We are given the polynomial function:

$f(x) = (x + 6)^2(2 - x)$

Step 1: Determine the degree of the polynomial

To analyze the end behavior, we need to first expand the polynomial to understand its leading term.

• $(x + 6)^2 = x^2 + 12x + 36$
• Now, multiply this with $(2 - x)$: $f(x) = (x^2 + 12x + 36)(2 - x)$

Expanding the product:

$f(x) = (x^2)(2 - x) + (12x)(2 - x) + (36)(2 - x)$

Distribute each term:

$f(x) = 2x^2 - x^3 + 24x - 12x^2 + 72 - 36x$

Simplify the expression:

$f(x) = -x^3 + 2x^2 - 36x + 72$

Step 2: Analyze the leading term for end behavior

The term with the highest degree is $-x^3$. The degree of the polynomial is 3, and the leading coefficient is $-1$.

Step 3: End behavior

For large values of $\lvert x \rvert$ (as $x \to \infty$ or $x \to -\infty$), the behavior of the graph is dominated by the leading term $-x^3$.

• As $x \to \infty$, $-x^3 \to -\infty$, so $f(x) \to -\infty$.
• As $x \to -\infty$, $-x^3 \to \infty$, so $f(x) \to \infty$.

Thus, the graph of $f(x)$ behaves like $y = -x^3$ for large values of $\lvert x \rvert$.

Final Answer:

The graph of $f(x)$ behaves like $y = -x^3$ for large values of $\lvert x \rvert$.

Would you like to see more details, or have any questions about this?

Here are five related questions to extend your understanding:

1. How do you find the roots of a polynomial function like this one?
2. What does the degree of a polynomial tell us about the shape of its graph?
3. How can we find the turning points of a polynomial function?
4. How do the coefficients of a polynomial affect its graph?
5. What role does symmetry play in the graph of polynomial functions?

Tip: Always check the leading term of a polynomial for an indication of the end behavior of its graph.