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Part 1
Analyze the polynomial function ​f(x)equals​(xplus6​)squared​(2minus​x) Determine the end behavior of the graph of the function.
The graph of f behaves like yequals

enter your response here for large values of StartAbsoluteValue x EndAbsoluteValue.

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.

Analyze the polynomial function f(x) = (x + 6)^2(2 - x). Determine the end behavior of the graph of the function. The graph of f behaves like y = ___ for large values of |x|.

Learn how to determine the end behavior of the polynomial function f(x) = (x + 6)^2(2 - x). This solution includes expanding the function, analyzing the leading term, and applying the End Behavior Theorem. Suitable for students in Grades 10-12.

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