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.

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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!
The function $$ f(x) = x^4 - 9x^2 $$ has $$ x $$-intercepts at $$ x = -3, 0, 3 $$. The graph crosses the $$ x $$-axis at $$ x = -3 $$ and $$ x = 3 $$, and touches and turns around at $$ x = 0 $$.

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