y=sin(7x) what is amplitude period and vertical shift

For the function \( y = \sin(7x) \), let's identify the amplitude, period, and vertical shift.

1. **Amplitude**: The amplitude is the maximum distance from the midline of the graph to its peak. In \( y = \sin(7x) \), the amplitude is given by the coefficient of \( \sin(7x) \), which is \( 1 \). So, the **amplitude is 1**.

2. **Period**: The period of a sine function \( y = \sin(bx) \) is calculated by \( \frac{2\pi}{|b|} \). Here, \( b = 7 \), so the period is:
   \[
   \frac{2\pi}{7}
   \]
   Thus, the **period is \( \frac{2\pi}{7} \)**.

3. **Vertical Shift**: The vertical shift is determined by any constant added outside the sine function. Since there is no constant added in \( y = \sin(7x) \), the **vertical shift is 0**.

### Summary
- **Amplitude**: 1
- **Period**: \( \frac{2\pi}{7} \)
- **Vertical Shift**: 0

Would you like more details or have any questions?

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1. How does the value of \( b \) in \( \sin(bx) \) affect the period of a sine function?
2. How would adding a constant to \( y = \sin(7x) \) change the vertical shift?
3. What is the effect of a negative amplitude on a sine function?
4. How does the frequency relate to the period in \( y = \sin(7x) \)?
5. What are the key differences between the graphs of \( y = \sin(7x) \) and \( y = \cos(7x) \)?

**Tip:** For transformations of trigonometric functions, use the general form \( y = a \sin(bx + c) + d \) to determine amplitude (\( |a| \)), period (\( \frac{2\pi}{|b|} \)), phase shift (\( -\frac{c}{b} \)), and vertical shift (\( d \)).

Discover how to determine the amplitude, period, and vertical shift of the trigonometric function y = sin(7x). This problem involves trigonometric transformations and is suitable for high school students, covering essential trigonometric concepts and formulas.

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