Vertical, Horizontal, and Slant Asymptotes

Aysmptotes of Rational Functions

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that is not part of a graph of a function but guides it for x-values “far” to the right and/or “far” to the left. The graph may cross it but eventually, for large enough or small enough values of x (approaching ), the graph would get closer and closer to the asymptote without touching it. A horizontal asymptote is a special case of a slant asymptote.

Multiple Choice

What is the equation of the horizontal asymptote of

$$f\left(x\right)=\frac{5}{x-7}$$  

• no aysmptote
  

• y=0
  

• y=5
  
  

• $$y=\frac{5}{7}$$  

Multiple Choice

What is the equation of the horizontal asymptote of

$$f\left(x\right)=\frac{x^5+2x^3+3}{x-4}$$  

• no aysmptote
  

• y=0
  

• y=1
  
  

• $$y=\frac{3}{4}$$  

Multiple Choice

What is the equation of the horizontal asymptote of

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

• no aysmptote
  

• y=0
  

• y=1
  
  

• $$y=\frac{1}{2}$$  

Slant Aymsptotes

Use long or synthetic division

Multiple Choice

Check whether the function has a Horizontal, Slant, or No Asymptote

$$f\left(x\right)=\frac{x^2-3x-28}{x^2-x-20}$$  

• Horizontal

• Slant

• neither

Multiple Choice

Check whether the function has a Horizontal, Slant, or No Asymptote

$$f\left(x\right)=\frac{x^2-81}{4x-36}$$  

• Horizontal

• Slant

• neither

Multiple Choice

Check whether the function has a Horizontal, Slant, or No Asymptote

$$f\left(x\right)=\frac{x^4-21x^3+110}{x^2-5x-66}$$  

• Horizontal

• Slant

• neither

Open Ended

Use long or synthetic division to write the equation of the slant asymptote

$$f\left(x\right)=\frac{x^2-4}{x}$$  

Open Ended

Use long or synthetic division to write the equation of the slant asymptote

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

Open Ended

Use long or synthetic division to write the equation of the slant asymptote

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

Vertical Asymptotes

Finding Vertical Asymptotes

Factor the denominator and solve!

Multiple Choice

Find the vertical asymptotes for $$f\left(x\right)=\frac{x^3+3x^2-4x}{x^3-9x}$$  

• $$x=0,\ x=\pm3$$

• $$x=0,\ x=1,\ x=-4$$

• $$x=1,\ x=-4$$

• $$x=\pm3$$

Multiple Choice

Find the vertical asymptotes for $$f\left(x\right)=\frac{x^3+2x^2-8x}{-2x^2-4x+6}$$

• $$x=\pm3$$

• $$x=1,\ x=-3$$

• $$x=0,\ x=2,\ x=-4$$

• $$x=-4$$