In mathematics, there are many branches of functions. You have Inverse, Linear, Quadratic, Cubic, Periodic, Monotonic, etc. In this paper we will cover Reciprocal and Rational Functions. Both can be plotted on a graph, both have asymptotes, and both have discontinuities. But just because they have a few similarities doesn’t mean that they are all the same. In this paper we will compare and contrast the two functions.

A rational function has the definition of, “Any function which can be written as the ratio of two polynomial functions.” Any time that the denominator is undefined, that means the function is undefined. Since a rational function is set up in a fraction, if the same term is in the numerator and the denominator then those two will “cancel” each other out. When two terms cancel each other out, a hole will occur at the point that both terms cancelled out at. The vertical asymptote is found by the term in the denominator. If you only have (x-2) in the denominator then you will have a vertical asymptote at 2 on the x axis, going vertical. It will be 2 because in the function the vertical asymptote is automatically given a negative sign, so the opposite of whatever your term is. The horizontal is found by any number outside of the fraction.

A reciprocal function can be defined as, “A function that models inverse variations.” Reciprocal functions can be graphed. They usually start at the top or sides of a graph; go inwards towards the origin, then bend past it and leave through the same quadrant that they entered. The x-axis is the horizontal asymptote, and the y-axis is the vertical asymptote. The vertical asymptote is found the same way as the rational function asymptote. As is the horizontal asymptote.

For both the Reciprocal and Rational Functions, you find the asymptotes the same way. You use the same formula, which is y=a/-x-h+k. The vertical asymptote is the opposite value because of the negative sign in the equation. The...

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