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)

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)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Degree", "Leading Coefficient", "End Behavior" ], "new_concepts": [], "current_concepts": [ "Polynomial Degree", "Leading Coefficient", "End Behavior" ] } ## Identify the leading term of the polynomial $$ \begin{aligned} f(x) &= x^5 - 8x^4 + 16x^3 \ \text{Leading Term} &= x^5 \end{aligned} $$ ## Determine degree and leading coefficient $$ \begin{aligned} \text{Degree } (n) &= 5 \quad (\text{odd}) \ \text{Leading Coefficient } (a_n) &= 1 \quad (\text{positive}) \end{aligned} $$ ## Apply end behavior rules $$ \begin{aligned} \lim_{x \to -\infty} f(x) &= -\infty \ \lim_{x \to +\infty} f(x) &= +\infty \end{aligned} $$

(A) $f(x) ightarrow -\infty \text{ as } x ightarrow -\infty; f(x) ightarrow -\infty \text{ as } x ightarrow +\infty$

(B) $f(x) ightarrow -\infty \text{ as } x ightarrow -\infty; f(x) ightarrow +\infty \text{ as } x ightarrow +\infty$

(C) $f(x) ightarrow +\infty \text{ as } x ightarrow -\infty; f(x) ightarrow -\infty \text{ as } x ightarrow +\infty$

(D) $f(x) ightarrow +\infty \text{ as } x ightarrow -\infty; f(x) ightarrow +\infty \text{ as } x ightarrow +\infty$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "End Behavior" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the leading term of the polynomial

Determine degree and leading coefficient

Apply end behavior rules

Answer: