Lunsider the following polynomial function. $$ f(x)=(x-2)(x+1)^{2}(x-1)^{2} $$ Answer the questions regarding the graph of $$ f $$. Thien, use this information to graph the function. $$ \begin{array}{l}\text { (a) Choose the end behavior of the graph of } f \text {. } \ \text { Choose One List each real zero of } f \text { according to the behavior of the graph at the } x \text {-axis near that } \ \text { zero. If there is more than one answer, separate them with commas. If there is no } \ \text { answer, click on "None". } \ \text { zero(s) where the graph crosses the } x \text {-axis: } \ \text { Zero(s) where the graph touches, but does not cross the } x \text {-axis: }\end{array} $$ (c) Find the $$ y $$-intercept of the graph of $$ f $$ : (d) Graph $$ f(x)=(x-2)(x+1)^{2}(x-1)^{2} $$ 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, solost th

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It seems that I am unable to generate a graph directly. However, I can guide you on how to plot the function $$ f(x) = (x-2)(x+1)^{2}(x-1)^{2} $$ using the information we have gathered.

### Steps to Graph the Function

1. **Plot the X-intercepts:**
   - Mark the points $$ (2, 0) $$, $$ (-1, 0) $$, and $$ (1, 0) $$ on the x-axis.
   - At $$ x = 2 $$, the graph crosses the x-axis.
   - At $$ x = -1 $$ and $$ x = 1 $$, the graph touches the x-axis.

2. **Plot the Y-intercept:**
   - Mark the point $$ (0, -2) $$ on the y-axis.

3. **Determine the Behavior:**
   - As $$ x 	o -\infty $$, the graph goes downwards (to $$ -\infty $$).
   - As $$ x 	o +\infty $$, the graph goes upwards (to $$ +\infty $$).
   - Between the intercepts, the graph will rise and fall according to the behavior at the intercepts:
     - From $$ (-\infty, -1) $$, the graph is below the x-axis.
     - It touches the x-axis at $$ -1 $$ and turns back up.
     - It rises to the x-axis at $$ (1, 0) $$ and turns back down.
     - It rises again to cross the x-axis at $$ (2, 0) $$.

4. **Sketch the Curve:**
   - Start from the left, draw a curve that approaches the x-axis from below, touches at $$ (-1, 0) $$, rises to touch at $$ (1, 0) $$, and finally crosses at $$ (2, 0) $$.
   - Ensure the curve goes through the y-intercept $$ (0, -2) $$.

### Conclusion
By following these steps, you can create an accurate sketch of the polynomial function $$ f(x) $$. If you have graphing software or a graphing calculator, you can input the function directly to visualize it more precisely.
- **End Behavior:** - As $$ x 	o -\infty $$, $$ f(x) 	o -\infty $$ - As $$ x 	o +\infty $$, $$ f(x) 	o +\infty $$ - **Zeros:** - Crosses the x-axis at $$ x = 2 $$ - Touches the x-axis at $$ x = -1 $$ and $$ x = 1 $$ - **Y-intercept:** - $$ (0, -2) $$ **Graph Description:** - The graph starts from the bottom left, rises to touch the x-axis at $$ x = -1 $$, continues upwards to touch the x-axis again at $$ x = 1 $$, then crosses the x-axis at $$ x = 2 $$. - It passes through the y-intercept at $$ (0, -2) $$. - As $$ x $$ approaches negative infinity, the graph goes down, and as $$ x $$ approaches positive infinity, it goes up. **Summary:** The function $$ f(x) = (x-2)(x+1)^{2}(x-1)^{2} $$ has a degree of 5 with a positive leading coefficient, resulting in the described end behavior. It crosses the x-axis at $$ x = 2 $$ and touches it at $$ x = -1 $$ and $$ x = 1 $$. The y-intercept is at $$ (0, -2) $$. The graph reflects the behavior of the zeros and the end behavior, providing a clear visual representation of the function.

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