5.4. Vertical Shift of Sinusoidal Functions http://www.ck12.org
- d=− 1
- Reflection overxaxis.
Note that it is critical that you know the shape of a regular sine graph and a negative sine graph.
Concept Problem Revisited
The following transformation can be described in basically two ways.
f(x) =sinx→g(x) =−3 sinx− 4
The first is to describe the stretching and reflecting first and then the vertical shift. This is the most logical way to
discuss the transformation verbally because then the numbers like 3 and -4 can be explicitly identified in the graph.
The second way to describe the transformation is to attempt to say the vertical shift first. In this case the vertical shift
would initially be−^43 , and then the vertical stretch would magnify this distance from thex-axis. This is significantly
less intuitive. If a description showed the vertical shift to be -4 initially followed by a stretch by a factor of 3, the
sinusoidal axis would move toy=12 which is incorrect.
The order in describing the transformation matters. When describing vertical transformations it is most intuitive to
simply describe the transformations in the same order as the order of operations.
Vocabulary
Thesinusoidal axisis the horizontal line that runs through the middle of the sine or cosine wave.
Vertical shiftis a rigid transformation that moves every point vertically by a set amount.
Guided Practice
- Transform the following sine graph in two ways. First, transform the sine graph by shifting it vertically up 1 unit
and then stretching it vertically by a factor of 2 units. Second, transform the sine graph by stretching it vertically by
a factor of 2 units and then shifting it vertically up 1 unit.