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 $\dfrac{2\pi}2=\pi$, so any range such that $\mathrm{Xmax}-\mathrm{Xmin}=2\pi$ will give two complete periods.

Answer 2

Answer:

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

Step-by-step explanation:

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

The given function is

$y=5+3\cos 2(x-\frac{\pi}{3})$

This function can be written as

$y=5+3\cos (2x-\frac{2\pi}{3})$          .... (1)

The general form of cosine function is

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

where, A is amplitude, $\frac{2\pi}{B}$, C is phase and D is midline.

From (1) and (2), we get

$A=3,B=2C=\frac{2\pi}{3},D=5$

$Period=\frac{2\pi}{2}=\pi$

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].