PARAMETRIC EQUATIONS: Math Art

Even for those who are not enamored by the intricacies of math, the potential for entering an equation into a grapher and watching an incredible design appear can be appealing.  Parametric equations are the artists of the math world.



Parametric equations appear every now and then on the SATII, in Precalculus classes, and in Calculus as well.  These equations express two variables (usually x and y) each in terms of a third variable called the parameter (often t).

THE OBVIOUS SOLUTIONS
Given the parametric equations....

    x = t^2 + 3
    y = 2t

....a graph can be plotted by calculating values of (x,y) in a three column chart.

    t      x     y
    0     3     0
    1     4     2
    2     7     4
    3    12    6
    4    19    8
    5    28    10

USING OUR ALGEBRA SKILLS
Many parametric equations can also be manipulated algebraically to form a single rectangular equation which may be easier to graph.   

    Solving for t in terms of either x or y in the example above and
    substituting into the alternate equation gives the function...

    y = 2 sqrt (x-3)


A SPECIAL CAUTION WHEN ELIMINATING THE PARAMETER
Caution should be taken to preserve the range of the original equations when eliminating a parameter.  In this example, the range of x is all real numbers greater than or equal to 3.  Luckily, this exclusion on x is also accounted for in the rectangular equation because of the square root.

But consider...

    x = t^2
    y = 3t^2 + 4

In this case, x doesn’t fall below 0 and y doesn’t fall below 4.  So the graph of the parametric equations is truncated below (0,4).

PARAMETERS CAN DO WHAT FUNCTIONS NEVER WILL
Many interesting shapes cannot be graphed as functions because they don’t pass the vertical line test.  Parametric Equations can be used to graph geometric figures like conics.

The Cartesian equation for a circle, x^2 + y^2 = r^2, can easily be converted to parametrics as
    x = r cos t
    y = r sin t

RECOGNIZE CHARACTERISTICS OF PARAMETRICS
Here are some “expectations” for parametric equations.

1.  x = sin t, y = cos t  represents a point moving clockwise around the unit circle.

2.  x = cos t, y = sin t describes the point moving counter-clockwise around the unit circle.
   
3.  x = t, y = t^2 signals a parabola

4.  x = t^2, y = t  in function form graphs as a square root

RELEASE THE ARTIST WITHIN
YouTube has a plethora of examples of exciting, artful parametric equations to program into a graphing calculator.  Here’s one that draws a butterfly......

    x = sin(t) (e^(cos(t)) - 2 cos (4t) - (sin (t/12))^5)
    y = cos(t) (e^(cos(t)) - 2 cos (4t) - (sin (t/12))^5)

Have fun!!