# How do you find the amplitude, period and graph y=3cos(1/2theta)?

Mar 30, 2017

The amplitude is $3$, the period is $4 \pi$, and the graph is shown below.

#### Explanation:

This is a graph in the form $y = a \cos \left(b \left(\theta - h\right)\right) + k$ where $a = 3$, $b = \frac{1}{2}$, and $h = k = 0$

For a graph in this form, the amplitude is $| a |$ and the period is $\frac{2 \pi}{|} b |$. In this case, the amplitude is $| 3 | = 3$ and the period is $\frac{2 \pi}{|} \frac{1}{2} | = 4 \pi$.

To graph it, it is helpful to find certain easy-to-find points. Because $h = k = 0$, there is no horizontal or vertical shift, so we can start with a regular cosine curve.
graph{cos(x) [-10, 10, -5, 5]}

Looking at the graph, the period is $2 \pi$ and the graph is maximum at the start of the period, minimum halfway through the period, and $0$ one quarter and three quarters through. Keeping these rules in mind but expanding the period to the one we want, $4 \pi$, we get:
graph{cos(.5x) [-10, 10, -5, 5]}

Currently the amplitude of the graph is $1$, but we need it to be 3 so we expand the maximum and minimum from $1$ and $\text{-} 1$ to $3$ and $\text{-} 3$:
graph{3cos(.5x) [-10, 10, -5, 5]}

This process is equivalent to performing the appropriate transformations to the parent cosine graph: a vertical stretch by 3 and a horizontal stretch by 2.