Find the average rate of change of the function \( f(x)=x^{2}+7 x \) from \( x_{1}=6 \) to \( x_{2}=9 \). The average rate of change is \( \square \). (Simplify your answer.)

To find the average rate of change of the function \( f(x) = x^2 + 7x \) from \( x_1 = 6 \) to \( x_2 = 9 \), we use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] First, we need to calculate \( f(6) \) and \( f(9) \). Calculating \( f(6) \): \[ f(6) = 6^2 + 7 \cdot 6 = 36 + 42 = 78 \] Calculating \( f(9) \): \[ f(9) = 9^2 + 7 \cdot 9 = 81 + 63 = 144 \] Now we can substitute these values into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{f(9) - f(6)}{9 - 6} = \frac{144 - 78}{9 - 6} = \frac{66}{3} = 22 \] Thus, the average rate of change of the function from \( x_1 = 6 \) to \( x_2 = 9 \) is \[ \boxed{22} \]