Question

graph the following function.
y = \frac{1}{5}\cos(\frac{1}{2}x - \frac{\pi}{8})
which transformations are needed to graph the function y = \frac{1}{5}\cos(\frac{1}{2}x - \frac{\pi}{8})? choose the correct answer below.
a. the graph of y = cos x should be horizontally shifted to the right by \frac{\pi}{4} units, horizontally stretched by a factor of 2, and vertically shrunk by a factor of \frac{1}{5}.
b. the graph of y = cos x should be horizontally shifted to the right by \frac{\pi}{4} units, horizontally shrunk by a factor of \frac{1}{5}, and vertically stretched by a factor of 2.
c. the graph of y = cos x should be horizontally shifted to the left by \frac{\pi}{4} units, horizontally stretched by a factor of 2, and vertically shrunk by a factor of \frac{1}{5}.
d. the graph of y = cos x should be horizontally shifted to the left by \frac{\pi}{4} units, horizontally shrunk by a factor of \frac{1}{5}, and vertically stretched by a factor of 2.

Explanation:

Step1: Analyze horizontal - shift

For the general form of a cosine function $y = A\cos(Bx - C)+D$, the horizontal shift is given by $\frac{C}{B}$. Here, $B=\frac{1}{2}$ and $C = \frac{\pi}{8}$, so the horizontal shift is $\frac{\frac{\pi}{8}}{\frac{1}{2}}=\frac{\pi}{4}$ units to the right (since for $y=\cos(Bx - C)$, a positive $C$ value gives a right - shift).

Step2: Analyze horizontal stretch/shrink

The horizontal stretch or shrink factor is related to the coefficient $B$. The period of $y = \cos x$ is $2\pi$, and the period of $y=\cos(Bx)$ is $\frac{2\pi}{B}$. Here, $B=\frac{1}{2}$, so the graph of $y = \cos x$ is horizontally stretched by a factor of $\frac{1}{B}=2$.

Step3: Analyze vertical stretch/shrink

The amplitude of the function $y = A\cos(Bx - C)+D$ is $|A|$. Here, $A=\frac{1}{5}$, so the graph of $y=\cos x$ is vertically shrunk by a factor of $\frac{1}{5}$.

Answer:

A. The graph of $y = \cos x$ should be horizontally shifted to the right by $\frac{\pi}{4}$ units, horizontally stretched by a factor of 2, and vertically shrunk by a factor of $\frac{1}{5}$.