Many of the explorations in mathematics is greatly enhanced by the use of graphing packages. These graphing packages may not readily be available to the student. Most computers come with a spreadsheet package. I will demonstrate the use of a typical spreadsheet to graph the path of a fixed point on a wheel as the wheel move in one horizontal direction; I will also graph the Polar form of Conic Equations.

First let us consider a fixed point on a wheel. We can develop parametric equations to describe the path of the point (see link below). We get

where t is our parameter and represents the angle the wheel has rotated as it rolls. The radius of the wheel is given by r .

To see this in our spreadsheet we first create a column for the parameter t. To do this I used a counter for the first column. The spreadsheet formulas I entered were:

In row 1 I gave a description to my columns. In row2 of column A I put 0 and in Row 3 of column A I put =A2+1. A2 stands for Column A, Row 2. I select cell A2 and copy it. I then select (highlight) Cells A3 through Cells A72 and with this highlighted I paste. This puts the formula =Aprevious + 1 in each cell. Then I compute the parameter t by A3*Pi()/10 which will give us data for each Pi()/10 rotations of our wheel. I then put the formulas for x in C and y in D. I copy the formulas and paste them up to row 72 as I did for the formula in A3. Notice how the spreadsheet automatically changes B2 to B3 in Column C and D in the rows below. This happens when you copy and paste a formula as we did to get our data before. If we don't want the spreadsheet to do this we put a $ in front of the row or column label we want to stay the same. Notice the E$3 in the formulas above. in cell E3 we have the radius, which remains the same for all the data and since I copied the formula for one column down many rows the E$3 means don't adjust the row number as I copy the formula to the rows below. Since I was not copying formulas across the columns, but just one column at a time I didn't need a $ in front of the E.

Now lets look at the data that results:

We can graph the data by highlighting the formulas that contain the formulas for x and y and clicking a graph button usually found on the tool bar. With the x-y columns highlighted you want to choose X-Y scatter plot for the type of graph.

We get the following:

This gives us the same result as the graphing calculator.

Theorem: Let F be a fixed point and l a fixed line in a plane. The set of all points P in the plane such that the ratio of the distance of the point to l and the distance of the point to F is a positive constant k is a conic section. Moreover, the conic section is a parabola if k =1,an ellipse if 0<k<1, and a hyperbola if k >1.

Using the definition we can derive a formula for the above definition of a conic in terms of k and p using polar coordinates. If l is defined as x = -p where p > 0 we get:

Again we can put this equation into a spreadsheet. We can put p and k into a cell and see how changing them affect the graph. Again In the first column I made a counter as in the last problem. I also defined an incrementing angle value for theta based using Pi and my counter. Next I put in the conic equation for r. Because I didn't have a polar coordinate graphing capability in my spreadsheet I used the equations and to convert my polar coordinates to Cartesian coordinates.

The following shows the formulas I entered:

A B C D

E F G H

Once this is done all that is left is to choose X-Y scatter-plot from the graph menu and watch how changing p,and k affect your graph.

The following graph is with p=1,k=1

Now when I change k= .25,p=1 I get:

Note: If you are having so much fun changing the parameters and making charts you may get a message telling you you can't create any more charts until you delete some previous ones. You can delete Charts from the Tools menu item at the top of the spreadsheet.

I have shown three ways to explore mathematical functions graphically. Two involved the use of a computer and one the use of a TI-83 calculator. The programs used on the computer are available for both Macintoshes and PC's.

RETURN To Cathryn Brooks