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Rational functions and Mobius transformations

In this video Norman Wildberger discusses rational functions and their special cases called Mobius transformations.
We’d now like to consider more general kinds of relations which are based upon inverse relations. Where we start with y equals 1 over x or y equals a over x and move that hyperbola around. What kind of new functions and new relations can we generate? Turns out that is a very interesting thing that goes back to a German geometer and astronomer called August Mobius — who’s also responsible for the Mobius band —- who introduced Mobius transformations. And this is something in the direction that we’re going to be talking about now to extend our algebra and our algebraic manipulations to translating and scaling hyperbolas around.
We’ll see some interesting connections with linear algebra and just a little bit of matrix theory, along with some very natural and interesting algebra. So let me remind you that when we started our course we were first considering direct proportionalities. Things of the form y equals a times x. Then after studying that and recognising that the number “a” was a very important concept of the geometrical slope, we considered more general lines that we can think about as being obtained from this kind of line by translating. So we get lines like y equals a x plus b which can be pretty well anywhere.
Then when we studied quadratic relations, we started with y equals a x squared, a basic quadratic relation, and ended up translating those around to get general quadratic functions, of the form y equals a x squared plus bx plus c. So the “a” here and the “a” here are the same. The shape of the parabola hasn’t changed, but its position in the plane has changed. And we saw in fact how to clearly understand where its position is in terms of where the vertex has gone to. So if the vertex is (h,k) we saw we could rewrite this problem in the form a times x minus h squared plus k.
This is intimately connected with completing the square, going from here to here. Alright so it’s natural for us to consider the same kind of situation here for our inverse relation. If we start off with the curve y equals a over x, and now “a” could be more general, not just one. So that’s the general inverse relation. It’s still a hyperbola that looks like the one over x hyperbola but it might be dilated out or dialated in towards the origin depending on the value of “a”. We can ask, well, what happens if we translate that? Say move it to there so that the centre of the hyperbola moves to the point (h,k)?
What is then the equation, and what kind of new family of functions do we get? So it turns out we get some very natural and interesting rational functions, basically studied by Mobius about 150 years ago. So let’s see what happens to an inverse relation given by our favourite hyperbola y equals a over x. If we translate it first in the x direction say by h and then by k in the y direction. So abstractly moving it h in the x direction replaces the x here with an x minus h. That’s the same thing that we saw when we were shifting parabolas around. We get y equals a over x minus h.
And then when we translate by k in the y direction, it means that we have to move it up or down depending on the value of k, by adding k to the y value. Now let’s look at what we get. This expression over here, if we take it over a common denominator, we get kx plus a minus hk over x minus h. This is a very particular kind of relation where we have the quotient of two linear expressions. The general form is alpha x plus beta over gamma x plus delta. This is an example of what’s called a rational function. It’s sort of one step up from talking about polynomials. It’s a polynomial divided by another polynomial.
That’s what a rational function is. But in this case there’s a very special kind of rational function because the two polynomials are both linear polynomials in x. So what we get is called a fractional linear transformation. A very natural term because it’s one linear transformation divided by another linear transformation. Or a Mobius transformation in honour of August Ferdinand Mobius who first studied these things. And conveniently the data of such a function can be expressed in this nice little two-by-two array because there’s really only four numbers, alpha, beta, gamma, delta, that are involved. And we can make a little matrix out of those four numbers.
We’ll see that in fact there’s a little bit of linear algebra connecting with the composition such rational functions. So let’s have a look at what happens in a specific example. We’ll start with y equals 1 over x, in blue there. And suppose that want to translate so that the centre moves over by 2 in the x direction. So we just have to translate the whole curve over by 2 and we get something that looks like this. And then this one also has to be shifted over by 2. We’ll get something that looks like this.
And what’s the equation of this thing? It will be y equals 1 over, x minus 2. So notice that this thing now has an asymptote not at x equals 0 but at x equals 2. That’s where it’s undefined. That’s the place where the denominator here would be zero. And suppose that we wanted to then translate it up by 1. So we’re going to move that whole thing up so that it’s centre is right there. Then we would get a curve that looks something like — one branch, and the other branch would do something like this.
So the effect is to — we get the equation y equals 1 over x minus 2. And now we have to add 1 in order to get this brown curve. And we can rewrite that as x minus 2 plus 1. So that’s x minus 1 over x minus 2. It still has an asymptote where x equals 2 and the other asymptote would be the line y equals 1. So that’s just a little bit of an introduction to an important family of functions obtained by just extending our understanding of the inverse relationship a little bit by adding some suitable translations in the x and y directions. It’s a very important algebraic kind of object that we’re connecting with here.
In this video we discuss rational functions and their special cases called Mobius transformations. We have seen that by translating simple proportions we get general lines, and by translating standard parabolas we get general quadratic functions. So it is natural to ask what kind of functions we get by scaling and translating the inverse relation \(\normalsize{y=1/x}\).

Mobius and his transformations

The answer leads us to an interesting family of rational functions introduced by the prominent German geometer August Ferdinant Mobius. Mobius also was an architect of the revival of projective geometry in the 19th century.
He introduced a family of functions or transformations that are quotients of one linear polynomial over another linear polynomial. An example of such a Mobius transformation, also called a fractional linear transformation, is
Both the numerator and denominator are linear, or constant, polynomials.
Graph of y=(5x-1)/(3x+2), showing asymptotes
We see that this graph indeed looks something like \(\normalsize y=1/x\), or more precisely \(\normalsize y=-1/x\), which in addition has been translated in the plane.
Why do we use the word transformation here and not function? Mobius, being a geometer more than an analyst, was interested in thinking about functions dynamically and geometrically, as moving points around. Hence the word transformation.

A connection with linear algebra

The general case of a Mobius transformation is
Such a function can be specified by a matrix
\[\Large{\begin{pmatrix} a & b\\ c & d \end{pmatrix}}\]
and there is a natural connection with elementary linear algebra, as we shall see when we investigate composition of Mobius transformations.
Mobius transformations are special cases of rational functions, which are quotients of polynomials.
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Maths for Humans: Linear, Quadratic & Inverse Relations

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