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Rational Function

4.1 Introduction: A rational function is a division of two polynomial functions. Or we can say that it is a polynomial divided by another polynomial.

In the case of one variable, x, a rational function is a function of the form

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f(x) = P (x)/ Q(x)

Where P and, Q are polynomial functions in x and Q is not the zero polynomial. The condition for a rational function to exist is that the domain of f is the set of all points x for which the denominator Q(x) is not zero.


f( x) = x2 + x – 20,

x2 – x -12

The polynomials in the numerator and the denominator of the above function would factor like this:

f(x) = (x + 5) (x – 4),

(x- 4) (x + 3)

Chapter 3: Rational Functions – Assignment

The objective of the project is to demonstrate your knowledge and understanding of the chapter key concepts of rational functions by creating your own rational function and graphing it BY HAND. Your function should meet the following criteria:

  • Numerator is a polynomial function with at least 1 x-intercept
  • Denominator is a polynomial function with at least 1 x-intercept
  • it has at least 1 vertical asymptote, and does NOT have a hole in the graph
  • It has a horizontal asymptote
  • Make sure you show your work in identifying key attributes listed below, and that your work is your own.
  • For these attributes show your work! For example, solve for x and y intercepts, and show your calculations for testing behaviours of the graph. Use N/A for any sections that do not apply for your graph.

Attributes of the graph:


Show solving work !

Vertical Asymptote(s)

Show how you know!


Show solving work!

Horizontal Asymptote

What rule did you use?


Use Interval Notation


Use Interval Notation

End Behaviour

Calculations for EB

Conclusions about EB:

As x → -∞,

As x → ∞,

Behaviour around

Vertical Asymptote(s)



Identify the interval(s) where the function is positive

Identify the interval(x) where the SLOPE is INCreasing

Identify the interval(s) where the function is negative

Identify the interval(x) where the SLOPE is DECreasing

Rational Functions Assignment D

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