Part 3 of 7 $$ \bigcirc $$ Points: 0 of 2 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 night B The graph of $$ f(x) $$ nises left and rises right C. The graph of $$ f(x) $$ falls left and falls night D. The graph of $$ f(x) $$ falls left and nses right b) Find the $$ x $$-intercepts. $$ x=0,3,-3 $$ (Type an integer or a decimal Usent 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. $$ x= $$ $$ \square $$ (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

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Let's work through parts **(a)** and **(b)** of the problem using the function $$ f(x) = x^{4} - 9x^{2} $$.

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### **Part (a): Leading Coefficient Test**

**Determine the graph's end behavior using the Leading Coefficient Test.**

1. **Identify the Degree and Leading Coefficient:**
   - The function $$ f(x) = x^{4} - 9x^{2} $$ is a **polynomial of degree 4**.
   - The **leading coefficient** (the coefficient of the highest degree term) is **positive** ($$ +1 $$).

2. **Apply the Leading Coefficient Test:**
   - **Even Degree & Positive Leading Coefficient:** For polynomials with an even degree and a positive leading coefficient, **both ends of the graph rise to positive infinity**.

3. **Choose the Correct Option:**
   - **Option B** states: *"The graph of $$ f(x) $$ rises left and rises right."*
   - This matches our analysis.

**Answer for Part (a):**
**B. The graph of $$ f(x) $$ rises left and rises right**

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### **Part (b): Finding the $$ x $$-Intercepts and Their Behavior**

**Find the $$ x $$-intercepts and determine where the graph crosses the $$ x $$-axis.**

1. **Find the $$ x $$-Intercepts:**
   - Set $$ f(x) = 0 $$:
     $$
     x^{4} - 9x^{2} = 0 \
     x^{2}(x^{2} - 9) = 0 \
     \Rightarrow x^{2} = 0 \quad \text{or} \quad x^{2} - 9 = 0 \
     \Rightarrow x = 0 \quad \text{or} \quad x = \pm 3
     $$
   - So, the $$ x $$-intercepts are at $$ x = 0 $$, $$ x = 3 $$, and $$ x = -3 $$.

2. **Determine the Behavior at Each Intercept:**
   - **Multiplicity:**
     - $$ x = 0 $$: **Multiplicity 2** (since $$ x^{2} $$)
       - Even multiplicity: **The graph **touches** the $$ x $$-axis and **does not cross**.
     - $$ x = 3 $$ and $$ x = -3 $$: **Multiplicity 1** each
       - Odd multiplicity: **The graph **crosses** the $$ x $$-axis**.

3. **Select the Correct Option:**
   - **Option A** allows you to list the specific $$ x $$-values where the graph crosses the $$ x $$-axis.
   - Since the graph crosses at $$ x = 3 $$ and $$ x = -3 $$, but **not** at $$ x = 0 $$, you fill in these values.

**Answer for Part (b):**
**A. $$ x = -3, 3 $$**

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**Summary of Answers:**

- **(a)** **B. The graph of $$ f(x) $$ rises left and rises right**
- **(b)** **A. $$ x = -3, 3 $$**
**(a) The graph of $$ f(x) $$ rises left and rises right.** **(b) The graph crosses the $$ x $$-axis at $$ x = -3 $$ and $$ x = 3 $$.**

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