Consider the following polynomial function. $$ f(x)=(x+1)^{2}(x-2) $$ Answer the questions regarding the graph of $$ f $$. Then, use this information to graph the function. $$ \begin{array}{l}\text { (a) Choose the end behavior of the graph of } f \ \text { Choose One } \ \text { (b) List each real zere of } f \text { according to the behavior of the graph at the } x \text {-axis near that } \ \text { zero. If there is more than one answer, separate them with commas, If there is no where the graph crosses the } x \text {-axis: } \ \text { answer, dick on "None". } \ \text { Zero(s) where the graph touches, but does not cross the } x \text {-axis: } \ \text { (c) Find the } y \text {-intercent of the graph of } f\end{array} $$ (d) Graph $$ f(x)=(x+1)^{2}(x-2) $$ by doing the following. - Plot all points where the graph of $$ f $$ intersects the $$ x $$-axis or $$ y $$-axis. - For each point on the $$ x $$-axis, select the correct behavior. - Click on the graph icon.

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UpStudy AI · Accepted Answer

**(a) End Behavior**

Since  
$$
f(x) = (x+1)^2 (x-2)
$$  
expands to a cubic polynomial with a positive leading coefficient, the end behavior is:

- As $$ x \to -\infty $$, $$ f(x) \to -\infty $$
- As $$ x \to \infty $$, $$ f(x) \to \infty $$

---

**(b) Real Zeros and Their Behavior**

Setting the function equal to zero gives:
$$
(x+1)^2 (x-2) = 0
$$

The zeros are:
- $$ x = -1 $$ (with multiplicity 2): The graph touches the $$ x $$-axis and turns around.
- $$ x = 2 $$ (with multiplicity 1): The graph crosses the $$ x $$-axis.

Thus:
- Zero(s) where the graph touches, but does not cross the $$ x $$-axis: $$ x = -1 $$
- Zero(s) where the graph crosses the $$ x $$-axis: $$ x = 2 $$

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**(c) $$ y $$-Intercept**

To find the $$ y $$-intercept, substitute $$ x = 0 $$ into $$ f(x) $$:
$$
f(0) = (0+1)^2 (0-2) = (1)^2 (-2) = -2
$$
So, the $$ y $$-intercept is $$ (0, -2) $$.

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**(d) Graphing the Function**

1. **Plot the Intercepts:**
   - $$ x $$-intercepts: $$ (-1, 0) $$ and $$ (2, 0) $$
     - At $$ x = -1 $$, the graph touches the $$ x $$-axis.
     - At $$ x = 2 $$, the graph crosses the $$ x $$-axis.
   - $$ y $$-intercept: $$ (0, -2) $$

2. **Plot Additional Points (if needed):**  
   You can calculate a few extra points on either side of the intercepts to better capture the curve.

3. **Indicate the End Behavior:**
   - As $$ x \to -\infty $$: the graph goes down (toward $$-\infty$$).
   - As $$ x \to \infty $$: the graph goes up (toward $$\infty$$).

4. **Sketch the Graph:**  
   - Begin in the third quadrant following the left end behavior.
   - Rise to touch the $$ x $$-axis at $$ x = -1 $$ (without crossing), then descend to the $$ y $$-intercept $$ (0, -2) $$.
   - Finally, rise again and cross the $$ x $$-axis at $$ x = 2 $$ following the right end behavior.

By following these steps, the graph of $$ f(x) = (x+1)^2 (x-2) $$ is fully determined.
**(a) End Behavior** - As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches negative infinity. - As $$ x $$ approaches positive infinity, $$ f(x) $$ approaches positive infinity. **(b) Real Zeros** - $$ x = -1 $$ (graph touches the $$ x $$-axis) - $$ x = 2 $$ (graph crosses the $$ x $$-axis) **(c) $$ y $$-Intercept** - $$ (0, -2) $$ **(d) Graphing the Function** 1. Plot the $$ x $$-intercepts at $$ (-1, 0) $$ and $$ (2, 0) $$. 2. Plot the $$ y $$-intercept at $$ (0, -2) $$. 3. Sketch the curve following the end behaviors: - Starts in the third quadrant. - Touches the $$ x $$-axis at $$ x = -1 $$ without crossing. - Passes through $$ (0, -2) $$. - Crosses the $$ x $$-axis at $$ x = 2 $$. 4. Ensure the graph reflects the end behaviors as $$ x $$ approaches positive and negative infinity.

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