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

Jul 3, 2015

Amplitude$= 4$
Period$= 4 \pi$
Vertical Translation$= - 2$

#### Explanation:

I would write your function isolating $y$ as:
$y = 4 \cos \left(\frac{x}{2}\right) - 2$
From this you can "see":
1] The amplitude of your cosine is $4$ (the number in front of $\cos$);
2] The period is $4 \pi$; by using the $\frac{1}{2}$ inside the argument of $\cos$ you get: $p e r i o d = \frac{2 \pi}{\textcolor{red}{\frac{1}{2}}} = 4 \pi$;
3] The vertical translation is given by $- 2$ telling you that your $\cos$ oscillates about the horizontal line passing through $y = - 2$ (instead of oscillating about the $x$ axis!).

Graphically:
graph{4cos(x/2)-2 [-16.02, 16.02, -8.01, 8.01]}

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.