Question

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

Answer 1

See below

Explanation:

When graphing any sinusoidal graph except for `tanx` and `cotx` the period is `(2pi)/w` where `w` is the value next to `x` in this case `2pi`.

So, our period is represented as:

Per. `\ T = (2pi)/w`

Per. `\ T = (2pi)/(2pi)`

Per. `\ 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 `[-1, 1]` our graph will range from `[-3, 3]`.

Once we have this information, the easiest way to graph for two periods is to go out four points on the graph's `x-"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, `1/2`, point, `1` (since `1/2` is half of `1`)

Then do it again:

Origin, `1/4`, `1/2`, point, `1`

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

Origin, `1/4`, `1/2`, `3/4`, `1`.

And since we know that positive `cos` graphs always start at `(0, "amplitude")`, we can now graph our equation, as seen below:

Hopefully, this helps you visualize it

For two full periods, just keep moving by `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.