Question

If the instructions for a problem ask you to use the smallest possible domain to completely graph two periods of y = 5 + 3 cos 2(x -pi/3), what should be used for Xmin and Xmax? Explain your answer.

Answer 1

The period of a sinusoid
`\cos 2(x-c)` is
`\frac{2\pi}2=\pi`, so any range such that
`\mathrm{Xmax}-\mathrm{Xmin}=2\pi` will give two complete periods.

Answer 2

The smallest possible domain to completely graph two periods is either [0, 2π] or [-π, π].

Explanation:

The period of cosine function [0, 2π].

The given function is

`y=5+3\cos 2(x-(\pi)/(3))`

This function can be written as

`y=5+3\cos (2x-(2\pi)/(3))` .... (1)

The general form of cosine function is

`y=A\cos (Bx+C)+D` .... (2)

where, A is amplitude,
`(2\pi)/(B)`, C is phase and D is midline.

From (1) and (2), we get

`A=3,B=2C=(2\pi)/(3),D=5`

`Period=(2\pi)/(2)=\pi`

The period of given function is [0,π]. So, the smallest possible domain to completely graph two periods is either [0, 2π] or [-π, π].