# What is the amplitude and period of y=3cos (1/2 x)?

Apr 29, 2015

Consider the basic form:
$y = \cos \left(\theta\right)$
which has an amplitude of $1$
($y$ has a range of $\left[- 1 , + 1\right]$)
and
a period of $2 \pi$
($\cos \left(0\right) \text{ to } \cos \left(2 \pi\right)$ forms one recurring cycle)

In $3 \cos \left(\frac{1}{2} x\right)$
the coefficient $3$ stretches the range of $y$ to $\left[- 3 , + 3\right]$
so its amplitude is $3$

The coefficient $\frac{1}{2} \text{ of } x$ forces $x$ to have to extend from
$0 \text{ to } 4 \pi$ to cover the set of argument values from $0 \text{ to } 2 \pi$
so the period of $y = 3 \cos \left(\frac{1}{2} x\right)$ is $4 \pi$