Find the average rate of change of \( g(x)=-2 x^{2}+10 \) from \( x=5 \) to \( x=7 \). Simplify your answer as much as possible.

To find the average rate of change of the function \( g(x) = -2x^2 + 10 \) from \( x = 5 \) to \( x = 7 \), we will use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \] where \( a = 5 \) and \( b = 7 \). First, we need to calculate \( g(5) \) and \( g(7) \): \[ g(5) = -2(5^2) + 10 = -2(25) + 10 = -50 + 10 = -40 \] \[ g(7) = -2(7^2) + 10 = -2(49) + 10 = -98 + 10 = -88 \] Now we can plug these values back into our formula: \[ \text{Average Rate of Change} = \frac{g(7) - g(5)}{7 - 5} = \frac{-88 - (-40)}{7 - 5} = \frac{-88 + 40}{2} = \frac{-48}{2} = -24 \] So, the average rate of change of \( g(x) \) from \( x = 5 \) to \( x = 7 \) is \( -24 \).