = 2x^2 - x - 4 over the interval -4 ≤ x ≤ 2. This solution involves applying the formula for the average rate of change, suitable for high school algebra students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation