Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex], let's break it down step by step:
### Amplitude
The amplitude of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is given by the absolute value of the coefficient [tex]\( A \)[/tex] in front of the [tex]\( \sin \)[/tex] function.
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The coefficient [tex]\( A \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Therefore, the amplitude is:
[tex]\[ \text{Amplitude} = \left| \frac{1}{2} \right| = 0.5 \][/tex]
### Period
The period of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex] in front of [tex]\( x \)[/tex]. The period is calculated using the formula:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex] since there is no coefficient explicitly written in front of [tex]\( x \)[/tex], so it is assumed to be [tex]\( 1 \)[/tex].
Therefore, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{1} = 2\pi \approx 6.283185307179586 \][/tex]
### Phase Shift
The phase shift of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is determined by the constant [tex]\( C \)[/tex] inside the sine function and is calculated using the formula:
[tex]\[ \text{Phase Shift} = -\frac{C}{B} \][/tex]
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The constant [tex]\( C \)[/tex] is [tex]\( 6 \)[/tex]
- The coefficient [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex]
Therefore, the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{6}{1} = -6 \][/tex]
This indicates a shift to the left by 6 units.
### Conclusion
Based on the above calculations, the characteristics of the trigonometric equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex] are:
- Amplitude: 0.5
- Period: [tex]\( 2\pi \approx 6.283185307179586 \)[/tex]
- Phase Shift: shifted to the left by 6 units
So, the answers are:
Amplitude: [tex]\( 0.5 \)[/tex]
Phase Shift: shifted to the left
### Amplitude
The amplitude of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is given by the absolute value of the coefficient [tex]\( A \)[/tex] in front of the [tex]\( \sin \)[/tex] function.
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The coefficient [tex]\( A \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Therefore, the amplitude is:
[tex]\[ \text{Amplitude} = \left| \frac{1}{2} \right| = 0.5 \][/tex]
### Period
The period of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex] in front of [tex]\( x \)[/tex]. The period is calculated using the formula:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex] since there is no coefficient explicitly written in front of [tex]\( x \)[/tex], so it is assumed to be [tex]\( 1 \)[/tex].
Therefore, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{1} = 2\pi \approx 6.283185307179586 \][/tex]
### Phase Shift
The phase shift of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is determined by the constant [tex]\( C \)[/tex] inside the sine function and is calculated using the formula:
[tex]\[ \text{Phase Shift} = -\frac{C}{B} \][/tex]
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The constant [tex]\( C \)[/tex] is [tex]\( 6 \)[/tex]
- The coefficient [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex]
Therefore, the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{6}{1} = -6 \][/tex]
This indicates a shift to the left by 6 units.
### Conclusion
Based on the above calculations, the characteristics of the trigonometric equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex] are:
- Amplitude: 0.5
- Period: [tex]\( 2\pi \approx 6.283185307179586 \)[/tex]
- Phase Shift: shifted to the left by 6 units
So, the answers are:
Amplitude: [tex]\( 0.5 \)[/tex]
Phase Shift: shifted to the left
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.