Excel: Parametrically

by

Christa Marie Nathe

Throughout the term we have been using Geometriors Sketchpad and the software program Graphing Calculator to investigate various mathematical concepts including, but not limited to, parabolas, constructing geometric figures and geometric tools as well as graphing various kinds of equations.  Even though these programs are excellent tools used to manipulate and understand various mathematical concepts, we are not strictly limited to these options. In the following investigation we are going to use the power of Microsoft Excel to graph some parametric equations.

The first set of parametric equations that we will graph using Excel is

x = t + 1

y = 2t -1

As you may remember, when we are working with parametric equations we use values of t to determine x and y. The first column is a list of the parameter values for t and subsequent increments for t up to 100. The second column is the formula for the x-coordinate and the third column is the formula for the y-coordinate. By using the fill down option in Excel we can determine the specific coordinates for x and y with respect to t.

 t x=t+1 y=2t-1 0 1 -1 10 11 19 20 21 39 30 31 59 40 41 79 50 51 99 60 61 119 70 71 139 80 81 159 90 91 179 100 101 199

Below is a graph of the values from the second and third columns

The next set of parametric equations that we will use Excel to graph is

x = a + t

y = b + kt

Below is a spreadsheet of our values.  The penultimate and final column state our x and y coordinates based on the parameter t and the variables a, b, and k. Beneath the list of values is the graph of the parametric equations via of the coordinates.

 t a b k x=a+t y=b+kt 0 0 0 0 0 0 10 1 1 1 11 11 20 2 2 2 22 42 30 3 3 3 33 93 40 4 4 4 44 164 50 5 5 5 55 255 60 6 6 6 66 366 70 7 7 7 77 497 80 8 8 8 88 648 90 9 9 9 99 819 100 10 10 10 110 1010

Finally, let us use Excel to graph a trigonometric parametric set of equations

x = cos(at)

y = sin(bt)

Using our spreadsheet to lay out the variety of values of a, b and t we obtain the following x-y coordinates.  Below you will find the graphical representation for this trigonometric expression.

 t a b x=cos(at) y=sin(bt) 0 0 0 1 0 10 1 1 -0.83907 -0.54402 20 2 2 -0.66694 0.745113 30 3 3 -0.44807 0.893997 40 4 4 -0.97563 0.219425 50 5 5 0.240988 -0.97053 60 6 6 -0.28369 0.958916 70 7 7 0.99609 -0.08834 80 8 8 0.633366 -0.77385 90 9 9 0.862349 -0.50631 100 10 10 0.562379 0.82688

Through this exploration of using Microsoft Excel to express various kinds of parametric equations, we have added an alternative to our mathematical resources.