Question

1. Calculate the average rate of change over the interval [2,4] given the function table for f(x) = −x^2 + 12.
X: 1, 2, 3, 4
f(x) = 11, 8, 3, -4

A) 2
B) -6
C) 6
D) -2


2. Calculate the average rate of change over the interval [3,4] of the given graph of the function.
A) 16
B) 1/16
C) -16
D) 8

Answer 1

To solve these problems, we will use the formula for the average rate of change of a function \( f(x) \) over the interval \([a, b]\):

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

### Problem 1:
We need to calculate the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\).

From the provided values:
- \( f(2) = 8 \)
- \( f(4) = -4 \)

Using the formula:

\[
\text{Average rate of change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-4 - 8}{4 - 2} = \frac{-12}{2} = -6
\]

So the answer for Problem 1 is **B) -6**.

### Problem 2:
For Problem 2, we need to calculate the average rate of change over the interval \([3, 4]\). However, we need the specific values from the given graph, which is not provided here. But if we assume we know the values from the graph:
- Let's assume \( f(3) \) has a certain value (we'll denote it as \( y_3 \)).
- Let's assume \( f(4) \) also has a certain value (we'll denote it as \( y_4 \)).

If for example (imaginary values), \( f(3) = 3 \) and \( f(4) = -4 \), then we can calculate the average rate of change as:

\[
\text{Average rate of change} = \frac{f(4) - f(3)}{4 - 3} = \frac{-4 - 3}{4 - 3} = \frac{-7}{1} = -7
\]

However, since we don't have the value of \( f(3) \) directly from your question, I cannot provide an exact answer for Problem 2 without that information.

If you have the values from the graph, you can plug them into the formula similarly to find the result.

If you provide those values, I can help you determine the exact average rate of change for Problem 2.