Question

How do you graph `y = cos pi x`?

Answer 1

See explanation below

graph{cos(pi*x) [-5, 5, -2, 2]}

Explanation:

Graph of function `y=f(x)`, by definition, is a set of all points `(A,B)` on the coordinate plane that satisfy the equation
`B=f(A)`.

Given a graph of a function `y=f(x)`, the graph of `y=f(Kx)`, where `K!=0`, can be obtained by "squeezing" the original graph horizontally towards the Y-axis in `K` times.

Here is why.
Consider a point `(A,B)` belongs to original graph. It means that `B=f(A)`.
Consider now a point `(A/K,B)`. Obviously, it belongs to a graph of function `y=f(Kx)` since
`f(KA/K)=f(A)=B`

So, for each point `(A,B)` that belongs to a graph of function `y=f(x)`, point `(A/K,B)` belongs to a graph of function `y=f(Kx)`.
The point `(A/K,B)` can be obtained from the point `(A,B)` by horizontal "squeezing" towards Y-axis.

Of course, if `K<0`, the whole graph is symmetrically reflected relative to Y-axis. If `|K|<1`, our "squeezing" is, actually, stretching.

To construct a graph of function `y=cos(pi x)`, we have to start from `y=cos(x)` and "squeeze" is horizontally towards Y-axis by a factor `pi`.
That means, the shape is preserved, but the periodicity will be in `pi` times smaller, that is `(2pi)/pi=2`.