The function given in the image is:

$f(x) = -4(x + 3)(x - 3)^3$

We are asked to determine the end behavior of the graph of this polynomial function.

Step 1: Expand the polynomial

The degree of a polynomial controls its end behavior. Let’s analyze the degree first by expanding $(x - 3)^3$:

$(x - 3)^3 = (x - 3)(x - 3)(x - 3)$

So the function becomes:

$f(x) = -4(x + 3)(x - 3)^3 = -4(x + 3)(x^3 - 9x^2 + 27x - 27)$

Step 2: Find the leading term

We only need the leading term to determine the end behavior, which comes from multiplying the highest degree terms in the expanded polynomial:

$-4(x)(x^3) = -4x^4$

Step 3: Determine the end behavior

Since the leading term is $-4x^4$, the degree is 4 (an even degree), and the leading coefficient is negative. For polynomials of even degree with a negative leading coefficient, the end behavior is:

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

Thus, the graph of $f(x)$ behaves like:

$y = -x^4$

for large values of $|x|$.

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

Related Questions:

1. What is the role of the degree of a polynomial in determining its end behavior?
2. How does the leading coefficient affect the direction of the graph?
3. What is the difference between odd and even degree polynomials in terms of end behavior?
4. Can you explain how to expand a cubic polynomial like $(x - 3)^3$?
5. How do we factor polynomials to simplify expressions like $f(x)$?

Tip:

Remember that for polynomials, the end behavior is determined entirely by the leading term (the term with the highest degree).