Sketch the graph of the function below, being sure to include at least two full periods. $y = \cos \left( x - \frac { \pi } { 2 } \right)$

Question

Quizplus · Accepted Answer

The function $\cos \left( x - \frac{\pi}{2} \right)$ can be understood as a cosine function that is shifted to the right by $\frac{\pi}{2}$ units. This means that the maximum value of the cosine function occurs at $x = \frac{\pi}{2}$ instead of $x = 0$. 
Since the period of cosine is $2\pi$, the graph will repeat every $2\pi$ units. Therefore, we can sketch two full periods by graphing the function on the interval $[0, 4\pi]$. 
Choice C correctly shows a cosine function that is shifted to the right by $\frac{\pi}{2}$ units, with maximum values at $x = \frac{\pi}{2}$ and $x = \frac{5\pi}{2}$, and minimum values at $x = \frac{3\pi}{2}$ and $x = \frac{7\pi}{2}$. The graph repeats every $2\pi$ units, as expected.

Sketch the Graph of the Function Below, Being Sure to Include