which of the following is the graph of $y = \\cos(2(x + \\pi))$?
which of the following is the graph of $y = \\cos(2(x + \\pi))$?
Answer
Answer:
The first - shown graph (the one with multiple full - cycles in the given x - range)
Explanation:
Step1: Simplify the function
Use the cosine addition formula $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here, $y = \cos(2(x+\pi))=\cos(2x + 2\pi)$. Since $\cos(\theta+2k\pi)=\cos\theta$ for any real number $\theta$ and integer $k$, then $y=\cos(2x + 2\pi)=\cos(2x)$.
Step2: Analyze the period
The general form of a cosine function is $y = A\cos(Bx - C)+D$. For $y=\cos(2x)$, $A = 1$, $B = 2$, $C = 0$, $D = 0$. The period of the cosine function $y=\cos(Bx)$ is given by $T=\frac{2\pi}{|B|}$. Substituting $B = 2$ into the formula, we get $T=\frac{2\pi}{2}=\pi$. The first graph shows multiple full - cycles of a cosine function with period $\pi$ in the given x - range from $-\pi$ to $\pi$, while the second graph does not match the period and shape of $y = \cos(2x)$.