Question

I need help finding the key points for the graph y=1/3 cos x and y=-sin 2piex/3

I just don't understand how to graph this and find keypoints.

Answer 1

y = 1/3 cos x.
You know that cos x = 0 when x = pi/2, 3pi/2, etc. All odd multiples of pi/2

It is 1 for all even multiples of pi
It is -1 for all odd multiples of pi.

But you have y = 1/3 cos x, so it is 1/3 or -1/3 at multiples of pi

Easy, right?

Now, for -sin 2pi/3 x, things do get a bit trickier.

You know that for sin x, graph crosses x-axis at all multiples of pi
y=1 for all 4k+1 * pi/2
y=-1 for all 4k-1 * pi/2

y = -sin x will look the same, but the max/min will be reversed.

But, we have -sin 2pi/3 x

So, graph will cross x-axis whenever 2pi/3 x is a multiple of pi
That is, 2pi/3 x = k*pi, so x = k*pi * 3/2pi = 3k/2 for all integer values of k
y = -1 when 2pi/3 x = (4k+1)pi
x = 4k+1 pi * 3/2pi = 3/2(4k+1) = 6k + 3/2
y = 1 when 2pi/3 x = 4k-1 pi, so x = 6k - 3/2

To do this kind of scaling/translating, it's usually easy in the beginning to work with what you know (sin x), and then step by step transform from the new variable.