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.