Answer :
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.
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.
