S3A3: Trigonometric function Y = A sin(Bx + C) + D

Trigonometric Function Y = A Sin (Bx + C) + D

A trigonometric function can be of the form:

   Y = A Sin (Bx + C) + D or Y = A Cos (Bx + C) + D 

   (A, B, C and D are numbers)

A : Amplitude => the height of the graph , A is always positive => |A| modulus 

B : Period => Number of Cycles within 360^o  or 2π

C : Phase Shift : C/B => Moving the graph Left/Right along the x-axis

D : Vertical Shift => Moving the graph Up/Down along the y-axis

A : Amplitude=> Height of the graph

Example

Draw the graph Y = 2SinX

=> A = 2, B = 1, C = 0 , D = 0, A => Height from centre to the maximum/minimum point

=> Y = 2 Sin (1x + 0) + 0

=> y = 2 Sin X

   

 |A|  = Max value - Min value / 2

  Y = 2sinX : The “height” of the graph increased by 2 times from the centre

B : Period => Number of Cycles within 360^o  

Y = Sin2X

=> A =1, B = 2, C = 0 , D = 0

=> Y = 1 Sin (2x + 0) + 0

=> y = Sin 2X

To find the number of cycles within 360^o

Period = 2π / B   or     360^o/B

            = 2π / 2           360^o / 2

             = π                  180^o

=> The entire cycle is within π, 180o

From the graph, there are 2 cycles within 360^o

=> B indicates the number of cycles within 360^o

      i.e. 2 => 2 cycles, 3 => 3 cycles , etc

C : Phase Shift : C/B => Moving the graph Left/Right

Y = Sin (X + 60^o)

=> A = 1, B = 1, C = 60^o , D = 0

=> Y = 1Sin (1x + 60^o) + 0

=> y =  Sin( X + 60^o)

Positive C “shifts” the graph to the left.

Y = Sin (X - 60^o)

=> A = 1, B = 1, C = -60^o , D = 0

=> Y = 1Sin (1x + 60^o) + 0

=> y =  Sin( X - 60^o)

Negative C “shifts” the graph to the Right.

D : Vertical Shift => Moving the graph Up/Down

Y = SinX + 1

=> A = 1, B = 1, C = 0 , D = 1

=> Y = 1Sin(X + 0) + 0

=> y =  SinX + 1

Positive D “shifts” the graph upward along the y-axis.

Y = SinX - 1

=> A = 1, B = 1, C = 0 , D = 1

=> Y = 1Sin(X + 0) + 0

=> y =  SinX + 1

Negative D “shifts” the graph downward along the y-axis.