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
x = rt - rsint
y = r - rcost
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 .
Click Here for derivation of equations
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.
To see the graphing calculator results and steps to poduce graph on a TI-83
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.
To see the graphing calculator results and he steps on a TI-83 to produce the graph
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