which is the graph of y = cos4(x - π)?

Explanation:

Step1: Simplify the function

Use the cosine - difference formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$. Here, $y = \cos4(x-\pi)=\cos(4x - 4\pi)$. Since $\cos(A - 2k\pi)=\cos A$ for any real number $A$ and integer $k$, and $4\pi = 2\times2\pi$, then $y=\cos(4x - 4\pi)=\cos4x$.

Step2: Find the period of the function

The general form of a cosine function is $y = A\cos(Bx - C)+D$, and its period is given by $T=\frac{2\pi}{|B|}$. For the function $y=\cos4x$, where $B = 4$, the period $T=\frac{2\pi}{4}=\frac{\pi}{2}$.

Step3: Analyze the key - points

The standard cosine function $y = \cos x$ has a maximum value of $y = 1$ at $x = 2k\pi,k\in\mathbb{Z}$, and a minimum value of $y=-1$ at $x=(2k + 1)\pi,k\in\mathbb{Z}$. For $y=\cos4x$, when $4x = 2k\pi$ (i.e., $x=\frac{k\pi}{2}$), $y = 1$; when $4x=(2k + 1)\pi$ (i.e., $x=\frac{(2k + 1)\pi}{4}$), $y=-1$.

The graph of $y = \cos4x$ oscillates 4 times as fast as the graph of $y=\cos x$ within the interval $[0,2\pi]$. The period of $y=\cos4x$ is $\frac{\pi}{2}$, so there are 4 full - cycles of the cosine wave in the interval $[0,2\pi]$.

The correct graph is the one with 4 full - cycles of the cosine wave in the interval $[0,2\pi]$.

Answer:

The second graph (the one with 4 full - cycles of the cosine wave in the interval $[0,2\pi]$)