# 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):

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.

The Phase Shift is how far the function is horizontally to the right of the usual position.

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

## All Together Now!

We can have all of them in one equation:

y = A sin(Bx + C) + D

• amplitude is A
• period is 2π/B
• phase shift is −C/B
• vertical shift is D

### 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:

### Example: 2 sin(4x − 2) + 3

• amplitude A = 2
• period 2π/B = 2π/4 = π/2
• phase shift −C/B = −(−2)/4 = 1/2
• vertical shift D = 3

In words:

• the 2 tells us it will be 2 times taller than usual, so Amplitude = 2
• the usual period is 2π, but in our case that is "sped up" (made shorter) by the 4 in 4x, so Period = π/2
• and it will be shifted to the right because of the − 2 (positive goes left, negative goes right), but because it is also "sped up" by 4 then it is shifted by only 1/2, so Phase Shift = 1/2
• lastly the +3 tells us the center line is y = +3, so Vertical Shift = 3

Note the Phase Shift formula −C/B has a minus sign:

• A positive value of C pushes the curve in the negative direction (to the left)
• A negative value of C pushes the curve in the positive direction (to the right)

Sometimes we have t instead of x (or maybe other variables):

### Example: 3 sin(100t + 1)

• amplitude is A = 3
• period is 2π/100 = 0.02 π
• phase shift is −C/B = −1/100 = −0.01
• vertical shift is D = 0

And we get:

## Frequency

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

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

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(100t + 1)

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".