Question

select the correct answer. which statement correctly compares the graph of function g with the graph of function f? f(x)=e^x - 4 g(x)=1/2 e^x - 4 a. the graph of function g is a vertical stretch of the graph of function f. b. the graph of function g is a vertical compression of the graph of function f. c. the graph of function g is a horizontal shift of the graph of function f to the right. d. the graph of function g is a horizontal shift of the graph of function f to the left.

Explanation:

Step1: Recall function transformation rules

For a function \(y = af(x)+k\), if \(0 < a<1\), it is a vertical compression. Here \(f(x)=e^{x}-4\) and \(g(x)=\frac{1}{2}e^{x}-4\). We can see that \(g(x)=\frac{1}{2}f(x)+0\) (since \(f(x) = e^{x}-4\)).

Step2: Analyze the transformation

The coefficient of \(e^{x}\) in \(g(x)\) is \(\frac{1}{2}\) (where the coefficient of \(e^{x}\) in \(f(x)\) is \(1\)). According to the rule \(y = af(x)\) (\(k = 0\) in our case as \(-4\) is the vertical shift for both functions and cancels out when comparing the non - shift part), when \(a=\frac{1}{2}\) (\(0<\frac{1}{2}<1\)), the graph of \(y = g(x)\) is a vertical compression of the graph of \(y = f(x)\).

Answer:

B. The graph of function \(g\) is a vertical compression of the graph of function \(f\).