Question

which steps will translate (f(x) = 3^x) to (g(x) = 3^{x + 1} + 4)?

shift (f(x) = 3^x) one unit up and four units to the right.
shift (f(x) = 3^x) one unit up and four units to the left.
shift (f(x) = 3^x) one unit to the right and four units up.
shift (f(x) = 3^x) one unit to the left and four units up.

Explanation:

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"concepts_used": [
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<reasoning>

Identify the transformation components

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &g(x) = f(x + h) + k \\ &g(x) = 3^{x + 1} + 4 \\ &h = 1, \quad k = 4 \end{aligned}$$

\]

Determine the horizontal shift

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &f(x + 1) \implies \text{Shift left by } 1 \text{ unit} \end{aligned}$$

\]

Determine the vertical shift

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &+ 4 \implies \text{Shift up by } 4 \text{ units} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>Shift \(f(x) = 3^x\) one unit up and four units to the right.</mcq-option>
<mcq-option>Shift \(f(x) = 3^x\) one unit up and four units to the left.</mcq-option>
<mcq-option>Shift \(f(x) = 3^x\) one unit to the right and four units up.</mcq-option>
<mcq-correct>Shift \(f(x) = 3^x\) one unit to the left and four units up.</mcq-correct>
</answer>

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

Answer:

<pre_analysis>
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"concepts_used": [
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<reasoning>

Identify the transformation components

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &g(x) = f(x + h) + k \\ &g(x) = 3^{x + 1} + 4 \\ &h = 1, \quad k = 4 \end{aligned}$$

\]

Determine the horizontal shift

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &f(x + 1) \implies \text{Shift left by } 1 \text{ unit} \end{aligned}$$

\]

Determine the vertical shift

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &+ 4 \implies \text{Shift up by } 4 \text{ units} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>Shift \(f(x) = 3^x\) one unit up and four units to the right.</mcq-option>
<mcq-option>Shift \(f(x) = 3^x\) one unit up and four units to the left.</mcq-option>
<mcq-option>Shift \(f(x) = 3^x\) one unit to the right and four units up.</mcq-option>
<mcq-correct>Shift \(f(x) = 3^x\) one unit to the left and four units up.</mcq-correct>
</answer>

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