Question

what is the general equation of a sine function with an amplitude of 2, a period of π and a horizontal shift of π units?
○ y = 2 sin(2(x - π))
○ y = 2 sin(4(x + π))
○ y = sin(0.5(x - π))
○ y = sin(2(x + π))

Explanation:

Step1: Recall sine function form

The general form of a sine function is \( y = A\sin(B(x - C)) + D \), where:

• \( |A| \) is the amplitude,
• \( \frac{2\pi}{|B|} \) is the period,
• \( C \) is the horizontal shift (right if \( C>0 \), left if \( C<0 \)),
• \( D \) is the vertical shift (we assume \( D = 0 \) here as no vertical shift is mentioned).

Step2: Determine \( A \) (amplitude)

Given amplitude is 2, so \( A = 2 \).

Step3: Determine \( B \) (period factor)

Period is \( \pi \). Using the period formula \( \text{Period} = \frac{2\pi}{|B|} \), we solve for \( B \):

$$ \pi = \frac{2\pi}{|B|} $$

Multiply both sides by \( |B| \): \( \pi|B| = 2\pi \)
Divide both sides by \( \pi \): \( |B| = 2 \), so \( B = 2 \) (we can take positive \( B \) as it affects the period's length).

Step4: Determine \( C \) (horizontal shift)

Horizontal shift is \( \pi \) units. The horizontal shift is \( C \) in the formula \( y = A\sin(B(x - C)) \). So if the shift is \( \pi \) units (assuming right shift, so \( C=\pi \)), the argument becomes \( B(x - \pi) \).

Step5: Construct the equation

Substitute \( A = 2 \), \( B = 2 \), \( C = \pi \), \( D = 0 \) into the general form:

$$ y = 2\sin(2(x - \pi)) $$

Answer:

\( y = 2\sin(2(x - \pi)) \) (corresponding to the first option)