2. Complete the expression to show the average rate of change of the function f(x)=2x^2+1 as x changes from 1 to 3. 2 3 13 : 16 38 19 (-3)/(-)

To find the average rate of change of the function $f(x) = 2x^2 + 1$ as $x$ changes from 1 to 3, we use the formula for the average rate of change:

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

where $a = 1$ and $b = 3$.

First, calculate $f(1)$:

$$f(1) = 2(1)^2 + 1 = 2 \times 1 + 1 = 3$$

Next, calculate $f(3)$:

$$f(3) = 2(3)^2 + 1 = 2 \times 9 + 1 = 18 + 1 = 19$$

Now, substitute these values into the formula:

$$\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{19 - 3}{3 - 1} = \frac{16}{2} = 8$$

Therefore, the average rate of change is 8.