Question

match each function with its graph.
$y = e^{x}$
$y = 3e^{x/3}$
$y = 3e^{x}$
$y = 3e^{-\frac{x}{3}}$

Explanation:

Step1: Analyze \(y = e^{x}\)

When \(x = 0\), \(y=e^{0}=1\). The function \(y = e^{x}\) is an exponential growth function.

Step2: Analyze \(y = 3e^{x/3}\)

When \(x = 0\), \(y = 3e^{0}=3\). Let \(t=\frac{x}{3}\), then \(y = 3e^{t}\). Compared to \(y = e^{x}\), it has a vertical stretch by a factor of \(3\) and a horizontal stretch by a factor of \(3\).

Step3: Analyze \(y = 3e^{x}\)

When \(x = 0\), \(y=3e^{0}=3\). Compared to \(y = e^{x}\), it has a vertical stretch by a factor of \(3\).

Step4: Analyze \(y = 3e^{-\frac{x}{3}}\)

When \(x = 0\), \(y = 3e^{0}=3\). Since the exponent is \(-\frac{x}{3}\), it is an exponential decay function.

Answer:

\(y = e^{x}\) matches the graph with \(y - \)intercept at \((0,1)\) (the first non - labeled graph among the four). \(y = 3e^{x/3}\) matches the graph which is less “steep” than \(y = 3e^{x}\) with \(y-\)intercept at \((0,3)\). \(y = 3e^{x}\) matches the steeper growth graph with \(y-\)intercept at \((0,3)\). \(y = 3e^{-\frac{x}{3}}\) matches the decay graph with \(y-\)intercept at \((0,3)\)