# 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 shifted **horizontally** from the usual position.

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

- amplitude is
**A** - period 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:

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:

### Example: 2 sin(4(x − 0.5)) + 3

- amplitude
**A = 2** - period
**2π/B**=**2π/4 = π/2** - phase shift
**= −0.5**(or**0.5**to the right) - 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 2
**π**, but in our case that is "sped up" (made shorter) by the**4**in 4x, so Period =**π/2** - and the
**−0.5**means it will be shifted to the**right**by**0.5** - lastly the
**+3**tells us the center line is y = +3, so Vertical Shift = 3

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:

- 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:

## Frequency

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

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

So the Frequency is 4

And the Period is \frac{1}{4}

In fact the Period and Frequency are related:

Frequency = \frac{1}{Period}

Period = \frac{1}{Frequency}

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

The period is 0.02**π**

So the Frequency is \frac{1}{0.02π} = \frac{50}{π}

Some more examples:

Period | Frequency |
---|---|

\frac{1}{10} | 10 |

\frac{1}{4} | 4 |

1 | 1 |

5 | \frac{1}{5} |

100 | \frac{1}{100} |

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

### Example: 50 Hertz means 50 times per second

The faster it bounces the more it "Hertz"!

## Animation