The goal behind this page is to give the reader an introduction to plotting parametric equations.
Just a reminder of the definition of parametric equations.
A definition of Parametric Equations :
Let f and g be two functions defined on an interval I. The relation of all ordered pairs (f(t),g(t)) for t in I is called curve C. The equations :
x = f ( t ) and y = g ( t ) ( for t in I )
are called parametric equations for C. and the variable t is called the parameter.
To plot a set of parametric equations by hand you need to follow the steps below :
First, you need to find or determine the range over which t varies. Sometimes this is given in the problem and other times you need to determine it by just picking some values and trying them in the equations.
Second, you need to go through each of the values for t and plug them into the equations x(t) and y(t) so that you find the ordered pairs ( x(t), y(t) ).
Third, the ordered pairs you have found in step two need to be plotted on a standard x-y coordinate plane.
If you have access to a graphing calculator, plotting parametric equations is very easy, just follow the steps below :
First, you need to make sure your mode is set to parametric equations, to do this on a TI graphing calculator go to the mode screen which should look something like this depending on which calculator you are using.
This is a TI-82 Mode Screen
This is a TI-85 Mode Screen
Second, you need to plug the equations into the equation editor (most likely found under the graph menu or by pushing a y= key on the calculator.)
Third, you need to adjust the viewing window so you can see the equation you are trying to graph. To do this locate the screen where you adjust the window dimensions, it might look something like this :
on the TI-82 (also the 81)
on the 85
If the equations you are given do not have a range included with them to plot over then the easiest way to start out is by using the built in Zoom windows that some calculators have built in them. If you are given a range of values plug those values into the calculator and pick nice step or scale (increment) values to start with, maybe one-tenth of your total ranges for each t, x, and y.
Fourth, choose to graph the function if you did not use the Zoom option, which will normally graph automatically when it is choosen.
Fifth, you can now use the trace key to locate special points on the graph (see below) or readjust your viewing window to get a better picture of the graph.