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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.

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/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Trigonometry

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