To graph the function $y = \cot\left(2x + \frac{\pi}{2}\right)$ over a two-period interval, let's break it down into its key components.

Step-by-Step Explanation:

1. General form of the cotangent function: The cotangent function is periodic with a period of $\pi$. In the given function, the period is modified by the factor of 2 inside the argument of the cotangent, which will affect its period.

2. Find the period: The period $T$ of a cotangent function $y = \cot(bx)$ is given by: $T = \frac{\pi}{|b|}$ Here, $b = 2$, so the period becomes: $T = \frac{\pi}{2}$ This means that the function will complete one cycle over an interval of $\frac{\pi}{2}$.

3. Phase shift: The phase shift is determined by the term inside the argument of the cotangent function. The general form $y = \cot(bx + c)$ has a phase shift of: $\text{Phase shift} = -\frac{c}{b}$ In this case, $c = \frac{\pi}{2}$, so: $\text{Phase shift} = -\frac{\frac{\pi}{2}}{2} = -\frac{\pi}{4}$ So, the graph is shifted to the left by $\frac{\pi}{4}$.

4. Graph over a two-period interval: Since the period is $\frac{\pi}{2}$, two periods will cover $\pi$. Hence, we will graph the function over the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, taking into account the phase shift.

I will now generate the graph for the function $y = \cot\left(2x + \frac{\pi}{2}\right)$ over this two-period interval. Let me show you the graph.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like further clarification on the steps, or should I assist you in graphing this function manually?

Here are some related questions you might find interesting:

1. What is the effect of changing the coefficient inside the cotangent function on the graph?
2. How does the phase shift affect the graph of trigonometric functions in general?
3. What is the difference between the cotangent and tangent graphs?
4. How can we find the vertical asymptotes for the cotangent function?
5. How would the graph look if the coefficient in front of $x$ was not 2?

Tip: Cotangent functions have undefined values (asymptotes) at certain points, usually multiples of $\pi$, due to division by zero.