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what is the amplitude, period, vertical translation, and any phase shift

Let's break down the given function to determine its characteristics:

The function is:  
\[ y = 10 - \sin \left( 3x + \frac{\pi}{4} \right) \]

### 1. **Amplitude**: 
The amplitude of a sine or cosine function is the coefficient in front of the sine or cosine. In this case, the amplitude is the absolute value of the coefficient of \(\sin\).

Here, the sine function is \(\sin \left( 3x + \frac{\pi}{4} \right)\), and there is no explicit coefficient other than 1. Therefore, the **amplitude** is:
\[
\text{Amplitude} = 1
\]

### 2. **Period**: 
The period of a sine function is given by:
\[
\text{Period} = \frac{2\pi}{B}
\]
where \(B\) is the coefficient of \(x\) inside the sine function. In this case, \(B = 3\), so the period is:
\[
\text{Period} = \frac{2\pi}{3}
\]

### 3. **Vertical Translation**: 
The vertical translation is represented by any constant added or subtracted from the sine function. Here, there is a constant 10 added to the sine function, which means the graph is shifted **10 units upward**.

### 4. **Phase Shift**: 
The phase shift is determined by the term inside the sine function, specifically by \(\frac{\pi}{4}\). The formula for phase shift is:
\[
\text{Phase Shift} = -\frac{C}{B}
\]
where \(C\) is the constant added inside the function, and \(B\) is the coefficient of \(x\). In this case, \(C = \frac{\pi}{4}\) and \(B = 3\). Therefore, the phase shift is:
\[
\text{Phase Shift} = -\frac{\frac{\pi}{4}}{3} = -\frac{\pi}{12}
\]
This means the graph is shifted **\(\frac{\pi}{12}\) units to the left**.

### Summary:
- **Amplitude**: 1
- **Period**: \(\frac{2\pi}{3}\)
- **Vertical Translation**: 10 units up
- **Phase Shift**: \(\frac{\pi}{12}\) units to the left

Do you need more details on any of these aspects or have any questions? 

Here are 5 relative questions to expand your understanding:
1. How would the period change if the coefficient of \(x\) were different?
2. What effect does a negative amplitude have on the sine graph?
3. How would you determine the phase shift if the sign inside the sine function were negative?
4. How would you find the amplitude if the sine function had a coefficient other than 1?
5. What happens to the vertical translation if the constant outside the sine function is negative?

**Tip**: When calculating the phase shift, always be mindful of both the coefficient of \(x\) and the constant inside the parentheses!

Find the amplitude, the period, any vertical translation, and any phase shift of the function y = 10 - sin(3x + π/4).

Learn how to find the amplitude, period, vertical translation, and phase shift of the function y = 10 - sin(3x + π/4). This solution covers trigonometric concepts, including the formula for amplitude, period, and phase shift, and explains the translation of sine functions. Suitable for students in Grades 10-12.

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