which of the following is the graph of y = cos(2(x + π))?
which of the following is the graph of y = cos(2(x + π))?
Answer
Explanation:
Step1: Simplify the function
Use the cosine addition - formula $\cos(A + B)=\cos A\cos B-\sin A\sin B$. First, expand $y = \cos(2(x+\pi))=\cos(2x + 2\pi)$. Since $\cos(A + 2k\pi)=\cos A$ for any real - number $A$ and integer $k$, then $\cos(2x + 2\pi)=\cos(2x)$.
Step2: Recall the properties of the cosine function $y = \cos(2x)$
The general form of a cosine function is $y = A\cos(Bx - C)+D$. For $y=\cos(2x)$, $A = 1$, $B = 2$, $C = 0$, and $D = 0$. The amplitude $|A|=1$, the period $T=\frac{2\pi}{|B|}$. Substituting $B = 2$ into the period formula, we get $T=\frac{2\pi}{2}=\pi$. The standard cosine function $y=\cos x$ has a period of $2\pi$, and $y = \cos(2x)$ is a horizontal compression of $y=\cos x$ by a factor of $\frac{1}{2}$.
Step3: Analyze the graph
The cosine function $y=\cos(2x)$ has a maximum value of $y = 1$ and a minimum value of $y=-1$. It starts at $y = 1$ when $x = 0$ (because $\cos(0)=1$), and has a period of $\pi$.
The graph shown in the first option is a cosine - type graph with amplitude $1$ and period $\pi$, which is the graph of $y=\cos(2x)$ (equivalent to $y = \cos(2(x+\pi))$).