Basic Rational Functions and Asymptotes

Rational Functions

Topic 4-2

Rational Functions

A Rational function is any function that looks like

 $r\left(x\right)\ =\ \frac{p\left(x\right)}{q\left(x\right)\ }$  where q(x) ≠ 0

Multiple Choice

What are the asymptotes for the following rational function?

$$f\left(x\right)\ =\ \frac{6x}{2x+1}$$  ?

• $$y\ =3,\ x\ =\ \frac{1}{2}$$  

• $$y=-\frac{1}{2},\ x\ =\ 3$$  

• $$y\ =3,\ x\ =-\frac{1}{2}$$  

• $$y=\frac{1}{2},\ x\ =\ 3$$  y=1/2

Multiple Choice

Rewrite the following rational expression to identify the asymptotes.

$$f\left(x\right)\ =\ \frac{2x}{x+4}$$  

• $$y=2,\ x\ =-4$$  

• $$y=-4,\ \ x\ =2$$  

• $$y=2,\ x=4$$  

• $$y=4,\ x=2$$  

Finding Vertical Asymptotes

To find the vertical asymptotes, factor the denominator to find the possible asymptotes

Use the zero product property to identify possible x-values for vertical asymptotes

Graph the function to find the vertical asymptotes

Finding Vertical Asymptotes

 $f\left(x\right)=\frac{3x-2}{x^2+7x+12}$  
Set denominator equal to zero
 $\ x^2+7x+12=0$  
 $\left(x+4\right)\left(x+3\right)=0$  
Possible asymptotes $x=-3,\ x=-4$  
Graph to find asymptotes

Multiple Choice

Find the Vertical Asymptotes for the following function

$$g\left(x\right)\ =\frac{2x^2+x-9}{x^2-2x-8}$$  

• $$x=-4,\ x=2$$  

• $$x=-2,\ x=4$$  

• $$x=-6,\ x\ =\ -2$$  

• $$x=\ 2,\ x=6$$  

Finding Horizontal Asymptotes

First, identify if the degree of the nominator is less than the degree of the denominator. If so, there is one asymptote and it is at:

 $y=0$  

Finding Horizontal Asymptotes

 $f\left(x\right)=\frac{x^2+1}{x+2}$  

Second, if the degree of the nominator is greater than the degree of the denominator there is no horizontal asymptote. Example: The nominator increasing faster than the denominator  so there are no horizonal asymptotes.

Finding Horizontal Asymptotes

 $f\left(x\right)=\frac{2x^2+x+1}{x^2-1}$  

The degrees of the variable in the nominator and the denominator are the same, dividing the denominator into the numerator we find that the ratio of the leading coefficients is the horizontal asymptote.

The ratio of the leading coefficients is 2. the horizontal asymptote is:
 $y=2$   
If the leading coefficients ratio is equal to one, the horizontal asymptote is:
 $y=0$  

Multiple Choice

What is the horizontal asymptote of the following function?

$$f\left(x\right)=\frac{3x^2}{4x^2-1}$$  

• $$y=\frac{3}{4}$$  

• $$y=\frac{4}{3}$$  

• $$y=0$$  

• No asymptote

Multiple Choice

What is the horizontal asymptote for the following function?

$$f\left(x\right)=\frac{4x+3}{x^2-4}$$  

• $$y=4$$  

• $$y=\frac{1}{4}$$  

• $$y=0$$  

• No Asymptote

Multiple Choice

What is the horizontal asymptote for the following function?

$$f\left(x\right)=\frac{5x^3+6}{x^2-9x+19}$$  

• $$y=5$$  

• $$y=\frac{1}{5}$$  

• $$y=0$$  

• No Asymptotes