An example of graphing:f(x) = cos3x from -pi to pi

Here is how I would go about graphing the function f(x) = cos3x from -pi to pi, using our grid with marks at multiples of pi/12. Make sure your calculator is in radians mode before you begin.

cos(3×0) = 1

cos(3×pi/12) = sqrt(2)/2 ~= .7071
cos(3×2pi/12) =0
cos(3×3pi/12) = -sqrt(2)/2 ~= -.7071
cos(3×4pi/12) = -1
cos(3×5pi/12) = -sqrt(2)/2 ~= -.7071
cos(3×6pi/12) =0
cos(3×7pi/12) = sqrt(2)/2 ~= .7071
cos(3×8pi/12) = 1
cos(3×9pi/12) = sqrt(2)/2 ~= .7071
cos(3×10pi/12) =0
cos(3×11pi/12) = -sqrt(2)/2 ~= -.7071
cos(3×12pi/12) = -1

Because cosine is an even function, the negative side of the graph is a mirror image of the positive side and I can use this information to make make work easier.

cos(3×-pi/12) = sqrt(2)/2 ~= .7071
cos(3×-2pi/12) =0
cos(3×-3pi/12) = -sqrt(2)/2 ~= -.7071
cos(3×-4pi/12) = -1
cos(3×-5pi/12) = -sqrt(2)/2 ~= -.7071
cos(3×-6pi/12) =0
cos(3×-7pi/12) = sqrt(2)/2 ~= .7071
cos(3×-8pi/12) = 1
cos(3×-9pi/12) = sqrt(2)/2 ~= .7071
cos(3×-10pi/12) =0
cos(3×-11pi/12) = -sqrt(2)/2 ~= -.7071
cos(3×-12pi/12) = -1

From those points, I know the graph of a sine or cosine function should be smooth instead of sharp, so I draw nice round curves going up to 1 and down to -1. Then, I draw the graph between sqrt(2)/2 and -sqrt(2)/2, making sure to pass through 0. As with everything I draw, it's never perfect, but it will give you a basic idea of what I expect on the homework and also on the next quiz. (Really big hint.)

Notice that the graph starts at its lowest point when x = -pi and repeats exactly 3 times. Whenever a graph is of the form f(x) = coskx or g(x) = sinkx, the distance from left to right in which the function repeats changes from 2pi to 2pi/|k|. If |k| > 1, this means the function repeats more quickly. If |k| < 1, this means the function repeat more slowly.