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.
Phase Shift of Sinusoidal Functions
The general sinusoidal function is:
It all depends on where you choose start and whether you see a positive or negative sine or cosine graph.
Given the following graph, identify equivalent sine and cosine algebraic models.
|Time (minutes)||Height (feet)|
William chooses to see a negative cosine in the graph. He identifies the amplitude to be 40 feet. The vertical shift of the sinusoidal axis is 42 feet. The horizontal shift is 5 minutes to the right.
Thus one equation would be:
Tide tables report the times and depths of low and high tides. Here is part of tide report from Salem, Massachusetts dated September 19, 2006.
|Time (hours : minutes)||Time (minutes)||Tide (feet)|
These numbers seem to indicate a positive cosine curve. The amplitude is four and the vertical shift is 5. The horizontal shift is 615 and the period is 720.
Thus one equation is:
There are four times within the 24 hours when the height is exactly 8 feet. You can convert these times to hours and minutes if you prefer.
Graph each of the following functions.
Give one possible sine equation for each of the graphs below.
Give one possible cosine function for each of the graphs below.
13. Use the equation from #12 to predict the temperature at 4:00 PM.
14. Use the equation from #12 to predict the temperature at 8:00 AM.
To see the Review answers, open this PDF file and look for section 5.6.