Let \( g(x)=5 x^{2}-3 \) (a) Find the average rate of change from -4 to 2 . (b) Find an equation of the secant line containing \( (-4, g(-4)) \) and \( (2, g(2)) \). (a) The average rate of change from -4 to 2 is \( \square \). (Simplify your answer.)

To find the average rate of change of the function \( g(x) = 5x^2 - 3 \) from \( x = -4 \) to \( x = 2 \), we use the formula: \[ \text{Average rate of change} = \frac{g(2) - g(-4)}{2 - (-4)} \] First, calculate \( g(-4) \): \[ g(-4) = 5(-4)^2 - 3 = 5(16) - 3 = 80 - 3 = 77 \] Now, calculate \( g(2) \): \[ g(2) = 5(2)^2 - 3 = 5(4) - 3 = 20 - 3 = 17 \] Now plug these values into the average rate of change formula: \[ \text{Average rate of change} = \frac{17 - 77}{2 + 4} = \frac{-60}{6} = -10 \] So, the average rate of change from -4 to 2 is \( -10 \). Now, let's find the equation of the secant line using the points \( (-4, g(-4)) \) and \( (2, g(2)) \). The slope \( m \) of the secant line is the average rate of change we just calculated: \[ m = -10 \] Using the point-slope form of the line, \( y - y_1 = m(x - x_1) \), we can use the point \( (2, 17) \): \[ y - 17 = -10(x - 2) \] Simplifying this, we get: \[ y - 17 = -10x + 20 \quad \Rightarrow \quad y = -10x + 37 \] Thus, the equation of the secant line is \( y = -10x + 37 \). In summary: (a) The average rate of change from -4 to 2 is \( -10 \). (b) The equation of the secant line is \( y = -10x + 37 \).