Question

question 4 points 1

if a linear function has \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ slope, then the graph is a vertical line.

zero

undefined

Explanation:

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<reasoning>

Analyze the definition of slope

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

Apply to a vertical line

For a vertical line, the \(x\)-coordinates of all points on the line are constant:
\[
x_2 - x_1 = 0
\]
Substituting this into the slope formula yields division by zero:
\[
m = \frac{y_2 - y_1}{0} \implies \text{undefined}
\]
</reasoning>

<answer>
If a linear function has <blank>undefined</blank> slope, then the graph is a vertical line.

<mcq-option>zero</mcq-option>
<mcq-correct>undefined</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,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
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"concepts_used": [
"Slope of a Linear Function"
],
"new_concepts": [],
"current_concepts": [
"Slope of a Linear Function"
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}
</pre_analysis>

<reasoning>

Analyze the definition of slope

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

Apply to a vertical line

For a vertical line, the \(x\)-coordinates of all points on the line are constant:
\[
x_2 - x_1 = 0
\]
Substituting this into the slope formula yields division by zero:
\[
m = \frac{y_2 - y_1}{0} \implies \text{undefined}
\]
</reasoning>

<answer>
If a linear function has <blank>undefined</blank> slope, then the graph is a vertical line.

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

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