# How do you graph y=3cos2pix and include two full periods?

Jun 5, 2018

See below

#### Explanation:

When graphing any sinusoidal graph except for $\tan x$ and $\cot x$ the period is $\frac{2 \pi}{w}$ where $w$ is the value next to $x$ in this case $2 \pi$.

So, our period is represented as:

Per. $\setminus T = \frac{2 \pi}{w}$

Per. $\setminus T = \frac{2 \pi}{2 \pi}$

Per. $\setminus T = 1$

Now let's find our amplitude, which is always the number to the left of the trigonometric function, in this case, $3$. This means that instead of having a vertical range of $\left[- 1 , 1\right]$ our graph will range from $\left[- 3 , 3\right]$.

Once we have this information, the easiest way to graph for two periods is to go out four points on the graph's $x - \text{axis}$ and mark our period: $1$.

So it would look like this: Origin, point, point, point, 1

(I know it's difficult to visualize but bear with me until the end)

Then take half of the period and put it at half of that point's distance:

Origin, point, $\frac{1}{2}$, point, $1$ (since $\frac{1}{2}$ is half of $1$)

Then do it again:

Origin, $\frac{1}{4}$, $\frac{1}{2}$, point, $1$

Now that we know each increment is by one fourth, we can find the missing point

Origin, $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, $1$.

And since we know that positive $\cos$ graphs always start at $\left(0 , \text{amplitude}\right)$, we can now graph our equation, as seen below:

Hopefully, this helps you visualize it

For two full periods, just keep moving by $\frac{1}{4}$ on the $x$-axis and $3$ up or down on the $y$-axis following the up and down pattern, it's very consecutive and easy to follow.