For the polynomial function $$ f(x)=x^{4}+4 x^{3}+4 x^{2} $$, answer the parts a through e a. Use the Leading Coefficient Tesi to delermine the graph's end behavior A. The graph of $$ f(x) $$ falls to the left and lails to the right B. The graph of $$ f(x) $$ rises to the leff and falls to the right C. The graph of $$ \mathrm{f}(\mathrm{x}) $$ rises to the lell and rises to the right D. The graph of $$ f(x) $$ falls to the left and rises to the right b. Find the $$ x $$-intercept(s). State whether the graph crosses the $$ x $$-axis, or louches the $$ x $$-axis and turns arcund, at each intercept The $$ x $$-infercopt(s) islaro $$ 0,-2 $$. (Iype an integer or a decimal, Use a comma to sepatate answers as needed. Type each answer only once.) At which $$ x $$-intercept(5) does the graph cross the $$ x $$-axis? Select the correct choice below and, if necessary. fill in the answer box to complele your choice. A. The $$ x $$-inlorcept $$ [s $$ ] at which the graph ciosses the $$ x $$-axis is/are $$ \square $$ . (Type an inleger or a decimal. Use a comma lo separate answers as needed. Type each answer only once.) B. There are no x-intercepts at which the graph crosses the $$ x $$-axis.

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UpStudy AI · Accepted Answer

**a. End Behavior**

The given function is  
$$
f(x)=x^4+4x^3+4x^2.
$$  
Factoring, we have  
$$
f(x)=x^2(x^2+4x+4)=x^2(x+2)^2.
$$  
The degree of the polynomial is 4 (even) and the leading coefficient is positive. Thus, as  
$$
x \to \infty,\; f(x)\to \infty,
$$  
and as  
$$
x \to -\infty,\; f(x)\to \infty.
$$  
This means the graph rises on both ends. Therefore, the correct answer is:

$$ \textbf{C. The graph of } f(x) \textbf{ rises to the left and rises to the right.} $$

---

**b. $$x$$-Intercepts and Behavior at the Intercepts**

The $$x$$-intercepts are found by setting $$ f(x)=0 $$:
$$
x^2(x+2)^2=0.
$$
This gives the solutions:
$$
x^2=0 \quad \Rightarrow \quad x=0,
$$
$$
(x+2)^2=0 \quad \Rightarrow \quad x=-2.
$$

Both roots have multiplicity 2 (even multiplicity). When an $$x$$-intercept has even multiplicity, the graph touches the $$x$$-axis and turns around instead of crossing it.

Thus, the $$x$$-intercepts are at:
$$
-2, \, 0.
$$
At both of these intercepts, the graph touches (or "louches") the $$x$$-axis without crossing it.

Since neither $$x$$-intercept features the graph crossing the $$x$$-axis, the correct choice for the intercept at which the graph crosses the $$x$$-axis is:

$$ \textbf{B. There are no x-intercepts at which the graph crosses the x-axis.} $$
**a. End Behavior** The graph of $$ f(x) $$ rises on both ends. **b. $$x$$-Intercepts and Behavior** The $$x$$-intercepts are at $$ -2 $$ and $$ 0 $$. At both intercepts, the graph touches the $$x$$-axis without crossing it.

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