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.