CK-12-Pre-Calculus Concepts

5.4. Vertical Shift of Sinusoidal Functions http://www.ck12.org

5.4 Vertical Shift of Sinusoidal Functions

Here you will explore the how a vertical shift of a sinusoidal function is represented in an equation and in a graph.
Your knowledge of transformations, specifically vertical shift, apply directly to sinusoidal functions. In practice,
sketching shifted sine and cosine functions requires greater attention to detail and more careful labeling than other
functions. Can you describe the following transformation in words?
f(x) =sinx→g(x) =−3 sinx− 4
In what order do the reflection, stretch and shift occur? Is there a difference?

Watch This

Watch the portions of this video focusing on vertical translations:

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/61179

http://www.youtube.com/watch?v=DswBtrtvR5M James Sousa: Horizontal and Vertical Translations of Sine and
Cosine

Guidance

The general form of a sinusoidal function is:
f(x) =±a·sin(b(x+c))+d
Recall thatacontrols amplitude and the±controls reflection. Here you will see howdcontrols the vertical shift.
The most straightforward way to think about vertical shift of sinusoidal functions is to focus on the sinusoidal axis,
the horizontal line running through the middle of the sine or cosine wave. At the start of the problem identify the
vertical shift and immediately draw the new sinusoidal axis. Then proceed to graph amplitude and reflection about
that axisas opposed to thexaxis.
Example A
Graph the following three functions.

f(x) =sinx+ 3

g(x) =sinx− 2

h(x) =sinx+^12