Posted by Math on Friday, September 30, 2011 at 11:17pm.
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.