Explorations With Graphs and Equations

By Donna Greenwood

Consider two points (3, 4) and (-5, -2). For any point (x, y) we can write the distance as

Distance 1 =

Distance 2 =

Explore graphs with these two distance equations. For example,

1. Consider when each set is set to a non-zero constant. Circles are graphed.

In this drawing, the distances are set to several different constants as follows:

Color	Distance
Blue	3
Aqua	4
Red	5
Magenta	10

It looks like the circles are tangent when the distance from each point is 5. Is that true? Looking at the distance between the two points, , equals 10, so the red circles are indeed tangent. Since the distance between the points is 10, it makes sense that circles graphed with a distance smaller than 5 do not intersect, while those with a distance greater than 5 do.

2. Consider the sum C =

For different values of C.

Color	Value of C
Magenta	10
Blue	15
Green	20
Aqua	25
Red	30

As the constant distance C gets larger, the graph appears to approach a circle.

Next, consider product as a constant distance:

Color	Value of C
Magenta	10
Green	20
Aqua	25
Red	30
Blue	50

Here is the graph when the difference of the distances is held constant:

Finally, here is the graph of the division of the distances: