Graphing Trig Functions

Date: 05/03/97 at 14:45:08
From: Nem
Subject: Trigonometric graphing

Dear Dr. Math,

To graph the equation y = 3sin4x-3, I know I have to find the 
amplitude, period, and vertical or phase shift. I know the amplitude 
is 3, and the period is pi/4, but do I have to factor out the 4 from 
4x-3 before I can find the vertical shift, or is it simply 3 down?

Thank you.

Date: 05/03/97 at 21:55:34
From: Doctor Scott
Subject: Re: Trigonometric graphing

Hi Nem:

You are absolutely right that in order to graph the equation you need 
to determine the amplitude, period, vertical, AND phase (or 
horizontal) shift.  Remember that the general form of a sinusoidal 
function (a sine or cosine curve) is:

        y = a + b*sin[c(x-d)]

In this form: a is the VERTICAL SHIFT
              b is the AMPLITUDE
              d is the HORIZONTAL SHIFT  (or PHASE SHIFT)   
              c is the FREQUENCY (which determines the period)

Now, your example, y = 3sin4x - 3 means the same as y = -3 + 3sin(4x).  
(Unless the 4x-3 is in parentheses, which we will discuss below).  
This means that we know that: 

   a = -3  (Vertical shift of 3 down)

   b =  3  (Amplitude of 3)

   c =  4  (The period is 2pi/4 = pi/2)  

The period of a "normal" sine curve is 2pi, so here we will see 4 
sine curves in the interval [0, 2pi), and thus the period of this 
curve is 2pi/4 or pi/2.

d =  0  (No phase shift)

If, however, the original function is y = 3sin(4x - 3), we could 
rewrite this as y = 3sin[4 (x - 3/4)] by factoring the 4 out of the 
(4x-3) expression.  Then,

   a = 0  (No vertical shift)

   b = 3  (Amplitude of 3)

   c = 4  (The period is 2pi/4 or pi/2)

   d = 3/4  (A horizontal shift of 3/4 unit to the right.)

Good luck!

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