Amplitude, Period, Phase Shift and Frequency

Some functions (like Sine and Cosine) repeat forever
and are called Periodic Functions.

The Period goes from one peak to the next (or from any point to the next matching point):

period and amplitude

The Amplitude is the height from the center line to the peak (or to the trough). Or we can measure the height from highest to lowest points and divide that by 2.


phase shift

The Phase Shift is how far the function is shifted horizontally from the usual position.

vertical shift

The Vertical Shift is how far the function is shifted vertically from the usual position.

All Together Now!

We can have all of them in one equation:

y = A sin(B(x + C)) + D

And here is how it looks on a graph:


Note that we are using radians here, not degrees, and there are 2π radians in a full rotation.

Example: sin(x)

This is the basic unchanged sine formula. A = 1, B = 1, C = 0 and D = 0

So amplitude is 1, period is 2π, there is no phase shift or vertical shift:

amplitude 1, period 2pi, no shifts

Example: 2 sin(4(x − 0.5)) + 3

amplitude 2, period pi/2, phase shift 0.5, vert shift 3

In words:

Instead of x we can have t (for time) or maybe other variables:

Example: 3 sin(100t + 1)

First we need brackets around the (t+1), so we can start by dividing the 1 by 100:

3 sin(100t + 1) = 3 sin(100(t + 0.01))

Now we can see:

And we get:

amplitude 3, period 0.02pi, phase shift -0.01, no vertical shift


Frequency is how often something happens per unit of time (per "1").

Example: Here the sine function repeats 4 times between 0 and 1:

period 1/4, frequency 4

So the Frequency is 4

And the Period is 1 4

In fact the Period and Frequency are related:

Frequency = 1 Period

Period = 1 Frequency

Example from before: 3 sin(100(t + 0.01))

amplitude 3, period 0.02pi, phase shift -0.01, no vertical shift

The period is 0.02π

So the Frequency is 1 0.02π = 50 π

Some more examples:

Period Frequency
1 10 10
1 4 4
1 1
5 1 5
100 1 100

When frequency is per second it is called "Hertz".

Example: 50 Hertz means 50 times per second

motocross bouncing
The faster it bounces the more it "Hertz"!