Question

what is the end behavior of the graph of the polynomial function \\(f(x) = -x^5 + 9x^4 - 18x^3\\)?

as \\(x \to -\infty\\), \\(y \to -\infty\\) and as \\(x \to \infty\\), \\(y \to -\infty\\)
as \\(x \to -\infty\\), \\(y \to -\infty\\) and as \\(x \to \infty\\), \\(y \to \infty\\)
as \\(x \to -\infty\\), \\(y \to \infty\\) and as \\(x \to \infty\\), \\(y \to -\infty\\)
as \\(x \to -\infty\\), \\(y \to \infty\\) and as \\(x \to \infty\\), \\(y \to \infty\\)

Explanation:

Identify the leading term

$$ f(x) = -x^5 + 9x^4 - 18x^3 \implies \text{Leading term} = -x^5 $$

Analyze degree and leading coefficient

$$
LATEXBLOCK0
$$

Determine limits as \(x \to \pm\infty\)

$$
LATEXBLOCK1
$$

Answer:

• As \(x \to -\infty\), \(y \to -\infty\) and as \(x \to \infty\), \(y \to -\infty\)
• As \(x \to -\infty\), \(y \to -\infty\) and as \(x \to \infty\), \(y \to \infty\)
• As \(x \to -\infty\), \(y \to \infty\) and as \(x \to \infty\), \(y \to -\infty\) (Correct answer)
• As \(x \to -\infty\), \(y \to \infty\) and as \(x \to \infty\), \(y \to \infty\)