Question

Question was provided via image upload.

Explanation:

Identify key points on the graph

The graph is an exponential growth curve passing through the y-intercept \((0, 0.5)\) or \((0, \frac{1}{2})\).

Test the given options at \(x = 0\)

Evaluate the visible options at \(x = 0\):

• For \(y = \frac{1}{8}e^x\): \(y(0) = \frac{1}{8}e^0 = \frac{1}{8} = 0.125\)
• For \(y = \frac{1}{2}e^x\): \(y(0) = \frac{1}{2}e^0 = \frac{1}{2} = 0.5\)

Confirm with another point

At \(x = 2\), the second option gives:

$$ y = \frac{1}{2}e^2 \approx \frac{1}{2}(7.389) \approx 3.69 $$

Looking at the graph, at \(x = 2\), the curve is indeed between \(3\) and \(4\), close to \(3.7\).

Answer:

• (A) \(y = \frac{1}{8}e^x\)
• (B) \(y = \frac{1}{2}e^x\) (Correct answer)