select the correct answer from each drop-down menu.

the table shows certain values of a cubic function.

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline x & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline y & 216 & 110 & 40 & 0 & -16 & -14 & 0 & 20 & 40 & 54 & 56 & 40 & 0 & -70 & -176 \\\hline\end{array}$

use the table to complete the statements.

the function has a relative maximum when $x$ is near choose an answer.
as $x$ approaches positive infinity, the value of the function approaches choose an answer.

Question

select the correct answer from each drop-down menu.

the table shows certain values of a cubic function.

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline x & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline y & 216 & 110 & 40 & 0 & -16 & -14 & 0 & 20 & 40 & 54 & 56 & 40 & 0 & -70 & -176 \\\hline\end{array}$

use the table to complete the statements.

the function has a relative maximum when $x$ is near choose an answer.
as $x$ approaches positive infinity, the value of the function approaches choose an answer.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Cubic Functions", "Relative Extrema", "End Behavior" ], "current_concepts": [ "Cubic Functions", "Relative Extrema", "End Behavior" ] } ## Identify local maximum from the table $$ \begin{aligned} &\text{For } x \in [1, 5]: \ &y(1) = 40 \ &y(2) = 54 \ &y(3) = 56 \quad (\text{maximum value in this interval}) \ &y(4) = 40 \ &y(5) = 0 \ &\implies \text{Relative maximum is near } x = 3 \end{aligned} $$ ## Determine end behavior as $x \to \infty$ $$ \begin{aligned} &y(4) = 40 \ &y(5) = 0 \ &y(6) = -70 \ &y(7) = -176 \ &\implies \text{As } x \to \infty, y \to -\infty \end{aligned} $$ The function has a relative maximum when $x$ is near 3. As $x$ approaches positive infinity, the value of the function approaches negative infinity. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Cubic Functions" ] }

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Question

Explanation:

Identify local maximum from the table

Determine end behavior as \(x \to \infty\)

Answer: