Question

if a linear function has a \\_\\_\\_\\_\\_\\_\\_\\_ slope, then the graph rises up as \\(x\\) increases.

• zero
• positive

Explanation:

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"Slope of a Linear Function"
],
"new_concepts": [],
"current_concepts": [
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</pre_analysis>

<reasoning>

Analyze the definition of slope

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Determine the sign of the slope

\[
\Delta x > 0 \implies x_2 > x_1
\]
\[
\text{Graph rises up} \implies y_2 > y_1 \implies \Delta y > 0
\]
\[
m = \frac{\Delta y}{\Delta x} > 0
\]
</reasoning>

<answer>
<mcq-option>zero</mcq-option>
<mcq-correct>positive</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Slope of a Linear Function"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
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"question_count": 1,
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
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"Slope of a Linear Function"
],
"new_concepts": [],
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]
}
</pre_analysis>

<reasoning>

Analyze the definition of slope

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Determine the sign of the slope

\[
\Delta x > 0 \implies x_2 > x_1
\]
\[
\text{Graph rises up} \implies y_2 > y_1 \implies \Delta y > 0
\]
\[
m = \frac{\Delta y}{\Delta x} > 0
\]
</reasoning>

<answer>
<mcq-option>zero</mcq-option>
<mcq-correct>positive</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Slope of a Linear Function"
]
}
</post_analysis>