Changing a Trigonometric GraphDate: 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/ |
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