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:<br /><br />\[<br />\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}<br />\]<br /><br />where \( a = 1 \) and \( b = 3 \).<br /><br />First, calculate \( f(1) \):<br /><br />\[<br />f(1) = 2(1)^2 + 1 = 2 \times 1 + 1 = 3<br />\]<br /><br />Next, calculate \( f(3) \):<br /><br />\[<br />f(3) = 2(3)^2 + 1 = 2 \times 9 + 1 = 18 + 1 = 19<br />\]<br /><br />Now, substitute these values into the formula:<br /><br />\[<br />\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{19 - 3}{3 - 1} = \frac{16}{2} = 8<br />\]<br /><br />Therefore, the average rate of change is 8.