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What is the general equation of a sine function with an amplitude of 2, a period of [tex]$\pi[tex]$[/tex], and a horizontal shift of [tex]$[/tex]\pi$[/tex] units?

A. [tex]y = 2 \sin(2(x - \pi))[/tex]
B. [tex]y = 2 \sin(4(x + \pi))[/tex]
C. [tex]y = \sin(0.5(x - \pi))[/tex]
D. [tex]y = \sin(2(x + \pi))[/tex]

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Answer :

GinnyAnswer
To determine the equation of a sine function with amplitude, period, and horizontal shift, let's break down the information given step-by-step.

### Given Parameters:
- Amplitude (A): 2
- Period (T): π
- Horizontal Shift (C): π units

### General Form of a Sine Function:
The general form of a sine function is:
[tex]\[ y = A \sin(B(x - C)) \][/tex]
Where:
- \( A \) is the amplitude.
- \( B \) is related to the period by the equation \( \text{Period} = \frac{2\pi}{B} \).
- \( C \) is the horizontal shift.

### Step-by-Step Solution:

1. Determine the Amplitude (A):
The amplitude is given as 2. So, \( A = 2 \).

2. Determine B Using the Period:
The period \( T \) is given as \( \pi \).
The period formula is:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
Given the period \( \pi \), we can set up the equation:
[tex]\[ \pi = \frac{2\pi}{B} \][/tex]
Solving for \( B \), we get:
[tex]\[ B = \frac{2\pi}{\pi} = 2 \][/tex]

3. Horizontal Shift (C):
The horizontal shift given is \( \pi \). Thus, \( C = \pi \).

4. Substitute these values into the general form:
Substitute \( A = 2 \), \( B = 2 \), and \( C = \pi \) into the general formula:
[tex]\[ y = 2 \sin(2(x - \pi)) \][/tex]

Thus, the correct equation is:
[tex]\[ y = 2 \sin(2(x - \pi)) \][/tex]

### Answer:
Among the given options, the correct equation is:
[tex]\[ y = 2 \sin(2(x - \pi)) \][/tex]

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