Mathematics

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.

2 Answer

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

    [tex]y=5+3\cos 2(x-\frac{\pi}{3})[/tex]

    This function can be written as

    [tex]y=5+3\cos (2x-\frac{2\pi}{3})[/tex]          .... (1)

    The general form of cosine function is

    [tex]y=A\cos (Bx+C)+D[/tex]              .... (2)

    where, A is amplitude, [tex]\frac{2\pi}{B}[/tex], C is phase and D is midline.

    From (1) and (2), we get

    [tex]A=3,B=2C=\frac{2\pi}{3},D=5[/tex]

    [tex]Period=\frac{2\pi}{2}=\pi[/tex]

    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].

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