Amplitude, Period, Phase Shift and Frequency

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

The Catamenia goes from one peak to the next (or from whatsoever 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 nosotros can measure out the top from highest to everyman points and carve up that by two.

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 At present!

We can have all of them in one equation:

y = A sin(B(10 + C)) + D

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

And hither is how it looks on a graph:

aa

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

Instance: sin(x)

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

So amplitude is 1, period is twoπ , there is no stage shift or vertical shift:

amplitude 1, period 2pi, no shifts

Example: 2 sin(iv(x − 0.five)) + 3

  • amplitude A = 2
  • period 2π/B = twoπ/four = π/2
  • stage shift = −0.five (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 us it will be two times taller than usual, and then Amplitude = two
  • the usual period is ii π , only in our case that is "sped upwardly" (made shorter) by the 4 in 4x, so Menstruation = π/2
  • and the −0.v ways information technology will be shifted to the right by 0.5
  • lastly the +3 tells united states of america the center line is y = +three, then Vertical Shift = iii

Instead of x nosotros tin take t (for fourth dimension) or maybe other variables:

Instance: 3 sin(100t + ane)

Start we need brackets effectually the (t+1), then we can beginning past dividing the 1 past 100:

3 sin(100t + ane) = three sin(100(t + 0.01))

Now we tin can see:

  • aamplitude is A = three
  • period is 2π/100 = 0.02 π
  • phase 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 often something happens per unit of time (per "one").

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

period 1/4, frequency 4

And so the Frequency is 4

And the Catamenia is i iv

In fact the Period and Frequency are related:

Frequency = 1 Period

Menses = ane Frequency

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

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

The menstruation is 0.02 π

So the Frequency is 1 0.02π = l π

Some more than examples:

Menses Frequency
1 10 10
1 4 4
1 1
v one 5
100 1 100

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

Case: 50 Hertz ways l times per second

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

Animation

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

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