Rational Functions

So here is the first of my 47 (if I can count….) blog posts as part of #summerblogchallenge – not quite as many as @MissNorledge!

I thought I would briefly write about rational functions which often come up in A Level and STEP papers. A rational function is any function \(y = f(x) = \frac{P(x)}{Q(x)} \) with \(P(x), Q(x) \) polynomials. Quite often you are asked to sketch such a function. To do this, the normal steps are to find the roots, i.e. where the function crosses the \(x-\)axis which is easily done by solving \(P(x) = 0 \). Then we can think about asymototes: For the horizontal asymptote consider the ratio of the leading coefficients of \(P(x)\) and \(Q(x)\), for the vertical asymptotes try to factorise \(Q(x)\) and find what values make \(Q(x)\) zero. It is the normal to mark on any local maxima and minima. Of course the usual way is to use calculus and differentiate, setting to zero and solving. However, sometimes we just need to know the approximate location of the maxima and minima  and the associated y values. This is due to the fact that we can often sketch the graph using this knowledge and examining what happens near the asymptotes.

I was shown this by another teacher who recently retired and had never considered it before, always using calculus. But with practise, this could be done mentally, pretty quickly I think.

The picture below shows the method for an example. Essentially you imagine a fixed horizontal line across the graph of your rational function, so \(y\) takes on a fixed value. We can then form a quadratic whose coefficients depend on \(y\). A maximum (or minimum) of our rational function will occur when the discriminant of our resulting quadratic is zero. Using this fact we can find the local maximum and minimum values of the rational function.

   

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