Sketch the graph y = 2sin(4x)

We know that y=sinx is the sinusoidal graph that starts at the origin, with maxima at x = pi/2 with a value of 1, and a minima at x = 3pi/2 with a value of -1.

The 4x inside the bracket means that all x values are magnified by a factor of 4, and hence these previous maxima/minima occur at values a quarter of what they currently are:

pi/8, 3pi/8.

Alongside this, as the sin function is sinusoidal and recurrent in 2pi, there are further maxima/minima at values multiples of 2pi/8 greater than these first two values.

Finally, the 2 outside the function gives us a scaling factor of two to all of the y values - so the maxima and minima are at 2 and -2 respectively.