Question

determine the amplitude, period, and phase shift of the function. then graph the function.
y = 3 cos (πx + 2π)

the amplitude is 3.
the period is 2.
(type an exact answer using π as needed. use integers or fractions for any numbers in the expression.)
the phase shift is -2
(type an exact answer using π as needed. use integers or fractions for any numbers in the expression.)
use the graphing tool to graph the function.
click to enlarge graph
(for any answer boxes shown with the grapher, type an exact answer. type the word pi to insert the symbol π as needed.)

Explanation:

Step1: Recall the general form of a cosine function

The general form of a cosine function is \( y = A\cos(Bx - C) + D \), where:

• Amplitude is \( |A| \)
• Period is \( \frac{2\pi}{|B|} \)
• Phase shift is \( \frac{C}{B} \)

Step2: Rewrite the given function in the general form

Given \( y = 3\cos(\pi x + 2\pi) \), we can rewrite it as \( y = 3\cos(\pi x - (-2\pi)) \). So, \( A = 3 \), \( B = \pi \), \( C = -2\pi \), and \( D = 0 \).

Step3: Calculate the amplitude

Amplitude is \( |A| \). Since \( A = 3 \), the amplitude is \( |3| = 3 \).

Step4: Calculate the period

Period is \( \frac{2\pi}{|B|} \). Here, \( B = \pi \), so the period is \( \frac{2\pi}{|\pi|} = \frac{2\pi}{\pi} = 2 \).

Step5: Calculate the phase shift

Phase shift is \( \frac{C}{B} \). Substituting \( C = -2\pi \) and \( B = \pi \), we get \( \frac{-2\pi}{\pi} = -2 \).

Answer:

Amplitude: \( 3 \), Period: \( 2 \), Phase Shift: \( -2 \)