Question

what is the end behavior of the graph of \\(f(x) = x^5 - 8x^4 + 16x^3\\)?

\\(f(x) \
ightarrow -\infty\\) as \\(x \
ightarrow -\infty\\); \\(f(x) \
ightarrow -\infty\\) as \\(x \
ightarrow +\infty\\)
\\(f(x) \
ightarrow -\infty\\) as \\(x \
ightarrow -\infty\\); \\(f(x) \
ightarrow +\infty\\) as \\(x \
ightarrow +\infty\\)
\\(f(x) \
ightarrow +\infty\\) as \\(x \
ightarrow -\infty\\); \\(f(x) \
ightarrow -\infty\\) as \\(x \
ightarrow +\infty\\)
\\(f(x) \
ightarrow +\infty\\) as \\(x \
ightarrow -\infty\\); \\(f(x) \
ightarrow +\infty\\) as \\(x \
ightarrow +\infty\\)

the graph touches, but does not cross, the \\(x\\)-axis at \\(x = \\)

the graph of the function crosses the \\(x\\)-axis at \\(x = \\)

Explanation:

Determine the end behavior of the function

$$
LATEXBLOCK0
$$

Find the factored form and roots of the polynomial

$$
LATEXBLOCK1
$$

Determine x-axis behavior at each root

$$
LATEXBLOCK2
$$

Answer:

Question 1

• (A) \(f(x) \to -\infty \text{ as } x \to -\infty; f(x) \to -\infty \text{ as } x \to +\infty\)
• (B) \(f(x) \to -\infty \text{ as } x \to -\infty; f(x) \to +\infty \text{ as } x \to +\infty\) (Correct answer)
• (C) \(f(x) \to +\infty \text{ as } x \to -\infty; f(x) \to -\infty \text{ as } x \to +\infty\)
• (D) \(f(x) \to +\infty \text{ as } x \to -\infty; f(x) \to +\infty \text{ as } x \to +\infty\)

Question 2

The graph touches, but does not cross, the \(x\)-axis at \(x =\) <blank>4</blank>.

Question 3

The graph of the function crosses the \(x\)-axis at \(x =\) <blank>0</blank>.