A rational function is any function that can be defined using a rational fraction, this is an algebraic faction which consist of polynomials for both the numerator and the denominator. An asymptote is a line or curve that acts as a limit for another line or curve as it approaches infinity. This mathematical investigation aims to explore the three different types of asymptotes, which are Vertical asymptotes, horizontal asymptotes and oblique asymptotes. They will be explored in four different types of rational function, in the form of ??(??) ??(??) , where ??(??) and ??(??) would be either linear or quadratic. This investigation divided in to three parts: Part A, which explores what vertical asymptotes are how they are found, Part B, which explores what horizontal asymptotes are how they are found and Part C, which explores what oblique asymptotes are and how they are found. The conceptual understanding and skills developed and demonstrated through the following parts of the investigation are applicable in the real-world context which require, for instance, equation representing direct, inverse and joint variation. The process used to solve the problem rely on the use of technology as well as mathematical concepts. Specifically, the Desmos graphing software is used to draw and display the functions, a Casio CG-50 graphics calculator was used to calculate the asymptotes as well verify results. Mathematical techniques and skills including long polynomial divisions, null factor law, algebra and substitution to get the result as needed.