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?
Watch This
Watch the portions of this video focusing on vertical translations:
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:
Recall that
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 axis as opposed to the
Example A
Graph the following three functions.
Solution: First draw the new sinusoidal axis for each graph. Then, draw a complete sine wave for each one. Remember to draw the five important points that separate each quadrant to get a clear sense of the graph. Right now every cycle starts at 0 and ends at
Example B
Identify the equation of the following transformed cosine graph.
Solution: Since there is no sinusoidal axis given, you must determine the vertical shift, stretch and reflection. The peak occurs at
Example C
Graph the following function:
Solution: Identify the important information. Then draw the sinusoidal axis.

a=1 
d=−1  Reflection over
x axis.
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.
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
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
The sinusoidal axis is the horizontal line that runs through the middle of the sine or cosine wave.
Vertical shift is a rigid transformation that moves every point vertically by a set amount.
Guided Practice
1. 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.
2. What equation models the following graph?
3. Graph the following function:
Answers:
1. 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.
2.
3. 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.
Practice
Graph each of the following functions that have undergone a vertical stretch, reflection, and/or a vertical shift.
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Find the minimum and maximum values of each of the following functions.
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Give the equation of each function graphed below.
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