Find the phase shift, amplitude and period of the function. y=1\2sin(2x-π/3)-3. Give exact values, not decimal approximations

To find the phase shift, amplitude, and period of the function \( y = \frac{1}{2} \sin(2x - \frac{\pi}{3}) - 3 \), we can analyze the standard form of the sine function, which is:

\[ y = a \sin(bx - c) + d \]

Here, each parameter has a specific interpretation:

- \( a \) is the amplitude.
- \( \frac{2\pi}{b} \) is the period.
- \( \frac{c}{b} \) is the phase shift.
- \( d \) is the vertical shift.

For the given function \( y = \frac{1}{2} \sin(2x - \frac{\pi}{3}) - 3 \):

1. **Amplitude**: The amplitude is the absolute value of the coefficient of the sine function, which is \( \left|\frac{1}{2}\right| = \frac{1}{2} \).

2. **Period**: The period of the sine function is given by:

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

   Here, \( b = 2 \), so:

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

3. **Phase Shift**: The phase shift is calculated by solving:

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

   In this case, \( c = \frac{\pi}{3} \) and \( b = 2 \), so:

   \[
   \text{Phase Shift} = \frac{\frac{\pi}{3}}{2} = \frac{\pi}{6}
   \]

   The phase shift is to the right because the expression inside the sine function is of the form \( (bx - c) \).

4. **Vertical Shift**: The vertical shift is \( d = -3 \). While not explicitly requested, it indicates the graph is shifted downward by 3 units.

In summary:
- **Amplitude**: \( \frac{1}{2} \)
- **Period**: \( \pi \)
- **Phase Shift**: \( \frac{\pi}{6} \) (to the right)

If you have further questions or need more details, feel free to ask!

Here are some related questions for further exploration:
1. How would you graph this sine function step by step?
2. What are the maximum and minimum values of this function?
3. How does the vertical shift affect the graph of the sine function?
4. How can we find the x-intercepts of this function?
5. How does a negative amplitude affect the graph of the sine function?
6. Can you explain the effect of different phase shifts on the sine graph?
7. How would a cosine function differ with the same parameters?
8. How do you determine the range of this function?

**Tip:** When identifying properties of trigonometric functions, it's helpful to rewrite them in their standard forms to easily identify amplitude, period, and phase shifts.

Learn how to find the phase shift, amplitude, and period of the function y = (1/2)sin(2x - π/3) - 3. Understand key concepts in trigonometry such as amplitude, period, and phase shift. Suitable for high school students studying trigonometric functions.

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