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

The function given is:

\[
y = \sin\left(x + \frac{\pi}{10}\right)
\]

We can now find the amplitude, period, vertical shift, and phase shift of the function:

1. **Amplitude**:  
   The amplitude of a sine function \( y = \sin(bx + c) \) is given by the coefficient of the sine function. Since there is no coefficient in front of the sine function here, the amplitude is **1**.

2. **Period**:  
   The period of a sine function is determined by the formula:

   \[
   \text{Period} = \frac{2\pi}{|b|}
   \]

   In this case, \( b = 1 \) (since the function is \( \sin(x + \frac{\pi}{10}) \)), so the period is:

   \[
   \text{Period} = \frac{2\pi}{1} = 2\pi
   \]

3. **Vertical Shift**:  
   The vertical shift occurs when there is an added constant outside the sine function. Since there is no constant added outside, the vertical shift is **0**.

4. **Phase Shift**:  
   The phase shift is determined by the horizontal translation of the sine wave and is calculated using:

   \[
   \text{Phase Shift} = \frac{-c}{b}
   \]

   Here, \( c = \frac{\pi}{10} \) and \( b = 1 \), so the phase shift is:

   \[
   \text{Phase Shift} = \frac{-\frac{\pi}{10}}{1} = -\frac{\pi}{10}
   \]

   Therefore, the graph is shifted **to the left** by \( \frac{\pi}{10} \).

### Summary:
- Amplitude: **1**
- Period: **\( 2\pi \)**
- Vertical Shift: **0**
- Phase Shift: **\( -\frac{\pi}{10} \)** (left by \( \frac{\pi}{10} \))

Would you like further details on any part of this explanation?

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Here are five related questions:
1. How does a coefficient in front of the sine function affect its amplitude?
2. How would the period change if the function was \( y = \sin(2x + \frac{\pi}{10}) \)?
3. What happens to the graph if we introduce a vertical shift?
4. How would you graph the function using the phase shift information?
5. What effect would multiplying the sine function by a negative value have on its graph?

**Tip**: To find the phase shift, always check the sign inside the parentheses with \(x\) for proper direction of shift.

What is the amplitude, period, vertical shift and phase shift for y = sin(x + π/10)?

Learn how to determine the amplitude, period, vertical shift, and phase shift for the trigonometric function y = sin(x + π/10). This guide covers key trigonometric concepts and provides step-by-step instructions on graphing and analyzing transformations of sine functions. Suitable for students in Grades 10-12.

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