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What is the amplitude, period, vertical shift and phase shift

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.

What is the amplitude, period, vertical shift, and phase shift of the function y = cos(x - π/2)?

Understand how to find the amplitude, period, vertical shift, and phase shift of the cosine function y = cos(x - π/2). This explanation breaks down each characteristic of the trigonometric function step by step. Ideal for high school students studying trigonometry.

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