Slant asymptote

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Intros
Lessons
  1. Introduction to slant asymptote

    i) What is a slant asymptote?

    ii) When does a slant asymptote occur?

    iii) Overview: Slant asymptote

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Examples
Lessons
  1. Algebraically Determining the Existence of Slant Asymptotes

    Without sketching the graph of the function, determine whether or not each function has a slant asymptote:

    1. a(x)=x23x10x5a(x) = \frac{x^{2} - 3x - 10}{x - 5}
    2. b(x)=x2x6x5b(x) = \frac{x^{2} - x - 6}{x - 5}
    3. c(x)=5x37x2+10x+1c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}
  2. Determining the Equation of a Slant Asymptote Using Long Division

    Determine the equations of the slant asymptotes for the following functions using long division.

    1. b(x)=x2x6x5b(x) = \frac{x^{2} - x - 6}{x - 5}
    2. p(x)=9x2+x4x38p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}
  3. Determining the Equation of a Slant Asymptote Using Synthetic Division

    Determine the equations of the slant asymptotes for the following functions using long division.

    1. b(x)=x2x6x5b(x) = \frac{x^{2} - x - 6}{x - 5}
    2. f(x)=x2+1x3f(x) = \frac{x^{2} + 1}{x - 3}
  4. Graphing Rational Functions Incorporating All 3 Kinds of Asymptotes

    Sketch the rational function

    f(x)=2x2x6x+2f(x) = \frac{2x^{2} - x - 6}{x + 2}

    by determining:

    i) points of discontinuity

    ii) vertical asymptotes

    iii) horizontal asymptotes

    iv) slant asymptote

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    Topic Notes
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    When the polynomial in the numerator is exactly one degree higher than the polynomial in the denominator, there is a slant asymptote in the rational function. To determine the slant asymptote, we need to perform long division.

    For a simplified rational function, when the numerator is exactly one degree higher than the denominator, the rational function has a slant asymptote. To determine the equation of a slant asymptote, we perform long division.