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.