Investigation of the graph y = a sin(bx+c)

By Na Young Kwon

5. Examine graphs of  y = a sin(bx + c)  for different values of a, b, and c.

Fix a and b (a=1,b=1) and consider the graph for different values of c=-2,-1,0,1,2.

A graph of y=sin(x+c) c=-2,-1,0,1,2 shows us similar shapes. They have the same periods

and a maximum and minimum value. They have same cycle per regular interval.

The regular interval is  2¹ and we call it a period. A maximum value of these graphs is 1

and a minimum value is Ð1. The graph of y=sinx  is the one that the graph of y=sin(x+1)

moves to x-axis for 1. Generally, graphs of y=sin(x+c) have a period 2¹, the maximum

value 1 and minimum value Ð1. If a graph of y=sin(x+c) moves to x-axis for c, it is the

graph of y=sinx.

Next, fix b and c (b=1, c=1) and consider the graph for different values of a (a=-2,-1,0,1,2).

A graph of y=a sin(x+1) (a=-2,-1,0,1,2) shows that the graphs have the same points in x-axis.

All graphs of y=a sin(x+1) meet the same points in x-axis and the point values are

-(2¹-1), -¹-1,-1, ¹-1, 2¹-1. These graphs also have a period 2¹. The different thing is

a maximum and minimum value. A maximum value of y=sin(x+1) and y=-sin(x+1) are

the same 1 and a maximum of y=2sin(x+1) and y=-2sin(x+1) are the same 2. A minimum

value of y=sin(x+1) and y=-sin(x+1) are the same  Ð1 and a minimum of y=2sin(x+1)

and y=Ð2sin(x+1) are the same Ð2. Generally, graphs of y=a sin(x+1) have a period 2¹,

the maximum value is a and the minimum value is Ða. If a graph of y=a sin(x+1)

moves to x-axis for 1, the graph corresponds to y=a sinx.

Next, fix a and c(a=1,c=1) and investigate the graph for different values of b(b=-3,-2,0,1,2).

A graphs of y=sin(bx+1) have different shapes. Only graphs of y=sin(x+1) and sin(-x+1)

have the same shapes and graphs of y=sin(2x+1) and y=sin(-2x+1) have the same shapes.

However, all graphs have a maximum value 1 and a minimum value Ð1.

Consider a period of graphs. Graphs of y=sin(x+1) and y=sin(-x+1) have the same period

2¹. Graphs of y=sin(2x+1) and y=sin(-2x+1) have the same period  ¹.

Generally, graphs of y=sin(ax+1) and y=sin(-ax+1) (a>0)have a same period  2¹/a.

All graphs of y=sin(ax+1) have a maximum value 1 and a minimum value Ð1.

Also, if a graph of y=sin(ax+1) (a>0) moves to x-axis for 1/a,  the graph corresponds

to a graph of y=sin(ax).

Finally, letÕs investigate a graph of y=a sin(bx+c).

A graph of y=sin x has a period  2¹, a maximum  value 1 and a minimum value Ð1.

In the case of y=a sin(bx+c), the graph has a period  2¹/b, a maximum value |a| and

a minimum value -|a|. In addition, this graph is the one that a graph of y=a sin(bx)

moves to x-axis for Ð(c/b).

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