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 2π
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