Vertical asymptote

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  1. Introduction to Vertical Asymptotes

    • How to determine vertical asymptotes of a rational function?


    For the rational function: f(x)=(2x+9)(x8)(6x+11)(x)(2x+9)(x+5)(3x7)(6x+11)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

    1. f(x)=52x+10f\left( x \right) = \frac{5}{{2x + 10}}
    2. g(x)=5x213x+62x2+3x+2g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}
    3. h(x)=x320x100h\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

    1. a(x)=x9x+9a(x) = \frac{x - 9}{x + 9}
    2. b(x)=x29x2+9b(x) = \frac{x^{2}-9}{x^{2}+9}
    3. c(x)=x2+9x29c(x) = \frac{x^{2}+9}{x^{2}-9}
    4. d(x)=x+9x29d(x) = \frac{x+9}{x^{2}-9}
    5. e(x)=x+3x29e(x) = \frac{x+3}{x^{2}-9}
    6. f(x)=x2+9x+9f(x) = \frac{x^{2}+9}{x+9}
    7. g(x)=x9x29g(x) = \frac{-x-9}{-x^{2}-9}
    8. h(x)=x29x2+9h(x) = \frac{-x^{2}-9}{-x^{2}+9}
    9. i(x)=x29x+3i(x) = \frac{x^{2}-9}{x+3}
    10. j(x)=x39x2x23xj(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}
Topic Notes

For a rational function: f(x)=numeratordenominatorf(x) = \frac{numerator}{denominator}

Provided that the numerator and denominator have no factors in common (if there are, we have "points of discontinuity" as discussed in the previous section), vertical asymptotes can be determined as follows:

\bullet equations of vertical asymptotes: x = zeros of the denominator

i.e.f(x)=numeratorx(x+5)(3x7)i.e. f(x) = \frac{numerator}{x(x+5)(3x-7)}; vertical asymptotes: x=0,x=5,x=75x = 0, x = -5, x = \frac{7}{5}