1. Sketching Rational Functions
CIE Centre A-level Further Pure Maths

2. Definition of rational function
In this section we will be looking at functions of
the form:
where p and q are polynomials in x. Almost all of our examples will be restricted to degree 1 and degree 2.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Keep in mind that we will continue to need
our basic theory (from P3) about division of
polynomials and partial fractions.

3. Sketching is not plotting
Curve sketching
Let’s review some basic points to remember about sketching the curve of a function.
Sketching is not plotting
Gather all information before starting the sketch
Mark on the intercepts - points where the curve crosses the coordinate axes (as best you can)
Mark on the asymptotes (I will show you how to find them)
Find the turning points and determine their nature (where possible)
Draw smooth curves between these key features.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

4. We start with the simplest kind of rational function:
Type 1: Linear/Linear
We start with the simplest kind of rational function:
Let’s try an example:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
The curve cuts the y-axis at …
-1/3
The curve cuts the x-axis at … -1

5. What are the asymptotes of these functions:-
What is an asymptote?
“An asymptote is a line which becomes a tangent to a curve as x or y tends to infinity” (p. 130)
What are the asymptotes of these functions:-
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
We will restrict our discussion to linear asymptotes

6. Going back to our example:
Type 1: Linear/linear
Going back to our example:
Finding the asymptotes.
There are three types of asymptote:-
Horizontal – of the form y = k, found by observing the behaviour as
Vertical – of the form x = k, occur at points where the function is undefined.
Oblique – of the form y = kx + c, found by observing the behaviour as
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
What kind of asymptotes does f(x) have?

7. A: One vertical asymptote at x=3, one horizontal asymptote at y=1.
Type 1: Linear/linear
A: One vertical asymptote at x=3, one horizontal asymptote at y=1.
Finding the vertical asymptotes is trivial for rational functions; they lie at the roots of the denominator.
Find the horizontal asymptotes and oblique asymptotes sometimes requires more thought.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

8. Technique: finding non-vertical asymptotes
This method of argument is fine for type 1 (linear/linear), but it will trick us into a mistake with quadratic functions. Note this alternative approach:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
By splitting into partial fractions, the asymptotic behaviour is observed directly.

9. Let’s sum up what we’ve learnt so far about this function:
Type 1: Linear/linear
Let’s sum up what we’ve learnt so far
about this function:
Intercepts the coordinate axes at: (0,-1/3) and (-1,0)
Has vertical asymptote at x = 3
Has horizontal asymptote at y = 1
You can easily prove that there are no stationary
points (how?).
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Let’s mark those details onto a graph…

10. Type 1: Linear/linear 1 -1 3
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

11. This more rigorous approach
Type 1: Linear/linear
Note! -> Type 1 rational functions never cross their asymptotes. This can help identifying the curves.
This more rigorous approach
will help when studying the asymptotes of more complicated functions.
It’s not necessary here.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
As x-> inf. , does the curve approach the horizontal
asymptote from above or below?

12. A: Recall that the partial fractions form is:
Type 1: Linear/linear
A: Recall that the partial fractions form is:
If x is greater than 3, f(x) > 1.
By the same reason, the curve approaches the
asymptote from below for x -> - inf.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
We are now ready to give the full sketch of the curve.

13. Type 1: Linear/linear
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

14. Type 2: Quadratic/linear
Find the points where the curve crosses the axes, and identify the vertical asymptote.
A: (0,-1), (1,0), (-3/2,0). The vertical asymptote
is x=-3.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Finding the vertical asymptotes is trivial for rational functions; they lie at the roots of the denominator.
Find the horizontal asymptotes and oblique asymptotes sometimes requires more thought.

15. Type 2: Quadratic/linear
Of course, y is “about” 2x for large x, but
we can make a better linear approximation.
There are two ways to prove it.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
First, we will use a binomial expansion.

16. Type 2: Quadratic/linear
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
According to this analysis y=2x-5 is an oblique asymptote.
Now we will use a different method.

17. Type 2: Quadratic/linear
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
We have again obtained the same oblique asymptote.
I hope you notice that polynomial division and partial
fractions is the most natural way to analyse asymptotic
behaviour. It also makes it easier to find stationary points.

18. Type 2: Quadratic/linear
It is easier to differentiate the function in partial fractions form than the original polynomial form (needs quotient rule).
Here you should find two stationary points have x-coordinates
Task:
Use the information we have gathered to
sketch the “skeleton” of the graph.
Also deduce the behaviour of the graph near the
asymptotes to sketch the complete curve. Don’t
forget to identify the stationary points and their
nature.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Intercepts: (0,-1), (1,0), (-3/2,0). The vertical asymptote
is x=-3. The oblique asymptote is y=2x-5. Stat. points as above.

19. Type 2: Quadratic/linear
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

20. Type 3: Quadratic/Quadratic
Find the points where the curve crosses the axes, and identify the vertical and horizontal asymptotes.
A: (0,-5/6), (5,0), (-2,0). The vertical asymptotes
are x=3, x=4. The horizontal asymptote is at y=1.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Note! We have a horizontal asymptote, not an oblique asymptote, just because the degree of the denominator and numerator are the same.
Q: What’s the easy way to find the stationary points?

21. Type 3: Quadratic/quadratic
A: Differentiation can be much simpler if we express
the function in partial fractions form:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Often you won’t be asked to do this.
Using this technique to find the range of f(x) is
often a troublesome and slow process. Is there another way?

22. Type 3: Quadratic/quadratic
Which values of y correspond to values of x?
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

23. Type 3: Quadratic/quadratic
Does the curve cross any of its asymptotes?
A: Yes. The type 3 curves will usually cross the
horizontal asymptote once. So, you should always
find out where this point is:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
So, at (5.5,1) the curve crosses the asymptote.

24. Type 3: Quadratic/quadratic
An accurate sketch can be constructed now. Let me
summarize all the information for you:
Axis crossings at (5,0), (-2,0), (0,-5/6).
Vertical asymptotes at x = 3, x = 4
Horizontal asymptote at y = 1.
Curve crosses asymptote at x = 11/2.
Turning points at
Stationary values corresponding are:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Draw the curve and mark on all the key features.

25. Type 3: Quadratic/quadratic
Large
Scale
view
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

26. Type 3: Quadratic/quadratic
Small
Scale
view
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

27. Type 3: Quadratic/quadratic
Bear in mind:
A type 3 (quadratic/quadratic) rational function normally has two vertical asymptotes; the asymptotic behaviour is usually opposite on each side of each asymptote (+inf vs. –inf.).
However if the roots of the denominator are repeated, these two asymptotes merge and so this rule is no longer obeyed.
Also, if the denominator does not have real roots, then there are no vertical asymptotes at all.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.