Graph of Rational Functions - Sketching

How to graph a rational function? A step by step tutorial. The properties such as domain, vertical and horizontal asymptotes of a rational function are also investigated. Free graph paper is available.

Definition

A rational function f has the form

where g (x) and h (x) are polynomial functions.

The domain of f is the set of all real numbers except the values of x that make the denominator h (x) zero.

In what follows, we assume that g (x) and h (x) have no common factors.

Vertical Asymptotes

Let

The domain of f is the set of all real numbers except 3, since 3 makes the denominator zero and the division by zero is not allowed in mathematics. However we can try to find out how does the graph of f behave close to 3.

let us evaluate function f at values of x close to 3 such that x < 3. The values are shown in the table below:

 x 1 2 2.5 2.8 2.9 2.99 2.999 2.99999 f (x) -1 -2 -4 -10 -20 -200 -2000 -2*105

Let us now evaluate f at values of x close to 3 such that x > 3.

 x 5 4 3.5 3.2 3.1 3.01 3.001 3.00001 f (x) 1 2 4 10 20 200 2000 2*105

The graph of f is shown below.

Notes

1 - As x approaches 3 from the left or by values smaller than 3, f (x) decreases without bound.

2 - As x approaches 3 from the right or by values larger than 3, f (x) increases without bound.

We say that the line x = 3, broken line, is the vertical asymptote for the graph of f.

In general, the line x = a is a vertical asymptote for the graph of f if f (x) either increases or decreases without bound as x approaches a from the right or from the left. This is symbolically written as:

Horizontal Asymptotes

Let

1 - Let x increase and find values of f (x).

 x 1 10 103 106 f (x) 3 2.1 2.001 2

2 - Let x decrease and find values of f (x).

 x -1 -10 -103 -106 f (x) 1 1.9 1.999 2

As | x | increases, the numerator is dominated by the term 2x and the numerator has only one term x. Therefore f(x) takes values close to 2x / x = 2. See graphical behavior below.

In general, the line y = b is a horizontal asymptote for the graph of f if f (x) approaches a constant b as x increases or decreases without bound.

How to find the horizontal asymptote?

Let f be a rational function defined as follows

Theorem

m is the degree of the polynomial in the numerator and n is the degree of the polynomial in the numerator.

case 1: For m < n , the horizontal asymptote is the line y = 0.

case 2: For m = n , the horizontal asymptote is the line y = am / bn

case 3: For m > n , there is no horizontal asymptote.

Example 1: Let f be a rational function defined by

a - Find the domain of f.

b - Find the x and y intercepts of the graph of f.

c - Find the vertical and horizontal asymptotes for the graph of f if there are any.

d - Use your answers to parts a, b and c above to sketch the graph of function f.

a - The domain of f is the set of all real numbers except x = 1, since this value of x makes the denominator zero.

b - The x intercept is found by solving f (x) = 0 or x+1 = 0. The x intercept is at the point (-1 , 0).

The y intercept is at the point (0 , f(0)) = (0 , -1).

c - The vertical asymptote is given by the zero of the denominator x = 1.

The degree of the numerator is 1 and the degree of the denominator is 1. They are equal and according to the theorem above, the horizontal asymptote is the line y = 1 / 1 = 1

e - Although parts a, b and c give some important information about the graph of f, we still need to construct a sign table for function f in order to be able to sketch with ease.

The sign of f (x) changes at the zeros of the numerator and denominator. To find the sign table, we proceed as in solving rational inequalities. The zeros of the numerator and denominator which are -1 and 1 divides the real number line into 3 intervals:

(- infinity , -1) , (-1 , 1) , (1 , + infinity).

We select a test value within each interval and find the sign of f (x).

In (- infinity , -1) , select -2 and find f (-2) = ( -2 + 1) / (-2 - 1) = 1 / 3 > 0.

In (-1 , 1) , select 0 and find f(0) = -1 < 0.

In (1 , + infinity) , select 2 and find f (2) = ( 2 + 1) / (2 - 1) = 3 > 0.

Let us put all the information about f in a table.

 x - inf -1 1 + inf f (x) + 0 x-intercepts - V.A +

In the table above V.A means vertical asymptote.

To sketch the graph of f, we start by sketching the x and y intercepts and the vertical and horizontal asymptotes in broken lines. See sketch below.

We now start sketching the graph of f starting from the left.

In the interval (-inf , -1) f (x) is positive hence the graph is above the x axis. Starting from left, we sketch f taking into account the fact that y = 1 is a horizontal asymptote: The graph of f is close to this line on the left. See sketch below.

Between -1 and 1 f (x) is negative, hence the graph of f is below the x axis. (0 , -1) is a y intercept and x =1 is a vertical asymptote: as x approaches 1 from left f (x) deceases without bound because f (x) < 0 in( -1 , 1). See sketch below.

For x > 1, f (x) > 0 hence the graph is above the x axis. As x approaches 1 from the right, the graph of f increases without bound ( f(x) >0 ). Also as x increases, the graph of f approaches y = 1 the horizontal asymptote. See sketch below.

We now put all "pieces" of the graph of f together to obtain the graph of f.

Matched Problem: Let f be a rational function defined by

f (x) = (-x + 2) / (x + 4)

a - Find the domain of f.

b - Find the x and y intercepts of the graph of f.

c - Find the vertical and horizontal asymptotes for the graph of f if there are any.

d - Use your answers to parts a, b and c above to sketch the graph of function f.

More references on graphing and rational functions.

• Graphing Functions

• Rational Functions - Applet

• Solver to Analyze and Graph a Rational Function