Curve sketching is an important part of the A Level JC H2 Math syllabus. The knowledge of curve sketching can be applied to other areas of A Level JC H2 Math such as Calculus. Most students are able to sketch rational functions accurately except for those involving improper fractions.

Using our JC H2 Maths methodology, most students  are able to sketch rational functions involving improper fractions easily.

In this post, we will discuss how we can determine the oblique asymptote when there are curve sketching involving improper fractions.

For improper fractions, whereby the degree of the polynomial of the numerator is larger than the degree of the polynomial of the denominator, an oblique (or a slant asymptote) will be formed on the graph of the function.

In this blog post, we will explore some of the common misconceptions students have on the sketching of functions with oblique asymptotes.

Let us consider the function 

To find the oblique asymptote, some students use this approach. They divide both the numerator and the denominator by x,  becomes,  . Many students assume that the slant asymptote is a line the graph tends towards as  . Hence students substitute  in the expression  and deduce that y = 2x is the oblique (or slant asymptote).

Do you spot any misconceptions in this answer? Why is this answer incorrect?

Let us explore the validity of the answer.

An oblique (or a slant asymptote) is defined as a line y = ax + b to the graph f at ∞ or –∞ if

Using this definition, we subtract, 2x from .

Since does not tend towards zero when x , y = 2x is not the oblique asymptote of the function  .

What is the correct approach to this question?

To find the oblique asymptote, we need to divide the numerator by the denominator using either long division or synthetic division.

Using the second method, we can see that y = 2x + 2 is the slant asymptote.

Let us subtract 2x + 2 from  to see if the condition  is fulfilled.

Since as x , y = 2x + 2 is the oblique asymptote to the graph of

Since  is the oblique asymptote to the graph of