To determine the amplitude, period, and phase shift of the trigonometric equation:
[tex]\[ y^{\prime}=4-\frac{4}{3} \sin (8 x-6) \][/tex]
we follow a series of steps to analyze the given function:
### 1. Amplitude:
The amplitude of a sine function, [tex]\( y = A \sin(Bx + C) \)[/tex], is determined by the coefficient in front of the sine function. For the given equation, the sine term is [tex]\(-\frac{4}{3} \sin(8x - 6)\)[/tex].
The amplitude is the absolute value of the coefficient of the sine function.
[tex]\[ \text{Amplitude} = \left| -\frac{4}{3} \right| = \frac{4}{3} \][/tex]
### 2. Period:
The period of a sine function, [tex]\( y = A \sin(Bx + C) \)[/tex], is computed by the formula:
[tex]\[ \text{Period} = \frac{2\pi}{|B|} \][/tex]
Here, [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the sine function. In the given equation, [tex]\( B = 8 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{8} = \frac{\pi}{4} \][/tex]
### 3. Phase Shift:
The phase shift of a sine function, [tex]\( y = A \sin(Bx + C) \)[/tex], occurs due to the term [tex]\( C \)[/tex]. The formula for the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{C}{B} \][/tex]
In the equation, the term [tex]\( 8x - 6 \)[/tex] can be compared to [tex]\( Bx + C \)[/tex], where [tex]\( C = -6 \)[/tex] and [tex]\( B = 8 \)[/tex].
[tex]\[ \text{Phase Shift} = -\frac{-6}{8} = \frac{6}{8} = \frac{3}{4} \][/tex]
Since the phase shift is positive, it means the graph is shifted to the right.
### Final Answer:
Combining all the information, we have:
- Amplitude: [tex]\( \frac{4}{3} \)[/tex]
- Period: [tex]\( \frac{\pi}{4} \)[/tex]
- Phase Shift: [tex]\( \frac{3}{4} \)[/tex] (shifted to the right)
