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?
In what order do the reflection, stretch and shift occur? Is there a difference?
Vertical Shift of Sinusoidal Functions
The general form of a sinusoidal function is:
The graphs of the following three functions are shown below:
Watch the portions of the following video focused on vertical translations:
Earlier, you were asked which order vertical shift and reflection should be performed in and if it matters. The following transformation can be described in basically two ways.
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.
Identify the equation of the following transformed cosine graph.
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.
When doing ordered transformations it is good to show where you start and where you end up so that you can effectively compare and contrast the outcomes. See how both transformations start with a regular sine wave. The two columns represent the sequence of transformations that produce different outcomes.
What equation models the following graph?
First draw the horizontal sinusoidal axis and identify the five main points for the cosine wave. Be careful to note that the amplitude is 2 and the cosine wave starts and ends at a low point because of the negative sign.
Graph each of the following functions that have undergone a vertical stretch, reflection, and/or a vertical shift.
Find the minimum and maximum values of each of the following functions.
Give the equation of each function graphed below.
To see the Review answers, open this PDF file and look for section 5.4.