Question

which set of steps will translate \\(f(x) = 6^x\\) to \\(g(x) = 6^{x - 5} - 7\\)?

shift \\(f(x) = 6^x\\) five units to the left and seven units up.
shift \\(f(x) = 6^x\\) five units to the right and seven units down.
shift \\(f(x) = 6^x\\) seven units to the right and five units up.
shift \\(f(x) = 6^x\\) seven units to the left and five units down.

Explanation:

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<reasoning>

Analyze horizontal translation

\[
g(x) = f(x-5) - 7 = 6^{x-5} - 7
\]
Replacing \(x\) with \(x-5\) shifts the graph \(5\) units to the right.

Analyze vertical translation

Subtracting \(7\) from the function shifts the graph \(7\) units down.

Combine transformations

The graph of \(f(x)\) is shifted \(5\) units to the right and \(7\) units down.
</reasoning>

<answer>
<mcq-option>Shift \(f(x) = 6^x\) five units to the left and seven units up.</mcq-option>
<mcq-correct>Shift \(f(x) = 6^x\) five units to the right and seven units down.</mcq-correct>
<mcq-option>Shift \(f(x) = 6^x\) seven units to the right and five units up.</mcq-option>
<mcq-option>Shift \(f(x) = 6^x\) seven units to the left and five units down.</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Transformations"
]
}
</post_analysis>

Answer:

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"step_cot",
"multiple_choice_formatter",
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"Function Transformations"
],
"new_concepts": [],
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</pre_analysis>

<reasoning>

Analyze horizontal translation

\[
g(x) = f(x-5) - 7 = 6^{x-5} - 7
\]
Replacing \(x\) with \(x-5\) shifts the graph \(5\) units to the right.

Analyze vertical translation

Subtracting \(7\) from the function shifts the graph \(7\) units down.

Combine transformations

The graph of \(f(x)\) is shifted \(5\) units to the right and \(7\) units down.
</reasoning>

<answer>
<mcq-option>Shift \(f(x) = 6^x\) five units to the left and seven units up.</mcq-option>
<mcq-correct>Shift \(f(x) = 6^x\) five units to the right and seven units down.</mcq-correct>
<mcq-option>Shift \(f(x) = 6^x\) seven units to the right and five units up.</mcq-option>
<mcq-option>Shift \(f(x) = 6^x\) seven units to the left and five units down.</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Transformations"
]
}
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