**What is Sinusoid ?**

A function is a sinusoid if it can be written in the form

f (x) = a sin (bx+c)+d

where a, b, c, and d are constants and neither a nor b is 0.

**How translations are associated with Sinusoids ?**

There is a special vocabulary used to describe some of our usual graphical transformations when we apply them to sinusoids.

Horizontal stretches and shrinks affect the period and the frequency.

Vertical stretches and shrinks affect the amplitude.

Horizontal translations bring about phase shifts.

All of these terms are associated with waves, and waves are quite naturally associated with sinusoids.

The graphs of

y = a sin (b(x-h)) + k and

y = a cos (b(x-h)) + k

(where a ≠ 0 and b ≠ 0) have the following characteristics:

amplitude = |a|

period = 2π/|b|

frequency = |b|/2π

When compared to the graphs of y = a sin bx and y = a cos bx, respectively, they also have the following characteristics:

a phase shift of h

a vertical translation of k.

**What is amplitude ?**

Graphically, the amplitude is half the height of the wave.

**What is period ?**

Graphically, the period is the length of one full cycle of the wave.

**What is phase shift ?**

In mathematics, a horizontal shift of a trigonometric function may be referred to as a phase shift.

For example :

- (x - 2) where 2 is a positive value being subtracted the shift is to the right.
- (x + 2) where 2 is a negative value being added the shift is to the left.

**What is vertical translation ?**

- When we have +k, we have to move the graph k units up.
- When we have -k, we have to move the graph k units down.

**Example 1 :**

Graph the following function

y = -2 sin (x - π/4) + 1

**Solution :**

Amplitude = 2

Period = 2π/|b| ==> 2π/|1| ==> 2π

Frequency = 1/2π

Phase shift = π/4 (π/4 units to the right)

Vertical shift = 1 (Move one unit to up)

First let us draw the curve for y = 2sinx

To get the graph of y = 2 sin (x - π/4), we move the graph of y = 2 sinx, π/4 to the right.

y = 2 sin (x - π/4)

Since we have -2 as the coefficient of sin, we do reflection.

Graph of y = -2 sin (x - π/4)

At the end, we have 1. So move the graph one unit up.

Graph of y = -2 sin (x - π/4) + 1

**Example 2 :**

Graph the following function

y = 5 cos (3x - π/6) + 0.5

**Solution :**

Amplitude = 5

Period = 2π/|b| ==> 2π/|3| ==> 2π/3

Frequency = 3/2π

Phase shift = π/6 (π/6 units to the right)

Vertical shift = 0.5 (Move 0.5 unit to up)

To get the graph of y = 5 cos (3x - π/6), we should move the graph horizontally π/6 to the right.

To get the graph of y = 5 cos (3x - π/6) + 0.5 from the graph of y = 5 cos (3x - π/6), we should move the graph vertically 0.5 units up.

After having gone through the stuff given above, we hope that you have understood the concept of graphing a sinusoidial functions.

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