Graphs of trig functions
See also transformations of functions.
Contents
Trig functions to graph
Using unit circle, , and can be extended to all real numbers (with some asymptotes for ) in a way that they are periodic, and that and are odd while is even.
passes through .
passes through .
Both and have a period of , and the pattern repeats for other values of . These are the five points you should consider when graphing or sketching their transformations.
are not usually used for transformations, but just remember the graph looks like lots of , with vertical asymptotes at every odd multiples of .
Example: Given
Find the coordinates of a maximum and a minimum.
Compare to base function , the transformations are
- horizontal stretch by
- vertical reflection across -axis and vertical stretch by a factor of .
- horizontal translation by to the left.
- vertical translation by up.
The base function has a maximum at and minimum at .
Upon a reflection, a maximum become a minimum, vice versa.
The corresponding minimum is
The corresponding maximum is
From graph or description to function
For sinusoidal functions in the forms
symbol | formula | notes |
---|---|---|
amplitude: distance between average and each max or min | ||
vertical shift: average: is the line about which the function is oscillating. | ||
horizontal shift: depends on a given point in the question | ||
angular frequency: the number of cycles in | ||
period: the difference in between consecutive complete cycles. |
In modelling, time is often used instead of .
HL: Inverse trig functions
See also domain and range of inverse functions and domain restrictions
HL students should also know the graphs of , commonly known as .
Note that all 3 are invertible functions, while , , and are not.