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.