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.)

Solve the equation by following steps: - step0: Solve for \(g\): \(g\times 2=5\times 2^{2}-3\) - step1: Reorder the terms: \(2g=17\) - step2: Divide both sides: \(\frac{2g}{2}=\frac{17}{2}\) - step3: Divide the numbers: \(g=\frac{17}{2}\) Solve the equation \( g(-4) = 5(-4)^{2} - 3 \). Solve the equation by following steps: - step0: Solve for \(g\): \(g\left(-4\right)=5\left(-4\right)^{2}-3\) - step1: Reorder the terms: \(-4g=77\) - step2: Change the signs: \(4g=-77\) - step3: Divide both sides: \(\frac{4g}{4}=\frac{-77}{4}\) - step4: Divide the numbers: \(g=-\frac{77}{4}\) To solve the problem, we will follow the steps outlined: ### Part (a): Find the average rate of change from -4 to 2 The average rate of change of a function \( g(x) \) from \( x = a \) to \( x = b \) is given by the formula: \[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \] In this case, \( a = -4 \) and \( b = 2 \). We have already calculated: - \( g(-4) = -19.25 \) - \( g(2) = 8.5 \) Now, we can substitute these values into the formula: \[ \text{Average Rate of Change} = \frac{g(2) - g(-4)}{2 - (-4)} = \frac{8.5 - (-19.25)}{2 + 4} \] Calculating the numerator: \[ 8.5 + 19.25 = 27.75 \] Calculating the denominator: \[ 2 + 4 = 6 \] Now, substituting these into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{27.75}{6} \] Now, let's simplify this fraction: \[ \text{Average Rate of Change} = 4.625 \] Thus, the average rate of change from -4 to 2 is: \[ \text{Average Rate of Change} = 4.625 \] ### Part (b): Find an equation of the secant line containing \( (-4, g(-4)) \) and \( (2, g(2)) \) The equation of a line can be expressed in point-slope form as: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope of the line, and \( (x_1, y_1) \) is one of the points on the line. The slope \( m \) can be calculated as: \[ m = \frac{g(2) - g(-4)}{2 - (-4)} = \frac{27.75}{6} = 4.625 \] Using the point \( (2, g(2)) = (2, 8.5) \): \[ y - 8.5 = 4.625(x - 2) \] Now, we can simplify this equation: \[ y - 8.5 = 4.625x - 9.25 \] Adding \( 8.5 \) to both sides: \[ y = 4.625x - 0.75 \] Thus, the equation of the secant line is: \[ y = 4.625x - 0.75 \] ### Summary of Results (a) The average rate of change from -4 to 2 is \( 4.625 \). (b) The equation of the secant line is \( y = 4.625x - 0.75 \).