Amplitude, Period, Phase Shift and Frequency

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

The Menstruum goes from one top to the next (or from whatsoever point to the next matching point):

period and amplitude

The Amplitude is the top from the centre line to the peak (or to the trough). Or nosotros can measure out the meridian from highest to lowest points and divide that by 2.

phase shift

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

vertical shift

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

All Together Now!

We tin take all of them in one equation:

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

  • amplitude is A
  • catamenia is 2π/B
  • phase shift is C (positive is to the left)
  • vertical shift is D

And here is how it looks on a graph:

aa

Annotation that we are using radians here, non degrees, and there are 2π radians in a total rotation.

Example: sin(x)

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

So amplitude is i, flow is 2π , in that location is no phase shift or vertical shift:

amplitude 1, period 2pi, no shifts

Example: 2 sin(four(x − 0.5)) + 3

  • aamplitude A = 2
  • period 2π/B = 2π/four = π/ii
  • phase shift = −0.5 (or 0.5 to the right)
  • vertical shift D = 3

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

In words:

  • the 2 tells united states it will be 2 times taller than usual, so Aamplitude = 2
  • the usual period is 2 π , only in our example that is "sped up" (made shorter) by the four in 4x, then Menses = π/2
  • and the −0.5 means information technology will be shifted to the right past 0.5
  • lastly the +3 tells us the eye line is y = +iii, so Vertical Shift = iii

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

Example: three sin(100t + ane)

First we need brackets around the (t+ane), so nosotros can start past dividing the i past 100:

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

At present we can meet:

  • amplitude is A = 3
  • period is 2π/100 = 0.02 π
  • stage shift is C = 0.01 (to the left)
  • vertical shift is D = 0

And we get:

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

Frequency

Frequency is how oft something happens per unit of time (per "ane").

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

period 1/4, frequency 4

Then the Frequency is four

And the Period is ane four

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 0.02 π

So the Frequency is 1 0.02π = 50 π

Some more examples:

Period Frequency
one 10 10
1 4 4
1 1
5 i 5
100 1 100

When frequency is per 2d it is called "Hertz".

Example: 50 Hertz means 50 times per second

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

Animation

../algebra/images/wave-sine.js

7784,7785,7788,7789,9863,7793,7794,7795,7796,7792