Twice a day, tides change on the coasts of the world. In the Bay of Fundy in eastern Canada, the water level changes by as much as 50 feet.
A person plotting the height of the water as the tide changes might get a picture that looks like this.
Tides are just one example of periodic phenomena in nature. There are also pendulums, heartbeats, pulsars, ocean waves, and many other things.
Trigonometry is how mathematics deals with such regular repeating phenomena.
The graphs of the trigonometric functions are subject to the same transformations as the graphs of other functions.
While, for the most part, the transformations work the same way, some of them have different names.
The vertical stretch and shrink transformation is called the amplitude. For the sine and cosine, the amplitude is half of the total height of the graph.
For the secant and cosecant, the amplitude is half of the vertical separation between the upper and lower branches.
For the tangent and cotangent, the amplitude is how far from the midline the quarter points are.
The amplitude appears in the function in the same manner as the vertical stretch and shrink parameter, as a multiplier for the functions as a whole.
The horizontal stretch or shrink is referred to as the period. Each of the trigonometric functions has a period.
The period of the untransformed sine, cosine, secant, and cosecant functions is 2π. The period of the untransformed tangent and cotangent functions is π.
The period doesn’t directly appear in the function. The value that does appear is the untransformed period of the function divided by the period of the transformed function.
The horizontal shift is renamed the phase shift.
The phase shift appears in the same place in the function as the horizontal shift.
The vertical shift is still called a vertical shift.
It still appears as a value added to the entire function.
With this tool, you can experiment with transformations of the graphs of the trigonometric functions.
You can change any of the transformation parameters one at a time or several at a time, including reflections in the x- and y-axes.
Instead of manipulating the horizontal stretch and shrink parameter, you can use a control to change the period of the graphs of the functions.
You can choose to examine the graphs of two functions simultaneously so that you may better understand the relationships between the various functions.
The second function will be graphed with the same transformation parameters as the first.
Use the two-function ability to explore the relationship between cofunctions or reciprocal functions.
Try to predict how changing parameters will affect the graph before actually doing it. Keep an open mind and you’ll be sure to maximize learning.