Question

which is the graph of \\(f(x) = 2(3)^x\\)?

Explanation:

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<reasoning>

Evaluate the y-intercept

Using the Y-Intercept and Exponential Functions knowledge points
\[
f(0) = 2(3)^0 = 2(1) = 2
\]

Evaluate the function at key points

Using the Function Evaluation knowledge point
\[

$$\begin{aligned} f(1) &= 2(3)^1 = 6 \\ f(2) &= 2(3)^2 = 18 \end{aligned}$$

\]

Match with the correct graph

Using the Graph Interpretation knowledge point

• The first graph has a y-intercept at \((0, 2)\) and passes through the point \((1, 6)\).
• The second graph passes through \((1, 9)\).
• The third graph passes through \((2, 8)\).
• The fourth graph shows a segment passing through \((2, 12)\).
• Therefore, the first graph represents \(f(x) = 2(3)^x\).

</reasoning>

<answer>
<mcq-correct>(A) The graph with y-intercept at \((0, 2)\) passing through the point \((1, 6)\)</mcq-correct>
<mcq-option>(B) The graph passing through the point \((1, 9)\)</mcq-option>
<mcq-option>(C) The graph passing through the point \((2, 8)\)</mcq-option>
<mcq-option>(D) The graph passing through the point \((2, 12)\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Evaluate the y-intercept

Using the Y-Intercept and Exponential Functions knowledge points
\[
f(0) = 2(3)^0 = 2(1) = 2
\]

Evaluate the function at key points

Using the Function Evaluation knowledge point
\[

$$\begin{aligned} f(1) &= 2(3)^1 = 6 \\ f(2) &= 2(3)^2 = 18 \end{aligned}$$

\]

Match with the correct graph

Using the Graph Interpretation knowledge point

• The first graph has a y-intercept at \((0, 2)\) and passes through the point \((1, 6)\).
• The second graph passes through \((1, 9)\).
• The third graph passes through \((2, 8)\).
• The fourth graph shows a segment passing through \((2, 12)\).
• Therefore, the first graph represents \(f(x) = 2(3)^x\).

</reasoning>

<answer>
<mcq-correct>(A) The graph with y-intercept at \((0, 2)\) passing through the point \((1, 6)\)</mcq-correct>
<mcq-option>(B) The graph passing through the point \((1, 9)\)</mcq-option>
<mcq-option>(C) The graph passing through the point \((2, 8)\)</mcq-option>
<mcq-option>(D) The graph passing through the point \((2, 12)\)</mcq-option>
</answer>

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