# Graphing transformations of trigonometric functions

### Graphing transformations of trigonometric functions

After learning all the graphs of basic trigonometric functions, in this lesson, we are going to go a little bit further on how the graphs will be transformed as the functions change. The general form for the equation of trig functions is y = f [B(x + c)] + D, where f refers the trig function; A refers to the amplitude/steepness; B represents the period of the graph; C refers to phase shift (left or right) and D represents vertical shift (up or down). We will learn how to graph the trig function for multiple periods; state the vertical displacement, phase shift, period and amplitude; and also find the domain and range of the transformed functions.

#### Lessons

• 1.
For each trigonometric function:
(i) Graph the trigonometric function for one period.
(ii) State the vertical displacement, phase shift, period, and amplitude.
(iii) State the domain and the range.
a)
$y = 2\sin \frac{\pi }{4}(x + 3) + 1$

b)
$y = 3\sec (\frac{\pi }{2}x - \pi ) - 1$

c)
$y = - 2\sin (4x + 4\pi ) - 3$

• 2.
For the trigonometric function: $y = - \tan \left( {\;\frac{x}{3} - \frac{\pi }{6}\;} \right)$
i) Graph the trigonometric function for two periods.
ii) State the domain and the range.