Find the average rate of change of \( f(x)=2 x^{2}+2 \) from \( x=4 \) to \( x=9 \). Simplify your answer as much as possible.

To find the average rate of change of the function \( f(x) = 2x^2 + 2 \) from \( x = 4 \) to \( x = 9 \), we can use the formula for average rate of change, which is given by: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \( a = 4 \) and \( b = 9 \). First, we need to calculate \( f(4) \) and \( f(9) \): \[ f(4) = 2(4)^2 + 2 = 2(16) + 2 = 32 + 2 = 34 \] \[ f(9) = 2(9)^2 + 2 = 2(81) + 2 = 162 + 2 = 164 \] Now we can plug these values into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{f(9) - f(4)}{9 - 4} = \frac{164 - 34}{9 - 4} = \frac{130}{5} = 26 \] Thus, the average rate of change of \( f(x) \) from \( x = 4 \) to \( x = 9 \) is \( \boxed{26} \).