Graph of Rational Functions - Sketching

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:

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

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).

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

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 = a_m / b_n

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.

Answer to Example 1

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.