Rational Functions and Asymptotes

Consider the rational function given by two polynomial functions;

What about these two polynomials decides the location of:

• The vertical asymptotes?
• The horizontal asymptotes?
• The solutions to f(x)=0?

I know the theory behind this, but what does it look like in motion?

The geogebra file linked here https://www.geogebra.org/m/m8eqsjzu explores this. What happens as I click and drag one of the curves? How does it make the graph of f(x) move?

Introducing Complex Numbers

 I was watching an old OU video and there was a demonstration there that needed an update to geogebra - no need for hand animation any more!

The geogebra file is https://www.geogebra.org/m/waggbjrr here!

The file allows you to take a polynomial curve and see what happens to the solution set as you move the curve vertically, demonstrating what happens to those roots in the complex plane.

I have used visualisation with many classes down the years, although I have got out of practice! The task below was suggested to me by some work I did at the recent ATM conference and concerns midpoints and triangles.

Spoiler images at the bottom of the post!

Visualising Midpoints

We are going to do a simple visualisation task. For this task you may not do any physical drawings, at first. You may have your eyes open or closed as you see fit.

• Imagine an infinite flat plane.
• A fixed point, A, appears.
• A second point, B, comes into the plane and is also fixed.
• Join the line segment AB.
• A third point C enters the scene. It is free to move.
• Watch as the point moves.
• Pause and describe what you see to me. 
• Right, now join AC and BC to make a triangle. Remember that A and B are fixed, but C is free to move.
• Find the midpoint of AC.
• Find the midpoint of BC.
• Join them. 
• Watch what happens to this line as C moves. 
• Tell me the story of what you see happening.
• What can you say about the line? 
• Tell me any conjectures you have. 
• Can we prove them?

Try the task now, you may come up with questions I have not!

At this point I would allow the students to draw diagrams and then in pairs choose a conjecture to prove. 

It is important that the questions are the students, but helpful hints toward parallelism, area or perimeter, may be needed.

My questions;

• Are the base and the mid-line parallel? Why?
• What is the area of the smaller triangle in terms of the first? Why?
• What is the perimeter of the smaller triangle in terms of the first? Why?

Extensions;

• Instead of midpoints split the lines AC and BC into the ratio 1:2 and ask all the same questions (Area is particularly fruitful).
• What about the ratio 2:3?
• What about the ratio x:y?
• What if we split one line in the ratio 2:3 and the other in the ratio 1:2?