Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 4 cos(2x − π)

Verified step by step guidance

1. Identify the standard form of the cosine function: \( y = a \cos(bx - c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). Here, \( a = 4 \), so the amplitude is \( |4| = 4 \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). In this case, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
4. Calculate the phase shift using \( \frac{c}{b} \). Here, \( c = \pi \) and \( b = 2 \), so the phase shift is \( \frac{\pi}{2} \) to the right.
5. Graph one period of the function by starting at the phase shift \( \frac{\pi}{2} \), marking the amplitude as 4, and completing one cycle over the period \( \pi \).

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its central axis. In the context of trigonometric functions like cosine, it is determined by the coefficient in front of the cosine term. For the function y = 4 cos(2x − π), the amplitude is 4, indicating that the graph will oscillate between 4 and -4.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. It can be calculated using the formula 2π divided by the coefficient of x in the function. For y = 4 cos(2x − π), the period is 2π/2 = π, meaning the function will repeat its values every π units along the x-axis.

Phase Shift

Phase shift refers to the horizontal shift of the graph of a trigonometric function. It is determined by the constant added or subtracted from the x variable inside the function. In y = 4 cos(2x − π), the phase shift can be found by setting 2x - π = 0, leading to a shift of π/2 units to the right, which alters the starting point of the cosine wave.