Question

the graph of a polynomial function is given. what is the smallest degree that the polynomial could have?

the smallest degree the polynomial could have is
(simplify your answer.)

Explanation:

<pre_analysis>
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"concepts_used": [
"Degree of Polynomial",
"Turning Points",
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"Turning Points",
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</pre_analysis>

<reasoning>

Identify the turning points of the graph

Using the Turning Points knowledge point
\[
\text{Number of local extrema (peaks and valleys)} = 3
\]

Determine the minimum degree from turning points

Using the Degree of Polynomial and Turning Points knowledge points
\[

$$\begin{aligned} &\text{Degree } n \ge \text{Number of turning points} + 1 \\ &n \ge 3 + 1 = 4 \end{aligned}$$

\]

Analyze end behavior for consistency

Using the End Behavior of Polynomials knowledge point
\[

$$\begin{aligned} &\lim_{x \to -\infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to \infty} f(x) = -\infty \\ &\text{Opposite end behaviors require an odd degree } n \\ &n \ge 4 \implies \text{Smallest odd integer is } 5 \end{aligned}$$

\]
</reasoning>

<answer>
The graph of a polynomial function is given. What is the smallest degree that the polynomial could have?

The smallest degree the polynomial could have is <blank>5</blank>
(Simplify your answer.)
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Degree of Polynomial"
]
}
</post_analysis>

Answer:

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"Degree of Polynomial",
"Turning Points",
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"Degree of Polynomial",
"Turning Points",
"End Behavior of Polynomials",
"Polynomial Graph Characteristics"
]
}
</pre_analysis>

<reasoning>

Identify the turning points of the graph

Using the Turning Points knowledge point
\[
\text{Number of local extrema (peaks and valleys)} = 3
\]

Determine the minimum degree from turning points

Using the Degree of Polynomial and Turning Points knowledge points
\[

$$\begin{aligned} &\text{Degree } n \ge \text{Number of turning points} + 1 \\ &n \ge 3 + 1 = 4 \end{aligned}$$

\]

Analyze end behavior for consistency

Using the End Behavior of Polynomials knowledge point
\[

$$\begin{aligned} &\lim_{x \to -\infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to \infty} f(x) = -\infty \\ &\text{Opposite end behaviors require an odd degree } n \\ &n \ge 4 \implies \text{Smallest odd integer is } 5 \end{aligned}$$

\]
</reasoning>

<answer>
The graph of a polynomial function is given. What is the smallest degree that the polynomial could have?

The smallest degree the polynomial could have is <blank>5</blank>
(Simplify your answer.)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Degree of Polynomial"
]
}
</post_analysis>