Question

The graph of a function is shown. Find the average rate of change between each of the given points.
points: (-8,-2) (-6,-3) (1,-3) (4,0)
x=−8 to x=−6:


x=1 to x=4:

Answer 1

To find the average rate of change of a function between two points, we use the formula:

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

where \( f(a) \) and \( f(b) \) are the function values at points \( a \) and \( b \) respectively.

Let's find the average rate of change for the two pairs of points you provided.

### 1. From \( x = -8 \) to \( x = -6 \):

Given points: (-8, -2) and (-6, -3)

Here, \( a = -8 \) and \( b = -6 \).

These points give us:

- \( f(-8) = -2 \)
- \( f(-6) = -3 \)

Now, applying the formula:

\[
\text{Average Rate of Change} = \frac{f(-6) - f(-8)}{-6 - (-8)}
\]
\[
= \frac{-3 - (-2)}{-6 + 8}
\]
\[
= \frac{-3 + 2}{2} = \frac{-1}{2} = -0.5
\]

### 2. From \( x = 1 \) to \( x = 4 \):

Given points: (1, -3) and (4, 0)

Here, \( a = 1 \) and \( b = 4 \).

These points give us:

- \( f(1) = -3 \)
- \( f(4) = 0 \)

Now, applying the formula:

\[
\text{Average Rate of Change} = \frac{f(4) - f(1)}{4 - 1}
\]
\[
= \frac{0 - (-3)}{4 - 1}
\]
\[
= \frac{0 + 3}{3} = \frac{3}{3} = 1
\]

### Summary of Results:
- The average rate of change from \( x = -8 \) to \( x = -6 \) is \(-0.5\).
- The average rate of change from \( x = 1 \) to \( x = 4 \) is \(1\).