Question

find an equation for a cosine function having an amplitude of 2, period of \\(\pi\\), horizontal shift of \\(2\pi\\) to the right, and a vertical shift of -3.\\(y = ? \cos(\quadx + \quad\pi) + \quad\\)

Explanation:

Step1: Recall Cosine Function Form

The general form of a cosine function is \( y = A\cos(B(x - C)) + D \), 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.

Step2: Determine Amplitude (\( A \))

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

Step3: Determine \( B \) from Period

Period is \( \pi \). Using \( \frac{2\pi}{|B|} = \pi \), solve for \( B \):
\( |B| = \frac{2\pi}{\pi} = 2 \), so \( B = 2 \) (we can take positive \( B \) for simplicity).

Step4: Determine Horizontal Shift (\( C \))

Horizontal shift is \( 2\pi \) to the right, so \( C = 2\pi \). The function \( B(x - C) \) becomes \( 2(x - 2\pi) = 2x - 4\pi \), which can be written as \( 2x + (-4)\pi \) (to match the given form \( Bx + [\ ]\pi \)).

Step5: Determine Vertical Shift (\( D \))

Vertical shift is -3, so \( D = -3 \).

Step6: Substitute into Given Form

The given form is \( y = A\cos(Bx + [\ ]\pi) + D \). From \( 2(x - 2\pi) = 2x - 4\pi \), we have \( B = 2 \), the coefficient of \( \pi \) is -4, \( A = 2 \), \( D = -3 \).

Answer:

\( y = \boldsymbol{2}\cos(\boldsymbol{2}x + \boldsymbol{(-4)}\pi) + \boldsymbol{(-3)} \) (or filling the boxes: 2, 2, -4, -3)