Question

Graph starts at 0,1.5 and goes through 8,17 what is the average rate of change of a function on the interval x:4,8

Answer 1

To find the average rate of change of a function on an interval \([x_1, x_2]\), you can use the formula:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

From your information, it seems that the function passes through the points \((0, 1.5)\) and \((8, 17)\). However, we need to determine the values of \(f(4)\) and \(f(8)\) to find the average rate of change specifically over the interval \([4, 8]\).

### Step 1: Find the function
Assuming the function is linear, we can find its equation using the two points given:

1. Determine the slope \(m\) between the two points \((0, 1.5)\) and \((8, 17)\):

\[
m = \frac{f(8) - f(0)}{8 - 0} = \frac{17 - 1.5}{8 - 0} = \frac{15.5}{8} = 1.9375
\]

2. With the slope \(m\) and the point \((0,1.5)\), we can find the equation of the line in point-slope form:

\[
f(x) = mx + b
\]

Using the point \((0, 1.5)\):

\[
f(x) = 1.9375x + 1.5
\]

### Step 2: Calculate \(f(4)\) and \(f(8)\)
Now, we can find the values of \(f(4)\) and \(f(8)\):

1. Calculate \(f(4)\):

\[
f(4) = 1.9375 \cdot 4 + 1.5 = 7.5 + 1.5 = 9
\]

2. Calculate \(f(8)\):

\[
f(8) = 17 \quad (\text{given})
\]

### Step 3: Calculate the average rate of change from \(x = 4\) to \(x = 8\)
Substituting \(f(4)\) and \(f(8)\) into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(8) - f(4)}{8 - 4} = \frac{17 - 9}{4} = \frac{8}{4} = 2
\]

### Conclusion
The average rate of change of the function on the interval \([4, 8]\) is \(2\).