Find the average rate of change of \( f(x)=8 x^{2}-9 \) on the interval \( [2, a] \). Your answer will be an expression involving \( a \)

To find the average rate of change of the function \( f(x) = 8x^2 - 9 \) on the interval \( [2, a] \), we can use the formula for the average rate of change, which is given by: \[ \text{Average Rate of Change} = \frac{f(a) - f(2)}{a - 2} \] First, we calculate \( f(2) \): \[ f(2) = 8(2^2) - 9 = 8(4) - 9 = 32 - 9 = 23 \] Next, we compute \( f(a) \): \[ f(a) = 8a^2 - 9 \] Now, substituting these values into the average rate of change formula gives us: \[ \text{Average Rate of Change} = \frac{(8a^2 - 9) - 23}{a - 2} \] Simplifying the numerator: \[ 8a^2 - 9 - 23 = 8a^2 - 32 \] So we have: \[ \text{Average Rate of Change} = \frac{8a^2 - 32}{a - 2} \] We can factor the numerator: \[ 8a^2 - 32 = 8(a^2 - 4) = 8(a - 2)(a + 2) \] Therefore, the expression simplifies to: \[ \text{Average Rate of Change} = \frac{8(a - 2)(a + 2)}{a - 2} \] For \( a \neq 2 \), this simplifies to: \[ \text{Average Rate of Change} = 8(a + 2) \] Thus, the average rate of change of \( f(x) = 8x^2 - 9 \) on the interval \( [2, a] \) is: \[ 8(a + 2), \quad a \neq 2 \]