Stephani Eckelkamp

 

Parametric Equations:  What does it all mean


 

Let us begin by defining parameter.

 

paŠramŠeŠter   (parametric adj.)

 

   1. Mathematics definition.

         1. A constant in an equation that varies in other equations of the same general form, especially such a constant in the equation of a curve or surface that can be varied to represent a family of curves or surfaces.

         2. One of a set of independent variables that express the coordinates of a point.

 

~~  Now that we know what it means, we can explore parametric equation and their graphs  ~~

 

Parametric equations in the plane is a pair of functions

 

x = f(t) and y = g(t)

 

which describe the x and y coordinates of the graph of some curve in the plane.

 

A very basic parametric equation is

 

x = cos (t)

y = sin (t)

 

this can also be written as

 


Explorations

 

                          

 

By increasing the coefficient of the x coordinate function and the y coordinate function we are able to see where the graphs cross the x and y axis respectively.

 

What happens if the x and y coefficient of these coordinate functions differ?

 

                                 

 

 

The coefficient attached to the x coordinate function is where the graph will cross the x axis, and the y coordinate function will cross the y axis.

We can see this in both the above and below examples.

 

                      

 

Further explorations:

 

                                   

 

 


 

What do you think will happen when a number is attached to t?

 

 

 

                  

 

By multiplying t by 2 the graph folds over itself in one place.  What do you think will happen for 3? 

 

 

 

 

              

 

                                                                                                                                          

Click here to see an animation for this parametric equation with the slider values from 1 to 10.

 

cos (nt)                                                                                                                                sin (nt)

 

What do you notice about how many times the graph crosses itself in relation to the n value given?  Does the coefficient of the x and y coordinate ideas that you previous had hold true in these cases?   Spend some time working with the link below, to see if your conjectures hold true for multiple cases.


 

For further explorations open this copy of graphing calculator

 

You can also graph these equations using a graphing calculator (ie TI series).


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