Sketching trigonometric graphs

Refresher basic trig graphs

CAST Diagram

 

A trig graph can be expressed in the form

where a = amplitude

b = no of cycles in 360°

c = shift in x direction

d = shift in y direction

 

 

To sketch the graph :

• Start with the basic shape.
• Use the number of cycles (b) to calculate the period
• Use the horizontal shift (c) to find slide the graph into position
• Use the amplitude (a) to scale vertically and calculate the starting minimum and maximum values
• Use the vertical shift (d) to move the graph and calculate the actual minimum and maximum values.

 

 

Example

Sketch the graph of y = 3sin( x -30 ) +1

A basic sine graph is translated 30° right, then scaled by a factor of 3 before being moved upwards one space.

The period is 1, since b = 1, so know points on a basic sine graph are (0,0) , (90°, 1), (180°, 0), (270°, -1 ) and (360°, 0)

Translating these points :

From these, it can be seen that the maximum value is 4 and the minimum value is -2.

To find the values where the graph cuts the x axis , set y = 0 and solve the trig equation 0 = 3sin( x -30 ) +1

 

Putting all of this together : -