consider the function $f(x) = -x^4 + 9$. determine which of the following is its graph, based on end behavior.

a)

b)

c)

Question

consider the function $f(x) = -x^4 + 9$. determine which of the following is its graph, based on end behavior.

a)

b)

c)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Polynomial End Behavior", "Leading Coefficient Test" ], "new_concepts": [], "current_concepts": [ "Polynomial End Behavior", "Leading Coefficient Test", "Polynomial Functions" ] } ## Identify degree and leading coefficient Using the *Polynomial End Behavior* and *Leading Coefficient Test* knowledge points $$ \begin{aligned} &f(x) = -x^4 + 9\ &\text{Degree } (n) = 4 \quad (\text{even})\ &\text{Leading Coefficient } (a_n) = -1 \quad (\text{negative}) \end{aligned} $$ ## Determine the end behavior Using the *Polynomial End Behavior* and *Leading Coefficient Test* knowledge points $$ \begin{aligned} &\lim_{x \to \infty} f(x) = -\infty\ &\lim_{x \to -\infty} f(x) = -\infty \end{aligned} $$ Both ends of the graph must point downwards. ## Match with the correct graph option Using the *Polynomial End Behavior* knowledge point - Option A: Left end points up, right end points down. - Option B: Left end points down, right end points up. - Option C: Both ends point down (parabola-like shape opening downwards with a y-intercept at 9). Therefore, Option C is the correct graph.

(A) Left end points up, right end points down

(B) Left end points down, right end points up

(C) Both ends point down

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

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QUESTION IMAGE

Question

Explanation:

Identify degree and leading coefficient

Determine the end behavior

Match with the correct graph option

Answer: