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.