Question

activity - comparing linear and non-linear functions

pair the answers.

graph - property

cubic function | linear function
quadratic function | maximum of three roots
no roots | slope
maximum of two roots | exponential function: y = 2^x

Explanation:

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"needs_drawing": false,
"concepts_used": [
"Linear Equations",
"Slope of a Linear Function",
"Quadratic Equations",
"Exponential Functions"
],
"new_concepts": [
"Cubic Function",
"Polynomial Roots"
],
"current_concepts": [
"Linear Equations",
"Slope of a Linear Function",
"Quadratic Equations",
"Exponential Functions",
"Cubic Function",
"Polynomial Roots"
]
}
</pre_analysis>

<reasoning>

Match Linear Function properties

Using the Linear Equations and Slope of a Linear Function knowledge points
\[
\text{Linear Function}
ightarrow \text{Slope}
\]

Match Quadratic Function properties

Using the Quadratic Equations knowledge point
\[
\text{Quadratic Function}
ightarrow \text{Maximum of two roots}
\]

Match Cubic Function properties

A cubic polynomial function of degree 3 can have at most 3 real roots.
\[
\text{Cubic Function}
ightarrow \text{Maximum of three roots}
\]

Match Exponential Function properties

Using the Exponential Functions knowledge point
\[
y = 2^x > 0 \implies \text{No roots}
\]
</reasoning>

<answer>

Graph	Property
Quadratic Function	Maximum of two roots
Cubic Function	Maximum of three roots
Exponential Function: \(y = 2^x\)	No roots

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Functions"
]
}
</post_analysis>

Answer:

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"step_cot",
"table_formatter",
"explain_then_answer",
"tutor",
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],
"needs_drawing": false,
"concepts_used": [
"Linear Equations",
"Slope of a Linear Function",
"Quadratic Equations",
"Exponential Functions"
],
"new_concepts": [
"Cubic Function",
"Polynomial Roots"
],
"current_concepts": [
"Linear Equations",
"Slope of a Linear Function",
"Quadratic Equations",
"Exponential Functions",
"Cubic Function",
"Polynomial Roots"
]
}
</pre_analysis>

<reasoning>

Match Linear Function properties

Using the Linear Equations and Slope of a Linear Function knowledge points
\[
\text{Linear Function}
ightarrow \text{Slope}
\]

Match Quadratic Function properties

Using the Quadratic Equations knowledge point
\[
\text{Quadratic Function}
ightarrow \text{Maximum of two roots}
\]

Match Cubic Function properties

A cubic polynomial function of degree 3 can have at most 3 real roots.
\[
\text{Cubic Function}
ightarrow \text{Maximum of three roots}
\]

Match Exponential Function properties

Using the Exponential Functions knowledge point
\[
y = 2^x > 0 \implies \text{No roots}
\]
</reasoning>

<answer>

Graph	Property
Quadratic Function	Maximum of two roots
Cubic Function	Maximum of three roots
Exponential Function: \(y = 2^x\)	No roots

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Functions"
]
}
</post_analysis>