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### Graphing Trig Functions

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

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