Question

question 5 of 12 step 2 of 2
graph the following function: $y = \frac{1}{2}\cos(x - 4\pi)$
step 2 of 2: determine how the general shape of the graph, chosen in the previous step, would be shifted, stretched, and reflected for the given function. graph the results on the axes provided.
answer
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x - axis reflection
reflect graph across x - axis
shift graph vertically
up down none

Explanation:

Step1: Analyze the coefficient of cosine

The coefficient of $\cos(x - 4\pi)$ is $\frac{1}{2}$. This causes a vertical stretch - compression. Since $0<\frac{1}{2}<1$, it vertically compresses the graph of $y = \cos(x)$ by a factor of $\frac{1}{2}$.

Step2: Analyze the phase - shift

The argument of the cosine function is $x - 4\pi$. For a cosine function $y = A\cos(Bx - C)+D$, the phase - shift is given by $\frac{C}{B}$. Here $B = 1$ and $C = 4\pi$, so the phase - shift is $\frac{4\pi}{1}=4\pi$. Since $4\pi$ is a multiple of $2\pi$, the graph of $y=\cos(x)$ is shifted to the right by $4\pi$ units, which is equivalent to no shift (because the cosine function is periodic with period $2\pi$).

Step3: Check for reflection and vertical shift

There is no negative sign in front of the cosine function, so there is no reflection across the $x$ - axis. Also, there is no constant added or subtracted outside the cosine function, so there is no vertical shift.

Answer:

$x$-Axis Reflection: No
Shift Graph Vertically: None