Rational Functions

Rational functions and the properties of their graphs such as domain, vertical and horizontal asymptotes, x and y intercepts are explored using an applet. The investigation of these functions is carried out by changing parameters included in the formula of the function. Each parameter can be changed continuously which allows a better understanding of the properties of the graphs of these functions. Once you finish with the present tutorial, you may want to another tutorial on rational functions to further explore the properties of these functions.

Definition and Domain of Rational Functions

A rational function is defined as the quotient of two polynomial functions.

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

Here are some examples of rational functions:

  • g(x) = (x2 + 1) / (x - 1)
  • h(x) = (2x + 1) / (x + 3)

The rational functions to explored in this tutorial are of the form

f(x) = (ax + b)/(cx + d)


where a, b, c and d are parameters that may be changed, using sliders, to understand their effects on the properties of the graphs of rational functions defined above.

Example: Find the domain of each function given below.

  1. g(x) = (x - 1) / (x - 2)
  2. h(x) = (x + 2) / x

Solution

  1. For function g to be defined, the denominator x - 2 must be different from zero or x not equal to 2. Hence the domain of g is given by
    (-infinity , 2)U(2,+infinity).

  2. For function h to be defined, the denominator x must be different from zero or x not equal to 0. Hence the domain of h is given by
    (-infinity , 0)U(0,+infinity).

Interactive Tutorial

Two ways to explore rational functions:

1) An HTML5 Applet to Explore Rational Functions and Their Asymptotes

or

2) a java applet
Your browser is completely ignoring the <APPLET> tag!

There is also another HTML5 Applet to Explore Rational Functions and Their Transforms.

  1. Click on the button "click here to start", above, to start the applet and maximize the window obtained.
  2. Set a to 1, b to -1, c to 1 and d to -2 in order to define function g given in part a) of the above example. Check that the graph is discontinuous at x = 2 (no graph at x = 2).
  3. Set a to 1, b to 2, c to 1 and d to 0 in order to define function h given in part b) of the above example. Check that the graph is discontinuous at x = 0 (no graph at x = 0).

Holes in the Graphs of Rational Functions

What if the zeros of the numerator and the denominator of the rational function are equal?

Example
f(x) = (2x + 2) / (x + 1)
= 2(x + 1) / (x + 1)
= 2 , for x not equal to -1.

The graph of function f is a horizontal line with a hole (function not defined) at x = -1.

Interactive Tutorial

  1. Go back to the applet window and set a to 2, b to 2, c to 1 and d to 1. Check that the graph is that of a horizontal line. It is not easy to observe the hole since the discontinuity (hole) in the graph has the dimension of one pixel which is very small to see.
  2. Define another rational function with equal zeros in the numerator and denominator and check that the graph is that of a horizontal line.

Vertical Asymptotes of Rational Functions

Let f(x) = 1/x. f(x) is not defined at x = 0 (division by zero is not allowed). However what is the behavior of the graph "close" to zero?

In the tables below are values of function f as x approaches zero from the right (x >0) and as x approaches zero from the left (x < 0).

table of values for f(x) as x approaches zero from right

We note that as x approaches zero from the right, f(x) takes larger values. Is there a limit to the values of f(x)? No, f(x) increases without bound.

table of values for f(x) as x approaches zero from left

We also note that as x approaches zero from the left, f(x) takes smaller values. Is there a limit to the values of f(x)? No, f(x) decreases without bound. The vertical line x = 0 is called the vertical asymptote and it is given by the zero of the denominator.

Interactive Tutorial

  1. Set parameters a to 0, b to 1, c to 1 and d to 0 (f(x) = 1/x). Observe the behaviour of the graph to the left and to the right of x = 0.
  2. Set parameters a to 0, b to 1, c to 1 and d to different values (0, 1, -1,.... Observe the behaviour of the graph to the left and to the right of x = d.

Horizontal Asymptotes of Rational Functions

Let f(x) = 1/x. What is the behavior of the graph of f as |x| becomes very large?

Tales below show values of f when x becomes very large, and when x becomes very small.

table of values for f(x) as x takes large values


table of values for f(x) as x takes small values

As x takes smaller values or as x takes larger values, f(x) takes values close to zero and the graph approaches the line horizontal line y = 0. This line is called the horizontal asymptote.

Interactive Tutorial

  1. Set parameters a to 0, b to 1, c to 1 and d to 0 (f(x) = 1/x). Observe the behaviour of the graph as x takes large values (right) and as x takes smaller values (left). Note that the graph gets closer to the x axis (y = 0). Zoom in or out if necessary.
  2. Set parameters a to 1, b to 1, c to 1 and d to 2. What is the equation of the horizontal asymptote? Change a and b only, set them to different non zero values and note that the equation of the horizontal asymptote is given by y = a/c.

Exercises: Find Equation Of Rational Function From Graph

Click on "click here to start" below to start the applet and generate graphs of rational functions of the form

f(x) = (ax - b) / (x - c)


Your browser is completely ignoring the <APPLET> tag!


The idea is to find the equation of the function from the graph. Click on the button "new graph" to generate a graph. Use the x intercept, the horizontal and the vertical asymptotes of the graph to find coefficients a, b and c. Use the zooming buttons if necessary. Once you have found a, b and c click on the button "show/hide" to show the answer, coefficients a, b and c.

More on topics related to rational functions

tutorial on rational functions.

Graphs of rational functions

tutorial on graphs of rational functions

self test on graphs of rational functions.



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Updated: 2 April 2013

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