Calculate the average rate of change over the interval [2, 4] given the function ta...
Check the final answer first, then review the worked steps.
Problem
Calculate the average rate of change over the interval [2, 4] given the function table for f(x) = -x^2 + 12.
Step-by-step solution
- Understand the concept of average rate of change: The average rate of change of a function $f(x)$ over an interval $[a, b]$ is the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$. It is calculated using the formula: $$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$ 2. Identify the interval and corresponding function values: The problem asks for the average rate of change over the interval $[2, 4]$. From the provided table, we can find the function values for $x=2$ and $x=4$. When $x=2$, $f(x) = 8$. When $x=4$, $f(x) = -4$. So, we have the points $(2, 8)$ and $(4, -4)$. 3. Apply the average rate of change formula: Using the formula from Step 1, we substitute $a=2$, $b=4$, $f(a)=8$, and $f(b)=-4$: $$ \text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-4 - 8}{4 - 2} $$ 4. Calculate the difference in function values and the difference in x-values: The difference in function values is $-4 - 8 = -12$. The difference in x-values is $4 - 2 = 2$. 5. Compute the average rate of change: Divide the difference in function values by the difference in x-values: $$ \text{Average Rate of Change} = \frac{-12}{2} = -6 $$
Answer
-6