Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-9 x^{2} \) a) Use the Leading coefficient Test to determine the graph's end behavior. A. The graph of \( f(x) \) rises left and falls right. B. The graph of \( f(x) \) rises left and rises right. C. The graph of \( f(x) \) falls left and falls right. D. The graph of \( f(x) \) falls left and rises right. b) Find the \( x \)-intercepts. \( x=\square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) ( 0 .

To solve the problem step by step, let's start with part (a) and then move on to part (b). ### Part (a): Leading Coefficient Test The function given is: \[ f(x) = x^4 - 9x^2 \] 1. **Identify the leading term**: The leading term of the polynomial is \( x^4 \). 2. **Determine the leading coefficient**: The leading coefficient is \( 1 \) (which is positive). 3. **Degree of the polynomial**: The degree of the polynomial is \( 4 \) (which is even). According to the Leading Coefficient Test: - If the leading coefficient is positive and the degree is even, the graph rises to the left and rises to the right. Thus, the correct answer for part (a) is: **B. The graph of \( f(x) \) rises left and rises right.** ### Part (b): Finding the \( x \)-intercepts To find the \( x \)-intercepts, we need to set \( f(x) = 0 \): \[ x^4 - 9x^2 = 0 \] 1. **Factor the equation**: We can factor out \( x^2 \): \[ x^2(x^2 - 9) = 0 \] 2. **Set each factor to zero**: - \( x^2 = 0 \) gives \( x = 0 \). - \( x^2 - 9 = 0 \) gives \( x^2 = 9 \), which results in \( x = 3 \) and \( x = -3 \). Thus, the \( x \)-intercepts are: \[ x = 0, 3, -3 \] So, the answer for part (b) is: \[ x = 0, 3, -3 \] In summary: - **Part (a)**: B. The graph of \( f(x) \) rises left and rises right. - **Part (b)**: \( x = 0, 3, -3 \)