Vertical asymptote

Lessons

• Introduction
  Introduction to Vertical Asymptotes

  • How to determine vertical asymptotes of a rational function?

  • Exercise:

  For the rational function: $f(x) = \frac{(2x+9)(x-8)(6x+11)}{(x)(2x+9)(x+5)(3x-7)(6x+11)}$

  i) Locate the points of discontinuity.

  ii) Find the vertical asymptotes.

---

• 1.
  Graphing Rational Functions

  Sketch each rational function by determining:

  i) vertical asymptote.

  ii) horizontal asymptotes

  a)

  $f\left( x \right) = \frac{5}{{2x + 10}}$
  
  
  b)

  $g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}$
  
  
  c)

  $h\left( x \right) = \frac{{{x^3}}}{{20x - 100}}$
  
  

---

• 2.
  Identifying Characteristics of Rational Functions

  Without sketching the graph, determine the following features for each rational function:

  i) point of discontinuity

  ii) vertical asymptote

  iii) horizontal asymptote

  iv) slant asymptote

  a)

  $a(x) = \frac{x - 9}{x + 9}$
  
  
  b)

  $b(x) = \frac{x^{2}-9}{x^{2}+9}$
  
  
  c)

  $c(x) = \frac{x^{2}+9}{x^{2}-9}$
  
  
  d)

  $d(x) = \frac{x+9}{x^{2}-9}$
  
  
  e)

  $e(x) = \frac{x+3}{x^{2}-9}$
  
  
  f)

  $f(x) = \frac{x^{2}+9}{x+9}$
  
  
  g)

  $g(x) = \frac{-x-9}{-x^{2}-9}$
  
  
  h)

  $h(x) = \frac{-x^{2}-9}{-x^{2}+9}$
  
  
  i)

  $i(x) = \frac{x^{2}-9}{x+3}$
  
  
  j)

  $j(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}$
  
  

---