http://www.ck12.org Chapter 5. Trigonometric Functions
5.6 Phase Shift of Sinusoidal Functions
Here you will apply all the different transformations, including horizontal shifting, to sinusoidal functions.
A periodic function that does not start at the sinusoidal axis or at a maximum or a minimum has been shifted
horizontally. This horizontal movement invites different people to see different starting points since a sine wave
does not have a beginning or an end.
What are five other ways of writing the functionf(x) = 2 ·sinx?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61262http://www.youtube.com/watch?v=wUzARNIkH-g James Sousa: Graphing Sine and Cosine with Various Trans-
formations
Guidance
The general sinusoidal function is:
f(x) =±a·sin(b(x+c))+d
The constantccontrols the horizontal shift. Ifc=π 2 then the sine wave is shifted left byπ 2. Ifc=−3 then the sine
wave is shifted right by 3. This is the opposite direction than you might expect, but it is consistent with the rules of
transformations for all functions.
Generallybis always written to be positive. If you run into a situation wherebis negative, use your knowledge of
even and odd functions to rewrite the function.
cos(−x) =cos(x)
sin(−x) =−sin(x)Example A
Graph the following function:f(x) = 3 ·cos(x−π 2 )+1.
Solution: First find the start and end of one period and sketch only that portion of the sinusoidal axis. Then plot
the 5 important points for a cosine graph while keeping the amplitude in mind.