Let \( g(x)=7 x^{2}-4 \) (a) Find the average rate of change from -5 to 2 (b) Find an equation of the secant line containing \( (-5, g(-5)) \) and \( (2, g(2)) \) (a) The average rate of change from -5 to 2 is -21 . (Simplify your answer.) (b) An equation of the secant line containing \( (-5, g(-5)) \) and \( (2, g(2)) \) is (Type your answer in slope-intercept form.)

To find the average rate of change of the function \( g(x) = 7x^2 - 4 \) from \( x = -5 \) to \( x = 2 \): (a) First, we calculate \( g(-5) \) and \( g(2) \): \[ g(-5) = 7(-5)^2 - 4 = 7(25) - 4 = 175 - 4 = 171 \] \[ g(2) = 7(2)^2 - 4 = 7(4) - 4 = 28 - 4 = 24 \] Now, we find the average rate of change: \[ \text{Average Rate of Change} = \frac{g(2) - g(-5)}{2 - (-5)} = \frac{24 - 171}{2 + 5} = \frac{-147}{7} = -21 \] So the average rate of change from -5 to 2 is indeed -21. (b) To find the equation of the secant line, we first need the slope, which we have already calculated as -21. Now we can use the point-slope form of the equation of a line: Using the point \( (2, g(2)) = (2, 24) \): \[ y - y_1 = m(x - x_1) \] \[ y - 24 = -21(x - 2) \] Distributing the slope: \[ y - 24 = -21x + 42 \] Now, solving for \( y \): \[ y = -21x + 42 + 24 \] \[ y = -21x + 66 \] Thus, the equation of the secant line is: \[ y = -21x + 66 \]