Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Join our knowledgeable community and get detailed, reliable answers to all your questions.
Sagot :
To determine the amplitude, period, and phase shift of the trigonometric equation:
[tex]\[ y = 8 - \frac{1}{2} \sin \left(\pi x + \frac{\pi}{4}\right) \][/tex]
we proceed as follows:
1. Amplitude:
The amplitude of a sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\(|A|\)[/tex]. In this equation, the coefficient of the sine function (without the negative sign and divided by the constant) is [tex]\(-\frac{1}{2}\)[/tex].
Thus, the amplitude is:
[tex]\[ \text{Amplitude} = \left| -\frac{1}{2} \right| = \frac{1}{2} = 0.5 \][/tex]
2. Period:
The period of the sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\( \frac{2\pi}{|B|} \)[/tex].
Here, the coefficient [tex]\( B \)[/tex] of [tex]\( x \)[/tex] inside the sine function is [tex]\(\pi\)[/tex].
Thus, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{\pi} = 2 \][/tex]
3. Phase Shift:
The phase shift of the sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is determined by the formula [tex]\( -\frac{C}{B} \)[/tex].
Here, [tex]\( C \)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\( B \)[/tex] is [tex]\(\pi\)[/tex].
Thus, the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{\frac{\pi}{4}}{\pi} = -\frac{1}{4} = -0.25 \][/tex]
Since the phase shift is negative, it means the graph of the sine function is shifted to the right by 0.25 units.
So, we summarize our answers:
- Amplitude: [tex]\( 0.5 \)[/tex]
- Phase Shift: shifted to the right
[tex]\[ y = 8 - \frac{1}{2} \sin \left(\pi x + \frac{\pi}{4}\right) \][/tex]
we proceed as follows:
1. Amplitude:
The amplitude of a sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\(|A|\)[/tex]. In this equation, the coefficient of the sine function (without the negative sign and divided by the constant) is [tex]\(-\frac{1}{2}\)[/tex].
Thus, the amplitude is:
[tex]\[ \text{Amplitude} = \left| -\frac{1}{2} \right| = \frac{1}{2} = 0.5 \][/tex]
2. Period:
The period of the sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\( \frac{2\pi}{|B|} \)[/tex].
Here, the coefficient [tex]\( B \)[/tex] of [tex]\( x \)[/tex] inside the sine function is [tex]\(\pi\)[/tex].
Thus, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{\pi} = 2 \][/tex]
3. Phase Shift:
The phase shift of the sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is determined by the formula [tex]\( -\frac{C}{B} \)[/tex].
Here, [tex]\( C \)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\( B \)[/tex] is [tex]\(\pi\)[/tex].
Thus, the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{\frac{\pi}{4}}{\pi} = -\frac{1}{4} = -0.25 \][/tex]
Since the phase shift is negative, it means the graph of the sine function is shifted to the right by 0.25 units.
So, we summarize our answers:
- Amplitude: [tex]\( 0.5 \)[/tex]
- Phase Shift: shifted to the right
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.