Sketch the graph of the function below, including correct signs, x-intercepts and y-intercepts.
f, of, x, equals, left parenthesis, x, plus, 3, right parenthesis, squared, left parenthesis, x, minus, 1, right parenthesis, squared, left parenthesis, x, minus, 3, right parenthesis
f(x)=(x+3) 
2
 (x−1) 
2
 (x−3)
Plot the y-intercept and the roots. Click on the graph to plot a point. Click a point again to delete it.

The function you provided is:

\[
f(x) = (x + 3)^2(x - 1)^2(x - 3)
\]

### 1. **Identify the roots (x-intercepts)**:
To find the roots, set \( f(x) = 0 \).

\[
(x + 3)^2(x - 1)^2(x - 3) = 0
\]

This occurs when any factor equals 0:
- \( x + 3 = 0 \Rightarrow x = -3 \)
- \( x - 1 = 0 \Rightarrow x = 1 \)
- \( x - 3 = 0 \Rightarrow x = 3 \)

So the **x-intercepts** are at \( x = -3 \), \( x = 1 \), and \( x = 3 \).

- The factors \( (x + 3)^2 \) and \( (x - 1)^2 \) are squared, so the graph touches the x-axis at \( x = -3 \) and \( x = 1 \) but does not cross it (since the roots have even multiplicity).
- The factor \( (x - 3) \) has odd multiplicity, so the graph crosses the x-axis at \( x = 3 \).

### 2. **Find the y-intercept**:
To find the y-intercept, set \( x = 0 \) in the function:

\[
f(0) = (0 + 3)^2(0 - 1)^2(0 - 3)
\]
\[
f(0) = (3)^2(-1)^2(-3) = 9 \cdot 1 \cdot (-3) = -27
\]

So, the **y-intercept** is at \( (0, -27) \).

### 3. **Behavior near the intercepts**:
- At \( x = -3 \) and \( x = 1 \), the graph touches but doesn't cross the x-axis.
- At \( x = 3 \), the graph crosses the x-axis.

### 4. **End behavior**:
Since the highest degree term of the polynomial is \( x^5 \), the function behaves like \( x^5 \) as \( x \to \infty \) and \( x \to -\infty \):
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to -\infty \).

Let me know if you'd like me to generate the graph for this! Would you also like to explore further details or questions about the function?

### 5 Relevant Questions:
1. How does the multiplicity of roots affect the behavior of the graph at those points?
2. Can you explain how the degree of a polynomial impacts its end behavior?
3. How would the graph change if one of the factors had an odd multiplicity instead of even?
4. What is the significance of the y-intercept in understanding the function?
5. How does the symmetry of the function impact its overall shape?

**Tip**: Always check the multiplicity of roots when sketching graphs—it determines whether the graph crosses or just touches the x-axis.

Sketch the graph of the function below, including correct signs, x-intercepts, and y-intercepts.
f(x) = (x + 3)^2(x - 1)^2(x - 3)

Learn how to graph the polynomial function f(x) = (x + 3)^2(x - 1)^2(x - 3) by identifying the roots, y-intercept, and analyzing the graph's behavior. Understand the impact of root multiplicity and the polynomial's end behavior. Suitable for high school students in Grades 10-12.

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