Question

y = \frac{1}{8}e^x

y = \frac{1}{2}e^x

y = 2e^x

y = 8e^x

Explanation:

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<reasoning>

Identify key points on the graph

Using the Exponential Function Graphing knowledge point
\[

$$\begin{aligned} &\text{At } x = 0, \quad y = 0.5 = \frac{1}{2}\\ &\text{At } x = 1, \quad y \approx 1.36\\ &\text{At } x = 2, \quad y \approx 3.69 \end{aligned}$$

\]

Test the general equation form

Using the Exponential Function Graphing knowledge point
\[

$$\begin{aligned} &y = a e^x\\ &\text{Substitute } (0, \frac{1}{2}):\\ &\frac{1}{2} = a e^0 \implies a = \frac{1}{2} \end{aligned}$$

\]

Verify with another point

Using the Exponential Function Graphing knowledge point
\[

$$\begin{aligned} &\text{For } y = \frac{1}{2}e^x:\\ &\text{At } x = 2, \quad y = \frac{1}{2}e^2 \approx \frac{1}{2}(7.389) \approx 3.69 \end{aligned}$$

\]
This matches the graph perfectly.
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{1}{8}e^x\)</mcq-option>
<mcq-correct>(B) \(y = \frac{1}{2}e^x\)</mcq-correct>
<mcq-option>(C) \(y = 2e^x\)</mcq-option>
<mcq-option>(D) \(y = 8e^x\)</mcq-option>
</answer>

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"Mathematics",
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Answer:

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<reasoning>

Identify key points on the graph

Using the Exponential Function Graphing knowledge point
\[

$$\begin{aligned} &\text{At } x = 0, \quad y = 0.5 = \frac{1}{2}\\ &\text{At } x = 1, \quad y \approx 1.36\\ &\text{At } x = 2, \quad y \approx 3.69 \end{aligned}$$

\]

Test the general equation form

Using the Exponential Function Graphing knowledge point
\[

$$\begin{aligned} &y = a e^x\\ &\text{Substitute } (0, \frac{1}{2}):\\ &\frac{1}{2} = a e^0 \implies a = \frac{1}{2} \end{aligned}$$

\]

Verify with another point

Using the Exponential Function Graphing knowledge point
\[

$$\begin{aligned} &\text{For } y = \frac{1}{2}e^x:\\ &\text{At } x = 2, \quad y = \frac{1}{2}e^2 \approx \frac{1}{2}(7.389) \approx 3.69 \end{aligned}$$

\]
This matches the graph perfectly.
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{1}{8}e^x\)</mcq-option>
<mcq-correct>(B) \(y = \frac{1}{2}e^x\)</mcq-correct>
<mcq-option>(C) \(y = 2e^x\)</mcq-option>
<mcq-option>(D) \(y = 8e^x\)</mcq-option>
</answer>

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