Determining end behavior and intercepts to graph a polynomial function Consider the following polynomial function. $$ f(x)=(x+3)(x+1)^{2}(x-1) $$ Answer the questions regarding the graph of $$ f $$. Then, use this information to graph the function. $$ \begin{array}{l}\text { (a) Choose the end behavior of the graph of } f . \ \text { Falls to the left and rises to the right } v \ \text { (b) List each real zero of } f \text { according to the behavior of the graph at the } \ \text { zero. If there is more than one answer, separate them with commas } \ \text { answer, click on "None". } \ \text { Zero(s) where the graph crosses the } x \text {-axis: } \ \text { Zero(s) where the graph touches, but does not cross the } x \text {-axis: }\end{array} $$ (c) Find the $$ y $$-intercept of the graph of $$ f: $$

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$$ \textbf{Step 1. Determine the End Behavior} $$

The given function is  
$$
f(x) = (x+3)(x+1)^2 (x-1).
$$  
The degree of the polynomial is the sum of the exponents:  
$$
1 + 2 + 1 = 4.
$$  
The leading term is obtained by multiplying the highest degree term from each factor:  
$$
x \cdot x^2 \cdot x = x^4.
$$  
Since the degree is even and the leading coefficient is positive, both ends of the graph rise. That is, as $$ x \to -\infty $$ and $$ x \to +\infty $$, we have  
$$
f(x) \to +\infty.
$$

\vspace{1mm}

$$\textbf{Answer for (a):}$$ Rises to the left and rises to the right.

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$$ \textbf{Step 2. Determine the Real Zeros and Their Behavior} $$

The zeros are found by setting each factor equal to zero:

1. $$ x+3=0 $$ gives $$ x=-3 $$ (multiplicity $$ 1 $$, so the graph crosses the axis).
2. $$ (x+1)^2=0 $$ gives $$ x=-1 $$ (multiplicity $$ 2 $$, so the graph touches the axis but does not cross).
3. $$ x-1=0 $$ gives $$ x=1 $$ (multiplicity $$ 1 $$, so the graph crosses the axis).

\vspace{1mm}

$$\textbf{Answer for (b):}$$

- Zero(s) where the graph crosses the $$ x $$-axis: $$ x = -3, \, 1 $$.
- Zero(s) where the graph touches, but does not cross the $$ x $$-axis: $$ x = -1 $$.

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$$ \textbf{Step 3. Find the $$ y $$-Intercept} $$

The $$ y $$-intercept is found by evaluating $$ f(0) $$:

$$
f(0) = (0+3)(0+1)^2(0-1) = 3 \cdot 1 \cdot (-1) = -3.
$$

Thus, the $$ y $$-intercept is $$ (0, -3) $$.

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$$\textbf{Final Answers:}$$

(a) The graph rises to the left and rises to the right.

(b) Zeros where the graph crosses the $$ x $$-axis: $$ x = -3, \, 1 $$.  
Zeros where the graph touches but does not cross the $$ x $$-axis: $$ x = -1 $$.

(c) The $$ y $$-intercept is $$ (0, -3) $$.
(a) The graph rises to the left and rises to the right. (b) Zeros where the graph crosses the $$ x $$-axis: $$ x = -3, \, 1 $$. Zeros where the graph touches but does not cross the $$ x $$-axis: $$ x = -1 $$. (c) The $$ y $$-intercept is $$ (0, -3) $$.

Determining end behavior and intercepts to graph a polynomial function
Consider the following polynomial function.

Answer the questions regarding the graph of .
Then, use this information to graph the function.

© Find the -intercept of the graph of

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Determining end behavior and intercepts to graph a polynomial function Consider the following polynomial function. Answer the questions regarding the graph of . Then, use this information to graph the function. © Find the -intercept of the graph of