Sure, let's break down the given equation [tex]\( 4y = 2\sin(5x - 3\pi) \)[/tex] to find the amplitude, period, and phase shift step-by-step.
### Step 1: Rewrite the equation in the standard form
First, we'll rewrite the given equation in the standard form of a sine function, [tex]\( y = a\sin(bx - c) \)[/tex].
Given:
[tex]\[ 4y = 2\sin(5x - 3\pi) \][/tex]
Divide both sides by 4 to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2}{4}\sin(5x - 3\pi) \][/tex]
[tex]\[ y = 0.5\sin(5x - 3\pi) \][/tex]
### Step 2: Identify the amplitude
The amplitude [tex]\( a \)[/tex] is the coefficient in front of the sine function.
From the equation [tex]\( y = 0.5\sin(5x - 3\pi) \)[/tex], we can see that the coefficient in front of the sine function is 0.5.
Thus, the amplitude is:
[tex]\[ \text{Amplitude} = 0.5 \][/tex]
### Step 3: Determine the period
The period of a sine function [tex]\( y = a \sin(bx - c) \)[/tex] is calculated using the formula:
[tex]\[ \text{Period} = \frac{2\pi}{b} \][/tex]
In our equation [tex]\( y = 0.5\sin(5x - 3\pi) \)[/tex], the coefficient of [tex]\( x \)[/tex] inside the sine function is 5.
So,
[tex]\[ b = 5 \][/tex]
[tex]\[ \text{Period} = \frac{2\pi}{5} \][/tex]
[tex]\[ \text{Period} \approx 1.2566 \][/tex] (rounded to four decimal places)
### Step 4: Identify the phase shift
The phase shift of a sine function [tex]\( y = a \sin(bx - c) \)[/tex] is calculated using the formula:
[tex]\[ \text{Phase Shift} = \frac{c}{b} \][/tex]
In our equation [tex]\( y = 0.5\sin(5x - 3\pi) \)[/tex], the constant subtracted from [tex]\( bx \)[/tex] inside the sine function is [tex]\( 3\pi \)[/tex].
So,
[tex]\[ c = 3\pi \][/tex]
[tex]\[ \text{Phase Shift} = \frac{3\pi}{5} \][/tex]
[tex]\[ \text{Phase Shift} \approx 1.885 \][/tex] (rounded to three decimal places)
And since [tex]\( 3\pi \)[/tex] is positive in the term [tex]\( 5x - 3\pi \)[/tex], the phase shift direction is to the right.
### Summary:
- Amplitude: [tex]\( 0.5 \)[/tex]
- Period: [tex]\( 1.2566 \)[/tex]
- Phase Shift: [tex]\( 1.885 \)[/tex], shifted to the right
