# Sine Graph

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Sine Graph: Trigonometric function sinx is a periodic functions.
Sine function has maximum value as +1 and minimum as -1.

Since sin(-x) = -sinx , sinx is an odd function.
The value of the functions sin x repeats in every 2π units of x . Therefore this function is periodic with a period of Function whose graph is the shape of sine curve is called sinusoidal functions and such graphs are called Sine Graph.

Some definitions used in sine curve are
Amplitude: This is half the distance between the maximum and minimum values.
The term a in the general form of the sinusoidal function represents the amplitude of this function.
The amplitude is a constant for sinusoidal functions.
Period: This is the smallest time needed for a function to execute one complete cycle.
From the general form of sinusoidal function, the period T can be written as` T = 2π/ b
Now let us come back to the simplest sinusoidal function .
y =a sin bx
where a = amplitude .
Period = 2π / b
Sin (0) = 0 so sine graph always start from zero. As b gets larger, the period decreases.

Phase shift of sine function
y = a sin (bx + c ) + d
Both b and c affect the phase shift or movement of graph
 If c -----------> Positive -----------> the shift is towards right If c -----------> Negative -----------> shift is towards left. If d ----------->Positive ----------->graph shifted up by d units. If d ----------->Negative -----------> graph shifted down by d units.

Phase shift = - c / b
Period = 2π / b
The phase shift is the amount that the curve is moved in a horizontal direction from its normal position.
Phase shift, Negative -----------> Left Displacement
Phase shift, Positive -----------> Right Displacement
To find the phase shift , just make
bx + c = 0 and solve.
Example :
y = 2 sin (2x +1 )
Here amplitude = a = 2
2x + 1 = 0 ⇒ x = -1/2 so Phase shift = -1/2.
Period = 2π / b
Period = 2π / b = 2π / 2 = π Sine graph

Graph Dictionary