Recently, I’ve been exploring the relationship between systems of linear functions and their reciprocals, namely:

The following are the two methodologies I used in this investigation:

1) Lo-tech method (longer way): Solve for all axis-intercepts, asymptotes and points of intersection for various examples.

2) Higher-tech method (shorter way): Graph examples using software.

After sketching several examples, I noticed that at the points of intersection (poi) y = ±1. This generalization was useful when calculating/checking the x-coordinate of the poi without the aid of graphing software.

After sketching the long way, similar results were confirmed through software.

Calculating the poi algebraically:

confirmed the observed coordinates for the poi:

I wondered if this relationship could be extended to a system of functions such as:

Plotting examples using graphing software didn’t yield any transparent relationship between the coefficients of the function and the poi coordinates.

So, again, calculating the poi algebraically:

resulted in the coordinates for the poi (with invariant point y = c^(1/2):

Same same but different!