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.

We see how to sketch different types of rational funcitons of the form \(f(x)=\frac { Linear }{ Linear } \) ; \(f(x)=\frac { Linear }{ Quadratic } \) ;  \(f(x)=\frac { Quadratic}{ Linear } \) ;  \(f(x)=\frac { Quadratic}{ Quadratic } \) ; \(f(x)=\frac { Cubical }{ Linear} \).

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.

 

 

Practice question: 2

Rewrite the function \(f(x)=\frac {-x^2 + 4x – 7 }{ x – 1} \) in the form of \(f(x)=(ax+b)+\frac { 1 }{ x – c } \), hence identify the asymptotes.

Solution:

Using long division the function \(f(x)=\frac { -x^2 + 4x – 7 }{ x – 1 } \)  can be written as \(f(x)= -x+3+\frac { -4 }{ x – 1 } \).

Hence the function f(x) has vertical asymptote x = 1and a oblique asymptote y = -x+3

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:

Sketch the graph of the rational funciton \(f(x)=\frac { 3x^3+1}{ x } \)

 The rational function \(f(x)=\frac { 3x^3+1}{ x } \) can be expressed using long division as \(f(x)=3x^2+\frac { 1 }{ x } \)

From the above expression its ovserved that the curved aysmptote is\(f(x)=3x^2 \) and the vertical asymptote is x = 0 whichs is y – axis.