# Amplitude, Period, Phase Shift, and Vertical Shift of Trigonometric Graphs

Some functions are like sine and cosine, which get repeated forever, and these are known as periodic functions.

## Period

The period is the amount of time it takes for the function to complete one cycle. The period goes from one point of a peak to another or says from one point to another matching point.

## Amplitude

The amplitude is the height from the center line to the point of peak or trough. We can find out the height from the highest points to the lowest points and divide it by 2.

## Phase Shift

This is how far the function is shifted horizontally (left and right) from the usual place.

## Vertical shift

The vertical shift is how far the function is shifted vertically (up and down) from the usual place.

A trigonometric function is written in the following form:

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

• The Amplitude is written as A.
• The Period is 2π/B.
• The phase shift is C. If it is + C, it shifts left. If the phase shift is – C, the function shifts right.
• The vertical shift is written as D. If it is + D, the functions move up. If it is – D, then the function moves down.

Notice that we use radians and not degrees and there are 2π radians in a full rotation.

Example 1 – Sin X

This is the basic changed formula of sine. A = 1, B = 1, C = 0 and D = 0. Thus, the amplitude is 1, its period is 2π, there is no phase shift or vertical shift.

Example 2 – 2 sin (4(x – 0.5)) + 3

Its amplitude A is 2.
Its period is 2π/B = 2π/4 = π/2
Its phase shift is -0.5 or 0.5 to its right
Its vertical shift D is 3

In other words, the number 2 tells us that it will be 2 times taller than usual, thus the amplitude is 2.

The usual period is 2π, but in our situation, which is sped up, makes it shorter by 4, thus period is π/2.

Example 3 – find out the 3 sin (100t + 1)

Firstly, there must be brackets around the t + 1, but we have to divide 1 by 100 –

3 sin (100t + 1) = 3 sin (100 (t + 0.01))

Now we can see that –

The amplitude of A is 3, its period is 2π/100 = 0.02 and the phase shift is C = 0.01 which is on the left and vertical shift is D = 0

## Frequency

It is the number of times something happens per unit of time.

Example – there is the sine function which is repeated 4 times between 0 and 1 –

Thus, the frequency is 4. The frequency and period are related to each other –
Frequency = 1 / period
Period = 1 / frequency

Example – 3 sin (100 (t + 0.01))

Here the period is 0.02π, thus the frequency will be 1 / 0.02π = 50π.

Thus, now you are clear with all the terms described and explained above, with examples.