y=sin(4x) has a range between -1 and 1 inclusive.
The solution to sin(4x)=1 tells us the period. 4x=π/2, 5π/2, 9π/2, ..., so x=π/8, 5π/8, 9π/8,...
The period is π/2. Other points of interest lie midway between these values of x: 3π/8, 7π/8, 11π/8,... where sin(4x)=-1. This gives us the local maxima and minima, which are on either side of the y-axis. Since sin(4x)=-sin(-4x), we can see that values of y for negative values of x are the negative of the corresponding positive values of x.
The zeroes are when 4x=0, π, 2π, 3π, etc., so x=0, π/4, π/2, 3π/4,...