Answered

Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

Determine the amplitude, period, and phase shift of the following trigonometric equation:

[tex]\[ 4y = 2 \sin(5x - 3\pi) \][/tex]

Amplitude:
Period:
Phase Shift:

Sagot :

GinnyAnswer
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
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.