Changing a Trigonometric Graph

Date: 08/06/98 at 04:20:57
From: eric Wang
Subject: trig

How do you graph:

   -cos 2x
   -------
      2

Please show me via e-mail, if possible.

Date: 08/06/98 at 13:28:17
From: Doctor Rick
Subject: Re: trig

Hi, Eric. 

You know what the graph of cos(x) looks like, I'm sure:

    1 **                                         **
      |   *                                   *
      |     *                               *
      |       *                           *
      |        *                         *
      |         *                       *
    0 +----------*----------+----------*----------+--    cos(x)
      |0          *         pi        *          2pi
      |            *                 *
      |             *               *
      |               *           *
      |                 *       *
   -1 +                    ***                     

Cos(x) is a periodic function with period 2pi; that is, it comes back 
to the same value whenever we add another 2pi to y. (You can't tell an 
angle of y degrees from an angle of y + 360 degrees, or in radians, 
you can't tell y from y + 2pi.) So the graph repeats forever to the 
left and right.

What will cos(2x) look like? When x = 0, cos(2x) = cos(0) = 1. That's 
easy. So what is the period of cos(2x)? 

   2x = 2pi if x = pi

which means the period of cos(2x) is pi: when x = pi, cos(2x) comes 
back up to 1. That's all that changes, so the graph looks like this:

    1 **                                         **
      |   *                                   *
      |     *                               *
      |       *                           *
      |        *                         *
      |         *                       *
    0 +----------*----------+----------*----------+--    cos(2x)
      |0          *        pi/2       *           pi
      |            *                 *
      |             *               *
      |               *           *
      |                 *       *
   -1 +                    ***                     

Now let's multiply the whole thing by 1/2. That means we change the 
vertical size of the graph: -cos(2x)/2 doesn't go between 1 and -1 
any more, but between 1/2 and -1/2.

    1/2 **                                         **
        |   *                                   *
        |     *                               *
        |       *                           *
        |        *                         *
        |         *                       *
      0 +----------*----------+----------*----------+--    cos(2x)/2
        |0          *        pi/2       *           pi
        |            *                 *
        |             *               *
        |               *           *
        |                 *       *
    -1/2 +                   ***                     

Finally, multiply the function by -1. That means turn it upside down.

    1/2 +                    ***                     
        |                 *       *
        |               *           *
        |             *               *
        |            *                 *
        |           *                   *
      0 +----------*----------+----------*----------+--    -cos(2x)/2
        |0        *          pi/2         *         pi
        |        *                         *
        |       *                           *
        |     *                               *
        |   *                                   *
   -1/2 **                                         **

That's it! The basic rules are:

 - Multiplication inside the cosine, cos(kx), means divide the period 
   by k. If k is negative, flip the graph left to right.

 - Addition inside the cosine, cos(x+a), means shift the curve left by 
   a. If a is negative, it will shift right by (-a).

 - Multiplication outside the cosine, b*cos(x), means to multiply the 
   height of the curve (called the amplitude) by b. If it's negative, 
   turn the graph upside down.

 - Addition outside the cosine, d + cos(x), means shift the curve up 
   by d. If d is negative, it will shift down by (-d).

With a more complicated expression, just do one step at a time, working 
from the inside out, in the order I listed here, unless there are 
parentheses that change the order of operations.

Sine works the same way; just start with a sine instead of a cosine. 
As an exercise, you should be able to see graphically that these 
functions are the same:

   cos(x - pi/2) = sin(x)
   cos(-x) = cos(x)
   sin(-x) = -sin(x)

And try this: 

   cos^2(x) = (1+cos(2x))/2

It can be proved using the double-angle rule:

   cos(2x) = cos^2(x) - sin^2(x),

and it's good to know what cosine squared looks like. Try plotting 
cos(x) and cos^2(x) on the same graph.

- Doctor Rick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/