the graph shows a cosine function $f(x)=4cos(x + \frac{pi}{2})$ and its transformed function, $g(x)$. which…

the graph shows a cosine function $f(x)=4cos(x + \frac{pi}{2})$ and its transformed function, $g(x)$. which value of $k$ is used to create the function $g(x)$? $k = 0.5$ $k=-1$ $k = 1$ $k = 0.33$

the graph shows a cosine function $f(x)=4cos(x + \frac{pi}{2})$ and its transformed function, $g(x)$. which value of $k$ is used to create the function $g(x)$? $k = 0.5$ $k=-1$ $k = 1$ $k = 0.33$

Answer

Explanation:

Step1: Analyze vertical - stretch/shrink

The general form of a vertical transformation of a function $y = f(x)$ is $y=kf(x)$. If $|k|> 1$, the graph is vertically stretched, if $0 < |k|<1$, the graph is vertically shrunk, if $k = - 1$, the graph is reflected about the $x$ - axis, and if $k = 1$, the graph remains the same.

Step2: Observe the graphs of $f(x)$ and $g(x)$

The amplitude of $f(x)=4\cos(x +\frac{\pi}{2})$ is $A_f = 4$. The amplitude of $g(x)$ is $A_g=2$. Since $A_g=\frac{1}{2}A_f$, and $g(x)=kf(x)$, we have $|k|=\frac{A_g}{A_f}$.

Step3: Calculate the value of $k$

Substitute $A_f = 4$ and $A_g = 2$ into the formula $|k|=\frac{A_g}{A_f}$, we get $|k|=\frac{2}{4}=0.5$. Also, since the graph of $g(x)$ is not reflected about the $x$ - axis (the peaks and troughs are in the same general vertical orientation as $f(x)$), $k = 0.5$.

Answer:

$k = 0.5$