Amplitude, Period, Phase Shift and Frequency

Some functions (similar Sine and Cosine) echo forever
and are called Periodic Functions.

The Period goes from 1 peak to the next (or from any point to the adjacent matching point):

period and amplitude

The Amplitude is the meridian from the middle line to the pinnacle (or to the trough). Or we can measure the elevation from highest to lowest points and carve up 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 office is shifted vertically from the usual position.

All Together Now!

We tin have all of them in i equation:

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

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

And here is how information technology looks on a graph:

aa

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

Example: sin(x)

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

And then amplitude is 1, flow is 2π , there is no phase shift or vertical shift:

amplitude 1, period 2pi, no shifts

Case: 2 sin(4(x − 0.five)) + 3

  • amplitude A = 2
  • period 2π/B = 2π/4 = π/2
  • phase shift = −0.five (or 0.five 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 exist 2 times taller than usual, so Amplitude = 2
  • the usual period is ii π , but in our case that is "sped upwards" (made shorter) by the iv in 4x, so Menses = π/2
  • and the −0.5 means information technology will be shifted to the right past 0.5
  • lastly the +iii tells united states the heart line is y = +3, and then Vertical Shift = 3

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

Case: 3 sin(100t + 1)

First we need brackets effectually the (t+1), so we tin can showtime by dividing the 1 by 100:

3 sin(100t + i) = iii sin(100(t + 0.01))

At present nosotros can see:

  • amplitude is A = 3
  • 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 fourth dimension (per "1").

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

period 1/4, frequency 4

So the Frequency is 4

And the Menstruum is one four

In fact the Menstruum and Frequency are related:

Frequency = 1 Period

Period = i Frequency

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

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

The flow is 0.02 π

So the Frequency is one 0.02π = 50 π

Some more examples:

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

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

Instance: 50 Hertz ways 50 times per second

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

Animation

../algebra/images/moving ridge-sine.js

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