1. **State the problem:** We have the function $g(x) = 6x^2 - 1$. (a) Find the average rate of change of $g$ from $x = -5$ to $x = 9$. (b) Find the equation of the secant line passing through the points $(-5, g(-5))$ and $(9, g(9))$. --- 2. **Recall the formula for average rate of change:** The average rate of change of a function $g$ from $x=a$ to $x=b$ is given by $$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$$ 3. **Calculate $g(-5)$ and $g(9)$:** $$g(-5) = 6(-5)^2 - 1 = 6 \times 25 - 1 = 150 - 1 = 149$$ $$g(9) = 6(9)^2 - 1 = 6 \times 81 - 1 = 486 - 1 = 485$$ 4. **Calculate the average rate of change:** $$\frac{g(9) - g(-5)}{9 - (-5)} = \frac{485 - 149}{9 + 5} = \frac{336}{14} = 24$$ This matches the given answer. --- 5. **Find the equation of the secant line:** The secant line passes through $(-5, 149)$ and $(9, 485)$ with slope $m = 24$. Use point-slope form: $$y - y_1 = m(x - x_1)$$ Using point $(-5, 149)$: $$y - 149 = 24(x - (-5)) = 24(x + 5)$$ 6. **Simplify to slope-intercept form:** $$y - 149 = 24x + 120$$ $$y = 24x + 120 + 149$$ $$y = 24x + 269$$ --- **Final answers:** (a) The average rate of change is $24$. (b) The equation of the secant line is $$y = 24x + 269$$.