Given the function f, of, x, equals, x, squared, minus, x, minus, 8f(x)=x
2
−x−8, determine the average rate of change of the function over the interval minus, 1, is less than or equal to, x, is less than or equal to, 5−1≤x≤5.

To determine the average rate of change of the function \( f(x) = x^2 - x - 8 \) over the interval \([-1, 5]\), we can use the formula for the average rate of change of a function over an interval \([a, b]\):

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

Here, \(a = -1\) and \(b = 5\).

### Step 1: Find \(f(-1)\) and \(f(5)\)
1. **Calculate \(f(-1)\):**
   \[
   f(-1) = (-1)^2 - (-1) - 8 = 1 + 1 - 8 = -6
   \]

2. **Calculate \(f(5)\):**
   \[
   f(5) = 5^2 - 5 - 8 = 25 - 5 - 8 = 12
   \]

### Step 2: Apply the formula
Substitute \(f(-1)\), \(f(5)\), \(a = -1\), and \(b = 5\) into the formula:
\[
\text{Average Rate of Change} = \frac{f(5) - f(-1)}{5 - (-1)} = \frac{12 - (-6)}{5 - (-1)} = \frac{12 + 6}{6} = \frac{18}{6} = 3
\]

### Conclusion:
The average rate of change of the function \( f(x) = x^2 - x - 8 \) over the interval \([-1, 5]\) is \( \boxed{3} \).

Would you like any more details, or do you have any questions?

Here are some related questions you might find interesting:
1. How would the average rate of change differ if the interval was \([0, 4]\)?
2. What is the significance of the average rate of change in terms of the function's graph?
3. Can you find the instantaneous rate of change of \(f(x)\) at \(x = 2\)?
4. How does the average rate of change compare to the slope of the secant line between \((-1, f(-1))\) and \((5, f(5))\)?
5. How would you determine if the function \(f(x)\) is increasing or decreasing over a specific interval?

**Tip:** The average rate of change of a function over an interval gives the slope of the secant line connecting the endpoints of the interval on the function's graph.

Learn how to calculate the average rate of change of the quadratic function f(x) = x^2 - x - 8 over the interval [-1, 5]. Follow a step-by-step solution using the average rate of change formula. 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