Graphing Rational Functions

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

f(x) = \dfrac{g(x)}{h(x)}
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) equal to zero.
In what follows, we assume that g (x) and h (x) have no common factors.
Vertical Asymptotes
Let
f(x) = \dfrac{2}{x-3}

The domain of f is the set of all real numbers except 3, since 3 makes the denominator equal to 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.
vertical asymptote
Notes that
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:
f(x) approaches increases without bound or decreases without bound as x approaches 3
Horizontal Asymptotes
Let
f(x) = \dfrac{2x + 1}{x}
1) Let x increase and find values of f (x).
x 1 10 103 106
f (x) 3 2.1 2.001 2.000001
2) Let x decrease and find values of f (x).
x -1 -10 -103 -106
f(x) 1 1.9 1.999 1.999999
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 behaviour below. horizontal asymptotes
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
f(x)=polynomial(1)/polynomial(2)

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
f(x) = \dfrac{x + 1}{x-1}

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:
(- ∞ , -1) , (-1 , 1) and (1 , + ∞).
We select a test value within each interval and find the sign of f (x).
In (- ∞ , -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 , + ∞) , select 2 and find f (2) = ( 2 + 1) / (2 - 1) = 3 > 0.
Let us put all the information about f in a table.
x

- ∞

-1 1

+ ∞

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.
vertical and horizontal asymptotes

We now start sketching the graph of f starting from the left.
In the interval (-∞ , -1) f (x) is positive hence the graph is above the x axis. Starting from left, we graph 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 graph below.

graph of f, left part


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 graph below.
graph of f, middle part

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 graph below.
graph of f, right part

We now put all "pieces" of the graph of f together to obtain the graph of f.
>graph of f
Matched Problem: Let f be a rational function defined by
f(x) = \dfrac{-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
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