Use the given function to complete parts a) through e) below. \( f(x)=-3(x-3)^{2}\left(x^{2}-4\right) \) a) Use the Leading Coefficient Test to determine the graph's end behavior The graph of \( f(x) \) rises left and falls night. The graph of \( f(x) \) falls left and falls right. The graph of \( f(x) \) falls left and rises right. b) Find the \( x \)-intercepts. The \( x \)-intercept(s) is/are \( \square \). \( \mathrm{\square} \). x . rises left and rises right. (Type an integer or a decimal. Use a comma to separate answers as needed )

Sure, let's work through parts **a)** and **b)** for the function: \[ f(x) = -3(x - 3)^2 (x^2 - 4) \] --- ### **a) Leading Coefficient Test: Determine the Graph's End Behavior** The **Leading Coefficient Test** helps us understand how the ends of the graph behave based on the leading term of the polynomial. 1. **Expand the Function to Identify the Leading Term:** \[ f(x) = -3(x - 3)^2 (x^2 - 4) \] First, expand \((x - 3)^2\): \[ (x - 3)^2 = x^2 - 6x + 9 \] Now, multiply this by \((x^2 - 4)\): \[ (x^2 - 6x + 9)(x^2 - 4) = x^4 - 6x^3 + 5x^2 + 24x - 36 \] Finally, multiply by \(-3\): \[ f(x) = -3x^4 + 18x^3 - 15x^2 - 72x + 108 \] The **leading term** is \(-3x^4\). 2. **Apply the Leading Coefficient Test:** - **Degree:** 4 (even) - **Leading Coefficient:** \(-3\) (negative) For polynomials with even degrees: - **Positive Leading Coefficient:** Both ends rise to \(+\infty\). - **Negative Leading Coefficient:** Both ends fall to \(-\infty\). Therefore, since the leading coefficient is negative and the degree is even: **End Behavior:** The graph **falls to the left** and **falls to the right**. --- ### **b) Find the \( x \)-Intercepts** To find the \( x \)-intercepts, set \( f(x) = 0 \) and solve for \( x \): \[ -3(x - 3)^2 (x^2 - 4) = 0 \] 1. **Set Each Factor Equal to Zero:** - \( (x - 3)^2 = 0 \) \(\Rightarrow x = 3\) (Multiplicity 2) - \( x^2 - 4 = 0 \) \(\Rightarrow x^2 = 4\) \(\Rightarrow x = 2 \) or \( x = -2 \) 2. **List the \( x \)-Intercepts:** \[ x = -2, \quad x = 2, \quad x = 3 \] **Note:** \( x = 3 \) is a repeated root, which means the graph **touches** the \( x \)-axis at this point but does not cross it. --- ### **Final Answers:** **a)** The graph of \( f(x) \) **falls left and falls right**. **b)** The \( x \)-intercepts are: \( -2, \ 2, \ 3 \).