# Rational Functions

A function is called rational function if it can be expressed as $$f(x)=\frac { p(x) }{ q(x) }$$ where P and Q are functions of the variable x.

This section is focussing on how to sketch the graphs of rational functions.

1.Vertical Asymptote

Consider a function of the form $$f(x)=\frac { 1 }{ x-c }$$

The graph of the function f(x) has a vertical asymptote  when the denominator function takes the value 0.

The vertical asymptote of $$f(x)=\frac { 1 }{ x-c }$$ is at some values of x where the denominator (x-c)=0.

Therefore This function f(x) has vertical asymptote at x = c.

Example:

Consider the function$$f(x)=\frac { 1 }{ x-3 }$$

This function has vertical asymptote where the denominator x – 3 = 0.

Therefore the function f(x) has vertical asymptote at x = 3 and the equation of vertical asymptote is x = 3.

From the interactive diagram given below, use the slider to change the values of ‘c’  in the function $$f(x)=\frac { 1 }{ x-c }$$

2. Horizontal Asymptote

Consider a rational function of the form $$f(x)=a+\frac { b }{ x-c }$$

The graph of f(x) has horizontal asymptote at y = a and the equation of horizontal asymptote is y = a.

Also as seen the the prevision section the graph f(x) has a vertical asymptote at x =c;

Example:

Consider a rational function $$f(x)=3+\frac { b }{ x-2 }$$

This function has vertical asymptote x = 2 and the horizontal asymptote y = 3

From the interactive diagram given below, use the slider to change the values of ‘ a and c’  in the function$$f(x)=a+\frac { b }{ x-c }$$

From the interactive diagram given below, use the slider to change the values of ‘ a ,b and c’  in the function$$f(x)=a+\frac { b }{ x-c }$$

3. Oblique ( Slant ) Asymptote

Consider a rational function of the form $$f(x)=(ax+b)+\frac { 1 }{ x-c }$$

The graph of f(x) has slant line as asymptote. The equation of oblique asymptote is y = ax+b, also the function has a vertical asymptote at x = c;

Example:

Consider a rational function $$f(x)=x+\frac { 1 }{ x-3 }$$

This function has vertical asymptote x = 3 and a oblique asymptote y = x

Practice question:1

Rewrite the function $$f(x)=\frac { x^2 + 4x + 2 }{ x + 4 }$$ in the form of $$f(x)=(ax+b)+\frac { 1 }{ x – c }$$, hence identify the asymptotes.

Solution:

Using long division the function $$f(x)=\frac { x^2 + 4x + 2 }{ x + 4 }$$  can be written as $$f(x)= x+\frac { 2 }{ x + 4 }$$.

Hence the function f(x) has vertical asymptote x = -4 and a oblique asymptote y = x.

The graph of f(x) is shown below.

3. Curved Asymptote

Consider a rational function of the form $$f(x)=(ax+b)+\frac { 1 }{ x-c }$$

The graph of f(x) has slant line as asymptote. The equation of oblique asymptote is y = ax+b, also the function has a vertical asymptote at x = c;

Example:

Consider a rational function $$f(x)=x+\frac { 1 }{ x-3 }$$

This function has vertical asymptote x = 3 and a oblique asymptote y = x

Practice question:1

Rewrite the function $$f(x)=\frac { x^2 + 4x + 2 }{ x + 4 }$$ in the form of $$f(x)=(ax+b)+\frac { 1 }{ x – c }$$, hence identify the asymptotes.

Solution:

Using long division the function $$f(x)=\frac { x^2 + 4x + 2 }{ x + 4 }$$  can be written as $$f(x)= x+\frac { 2 }{ x + 4 }$$.

Hence the function f(x) has vertical asymptote x = -4 and a oblique asymptote y = x.

The graph of f(x) is shown below.