Question

How do you graph `y=1/3 cosx`?

Answer 1

You are modifing a function by scalar multiplication, i.e. you're going from `f(x)` to `lambda f(x)`, being `lambda` some real number.

This kind of changing only affects the amplitude of the function. In fact, if `cos(x)` ranges from `-1` to `1`, then `1/3cos(x)` will range from `-1/3` to `1/3`.

You can see that all the rest remains untouched, if `f(x_0)=0` for some `x_0`, then also `lambda f(x_0)` will be zero.

Moreover, also the derivatives are in the same relation, since `(lambda f(x))' = lambda f'(x)`, so `lambdaf(x)` is growing if and only if `f(x)` was growing too, and vice versa.

Of course, all I wrote applies when `lambda` is positive, otherwise you can switch `lambda` to `-lambda` and repeat everything about `-f(x)`.