Parametric curves
The co-ordinates (x, y) defining a function can be expressed in terms of a third variable, e.g. time or an angle.
The first example below is a circle defined by x(t) = rcos(t) and y(t) = rsin(t).
We have the added advantage that we can trace the curve ...
Note that the "time" interval runs from 0 to 2pi, i.e. one revolution and the "speed" (the number of intervals) is the same for all examples.
Click on the drop-down menu at the top and Trace to illustrate how the graph is influenced by
- amplitude (or vertical scaling), e.g. 5cos(t),
- period (or horisontal scaling), e.g. 10cos(5t),
- phase (or horisontal translation), e.g. 10cos(t+1), and
- vertical translation, e.g. 10cos(t)+5
The challenge is to see the connections between the diagram, its defining formulae, and the separate graphs of x(t) and y(t),
Activity: For each example in the drop-down menu, click Trace Curve!, study the movement and then describe and graph x(t) and y(t).
Remember, x(t) is the projection of the movement of the point onto the X-axis, and y(t) is the projection of the path of the point onto the Y-axis.
Note: you can experiment further by entering your own formulae and changing the values of t (defining the domain and range!).
In the launcher below we give the same set of functions, but without the parametric formulae. The challenge is to formulate the defining formulae and draw their graphs.
We have emphasized before that trigonometry describes periodic phenomena and this does not necessarily mean just circular motion! The examples below are very interesting curves, and you can experiment by entering your own formulae below ...
The movemments may seem complicated, but, still, x(t) merely describes the projection of the movement of the point onto the X-axis and y(t) the movement onto the Y-axis, and we should be able to describe and graph this projection! For example, in Connected hearts below, despite the complicated movement, x(t) simply is a cosine graph with amplitude 10 and period 2pi/3.
Of course, we can check the form of the graphs by using a Simple Graph applet.
The activity would be more challenging if we hid the formulae and deduced the form of the graphs from analysing the x- and y-movements!!