Let's break down the function you provided:

$y = \cos\left(x - \frac{\pi}{2}\right)$

This is a cosine function with a horizontal (phase) shift. Let's analyze the various characteristics of the graph:

1. Amplitude:

The amplitude of a cosine function of the form $y = A\cos(Bx + C) + D$ is given by $|A|$. In your case, there is no coefficient in front of the cosine function, so the amplitude is: $\text{Amplitude} = 1$

2. Period:

The period of a cosine function is calculated using the formula: $\text{Period} = \frac{2\pi}{|B|}$ Here, the function is $\cos(x - \frac{\pi}{2})$, and since there is no coefficient multiplying $x$ (i.e., $B = 1$), the period remains: $\text{Period} = \frac{2\pi}{1} = 2\pi$

3. Vertical Shift:

The vertical shift $D$ is the constant added or subtracted outside the cosine function. Since there is no constant added, the vertical shift is: $\text{Vertical Shift} = 0$

4. Phase Shift:

The phase shift $C$ is determined by the expression inside the cosine function, $(x - \frac{\pi}{2})$. To find the phase shift, use the formula: $\text{Phase Shift} = -\frac{C}{B}$ Here, $C = -\frac{\pi}{2}$ and $B = 1$, so the phase shift is: $\text{Phase Shift} = \frac{\pi}{2} \text{ to the right.}$

Summary:

• Amplitude: 1
• Period: $2\pi$
• Vertical Shift: 0
• Phase Shift: $\frac{\pi}{2}$ units to the right.

Would you like to see the graph or further details on how to plot it?

Here are 5 related questions for further practice:

1. How does the amplitude affect the graph of a cosine function?
2. What happens to the period of the function if the equation is $\cos(2x - \frac{\pi}{2})$?
3. How would the graph change if there were a vertical shift, say $y = \cos(x - \frac{\pi}{2}) + 2$?
4. What is the period of the function $y = \sin(x - \pi)$?
5. How can you calculate the phase shift for the function $y = \cos(3x - \frac{\pi}{3})$?

Tip: When identifying shifts in a function, always look for values added/subtracted inside and outside the trigonometric function. These indicate phase and vertical shifts, respectively.