https://file.aiquickdraw.com/mathbot/file/17299509904396xisxd7l.jpg

To find the average rate of change of \( f(x) = 2x^2 - x - 4 \) over the interval \( -4 \leq x \leq 2 \), we can use the formula for the average rate of change over an interval \( [a, b] \):

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

In this case:
- \( f(x) = 2x^2 - x - 4 \)
- \( a = -4 \)
- \( b = 2 \)

### Step 1: Calculate \( f(2) \)
\[
f(2) = 2(2)^2 - (2) - 4 = 2 \cdot 4 - 2 - 4 = 8 - 2 - 4 = 2
\]

### Step 2: Calculate \( f(-4) \)
\[
f(-4) = 2(-4)^2 - (-4) - 4 = 2 \cdot 16 + 4 - 4 = 32 + 4 - 4 = 32
\]

### Step 3: Apply the Average Rate of Change Formula
\[
\text{Average rate of change} = \frac{f(2) - f(-4)}{2 - (-4)} = \frac{2 - 32}{2 + 4} = \frac{-30}{6} = -5
\]

### Answer
The average rate of change is \(-5\), so the correct answer is:

**B. -5**

Would you like further explanation on any part of this process?

---

Here are 5 related questions to deepen your understanding:
1. How do you calculate the average rate of change for other types of functions, like exponential or trigonometric functions?
2. What does the average rate of change represent graphically for a quadratic function?
3. How can we interpret the average rate of change in real-world applications?
4. How does the interval chosen impact the average rate of change for a function?
5. What is the difference between the average rate of change and the instantaneous rate of change?

**Tip:** For quadratic functions like \( f(x) = ax^2 + bx + c \), the average rate of change over any interval will generally vary depending on the endpoints due to the curve's slope changing at different points.

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

This problem explores the average rate of change for the function f(x) = 2x^2 - x - 4 over the interval -4 ≤ x ≤ 2. Step-by-step calculations are provided, applying the average rate of change formula for a quadratic function. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation