To analyze the function \( y = 3\sin(x - 30^\circ) \), we need to determine the amplitude, period, and phase shift.
1. Amplitude:
- The amplitude of a sine wave \( y = A\sin(Bx - C) \) is given by the absolute value of the coefficient \( A \) in front of the sine function.
- In the given function \( y = 3\sin(x - 30^\circ) \), the coefficient \( A \) is 3.
- Therefore, the amplitude is \( 3 \).
2. Period:
- The period of a sine wave \( y = A\sin(Bx - C) \) is determined by the coefficient \( B \) in front of \( x \).
- The period is calculated using the formula: \( \text{Period} = \frac{360^\circ}{B} \).
- In the given function, \( B \) is 1 (since there is no coefficient explicitly written in front of \( x \), it is understood to be 1).
- Therefore, the period is \( \frac{360^\circ}{1} = 360^\circ \).
3. Phase Shift:
- The phase shift of a sine wave \( y = A\sin(Bx - C) \) is given by the formula: \( \text{Phase Shift} = \frac{C}{B} \).
- In the given function, \( C \) is 30°.
- Since \( B \) is 1, the phase shift is \( \frac{30^\circ}{1} = 30^\circ \) to the right.
To sketch the graph of \( y = 3\sin(x - 30^\circ) \) within the domain \( 0 < x < 360^\circ \):
- Amplitude: The maximum value of \( y \) is 3, and the minimum value of \( y \) is -3.
- Period: One complete cycle occurs over \( 360^\circ \).
- Phase Shift: The graph is shifted 30° to the right.
### Key Points to Plot:
1. Start the sine function 30° to the right of the origin. This means that \( y = 0 \) at \( x = 30° \).
2. At \( x = 30° + 90° = 120° \), the sine function reaches its maximum amplitude, so \( y = 3 \).
3. At \( x = 30° + 180° = 210° \), the sine function goes back to zero, so \( y = 0 \).
4. At \( x = 30° + 270° = 300° \), the sine function reaches its minimum amplitude, so \( y = -3 \).
5. At \( x = 30° + 360° = 390° \), the sine function goes back to zero again, completing one full cycle. But since 390° exceeds our 0° to 360° range, end the cycle at 360° where the sine wave returns close to its zero value after the complete period.
Sketch:
```
y
↑ • •
| /|\ /|\
| / | \ / | \
| / | \ / | \
|/ | \/ | \
| | \ | \
| | / | /
| | / \ | /
| | / \ | /
| | / \| /
+--------------------------> x
30° 120° 210° 300° 360°
```
In summary:
- Amplitude: 3
- Period: 360°
- Phase Shift: 30° to the right.
Proceed to mark key points and sketch the shape accordingly, ensuring that the graph captures the periodic and oscillatory nature of the sine function within the specified domain.
