# Graph of y equal sin bx(y = a sin bx)

The graph of y equal sin bx which is y = a sin bx where 'a' is the amplitude and 'b' affects the period of the graph. The period is the distance that it takes for the sine curve to begin repeating again.
As we know that the period for sine curve is 2π. So the relation between period (2π) and 'b' is given by
Period =$\frac{2π}{b}$
As the value of 'b' increases the period decreases.
For Example : 1) y = 2 sin 3x
Here the amplitude = a = 2
Period =$\frac{2π}{b}$
b= 3
⇒ Period =$\frac{2π}{3}$
So the sine curve cut the x-axis at (π/3,0), (2π/3),(π,0),...
The graph will look like 2) y = 3 sin 2x
Here the amplitude = a = 3
Period =$\frac{2π}{b}$
b= 2
⇒ Period =$\frac{2π}{2} = π$
So the sine curve cut the x-axis at (π/2,0), (π,0),(3π/2,0),...

3) From the given graph, find the equation. From the above graph, we can see that the graph is increasing curve so its sine curve.
So the equation will be y = a sin bx
Now here the amplitude = a = 2
To find the period,
one sine cycle is at π, so
Period =$\frac{2π}{b}$ = π
So b = $\frac{2π}{π}$
b = 2
So the equation of the above graph is y = 2 sin(2x).

3) Determine the equation for the graph given below: From the above graph, we can see that the graph is increasing curve so its sine curve.
So the equation will be y = a sin bx
Now here the amplitude = a = 1
To find the period,
Here we can not find the time period for 1 cycle.
But from the graph we can see that one and half cycle is π, so
Period =$\frac{3}{2} \times\frac{2π}{b}$ = π
So b = $\frac{3}{2} \times\frac{2π}{π}$
b = 3
So the equation of the above graph is y = sin(3x).

## Practice on graph of y equal sin bx

1) State the amplitude and period for y = 3 sin 2x, and graph the equation for the interval $-π\leq x \leq 2π$
2) Compare the amplitudes of y = 1/2 sin x and y = 2 sin x with the amplitude of y = sin x and sketch a graph of each on the same coordinate system for $0 \leq x \leq 2π$

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