# How do you determine the amplitude, period and vertical translation of y+2=4cos(x/2)

Ethen Frey 2022-07-23 Answered
How do you determine the amplitude, period and vertical translation of $y+2=4\mathrm{cos}\left(\frac{x}{2}\right)$?
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## Answers (1)

Cheyanne Charles
Answered 2022-07-24 Author has 13 answers
I would write your function isolating y as:
$y=4\mathrm{cos}\left(\frac{x}{2}\right)-2$
From this you can "see":
1] The amplitude of your cosine is 4 (the number in front of $\mathrm{cos}$);
2] The period is $4\pi$ ; by using the $\frac{1}{2}$ inside the argument of $\mathrm{cos}$ you get:
$period=\frac{2\pi }{\frac{1}{2}}=4\pi$
3] The vertical translation is given by -2 telling you that your $\mathrm{cos}$ oscillates about the horizontal line passing through y=-2 (instead of oscillating about the x axis).
Graphically:
graph{$4\mathrm{cos}\left(x/2\right)-2\left[-16.02,16.02,-8.01,8.01\right]$}
As you can see you have a cosine oscillating between 2 and -6 about the horizontal line at y=-2 and one complete oscillation now takes $4\pi$ to be completed.
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