pearson.com/Student/PlayerHomework.aspx?homeworkld=689484221\&questionld=11\&flushed=false\&cld=8129566\&\¢erwin... 1901 Eric Walden 02/03/25 6:20 PM Question 47, 2.3.43 Part 4 of 7 HW Score: 32.44\%, 36.33 of HW \( 2 B \) 112 points Points: 0 of 2 Save Use the given function to complete parts (a) through (e) below. \[ f(x)=x^{4}-9 x^{2} \] 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. \[ x=0,3,-3 \] (Type an integer or a decimal. Use a comma to separate answers as needed.) At which zeros does the graph of the function cross the \( x \)-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x=-3,3 \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which zeros does the graph of the function touch the x-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 48 A. \( x= \) \( \square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) 49 B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around.

Let's analyze the function \( f(x) = x^4 - 9x^2 \) to determine its \( x \)-intercepts and understand the behavior of the graph at those points. ### **1. Finding the \( x \)-Intercepts** To find the \( x \)-intercepts, set \( f(x) = 0 \): \[ x^4 - 9x^2 = 0 \] Factor the equation: \[ x^2(x^2 - 9) = 0 \\ x^2(x - 3)(x + 3) = 0 \] So, the solutions are: \[ x = 0 \quad (\text{with multiplicity } 2) \\ x = 3 \\ x = -3 \] **\( x \)-Intercepts:** \[ x = -3, \quad x = 0, \quad x = 3 \] ### **2. Behavior of the Graph at the \( x \)-Intercepts** - **Multiplicity of Roots:** - **\( x = 0 \):** This root has a multiplicity of **2** (even), which means the graph **touches** the \( x \)-axis at this point and **turns around**. - **\( x = 3 \) and \( x = -3 \):** These roots have a multiplicity of **1** (odd), which means the graph **crosses** the \( x \)-axis** at these points. ### **3. Answering the Specific Questions** #### **a) At which zeros does the graph of the function cross the \( x \)-axis?** **Correct Choice:** **A. \( x = -3, 3 \)** - **Explanation:** At \( x = -3 \) and \( x = 3 \), the graph crosses the \( x \)-axis because these roots have an odd multiplicity. #### **b) At which zeros does the graph of the function touch the \( x \)-axis and turn around?** **Correct Choice:** **A. \( x = 0 \)** - **Explanation:** At \( x = 0 \), the graph touches the \( x \)-axis and turns around because this root has an even multiplicity. ### **Summary** - **\( x \)-Intercepts:** \( x = -3, 0, 3 \) - **Crosses the \( x \)-axis at:** \( x = -3 \) and \( x = 3 \) - **Touches and turns around at:** \( x = 0 \) If you have any further questions or need additional explanations, feel free to ask!