Horizontal asymptote

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Intros
Lessons
  1. Introduction to Horizontal Asymptote

    • Horizontal Asymptotes define the right-end and left-end behaviors on the graph of a function.

    • 3 cases of horizontal asymptotes in a nutshell…
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Examples
Lessons
  1. Algebraic Analysis on Horizontal Asymptotes

    Let's take an in-depth look at the reasoning behind each case of horizontal asymptotes:

    1. Case 1:

      if: degree of numerator < degree of denominator

      then: horizontal asymptote: y = 0 (x-axis)

      i.e.f(x)=ax3+......bx5+......i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......} → horizontal asymptote: y=0y = 0

    2. Case 2:

      if: degree of numerator = degree of denominator

      then: horizontal asymptote: y = leading  coefficient  of  numeratorleading  coefficient  of  denominator\frac{leading\; coefficient \;of\; numerator}{leading\; coefficient\; of\; denominator}

      i.e.f(x)=ax5+......bx5+......i.e. f(x) = \frac{ax^{5}+......}{bx^{5}+......} → horizontal asymptote: y=aby = \frac{a}{b}

    3. Case 3:

      if: degree of numerator > degree of denominator

      then: horizontal asymptote: NONE

      i.e.f(x)=ax5+......bx3+......i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......} NO  horizontal  asymptote NO\; horizontal\; asymptote

  2. 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)=5x2−13x+6−2x2+3x+2g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}
    3. h(x)=x320x−100h\left( x \right) = \frac{{{x^3}}}{{20x - 100}}
  3. 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)=x−9x+9a(x) = \frac{x - 9}{x + 9}
    2. b(x)=x2−9x2+9b(x) = \frac{x^{2}-9}{x^{2}+9}
    3. c(x)=x2+9x2−9c(x) = \frac{x^{2}+9}{x^{2}-9}
    4. d(x)=x+9x2−9d(x) = \frac{x+9}{x^{2}-9}
    5. e(x)=x+3x2−9e(x) = \frac{x+3}{x^{2}-9}
    6. f(x)=x2+9x+9f(x) = \frac{x^{2}+9}{x+9}
    7. g(x)=−x−9−x2−9g(x) = \frac{-x-9}{-x^{2}-9}
    8. h(x)=−x2−9−x2+9h(x) = \frac{-x^{2}-9}{-x^{2}+9}
    9. i(x)=x2−9x+3i(x) = \frac{x^{2}-9}{x+3}
    10. j(x)=x3−9x2x2−3xj(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}
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Topic Notes
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There are 3 cases to consider when determining horizontal asymptotes:

1) Case 1:

if: degree of numerator < degree of denominator

then: horizontal asymptote: y = 0 (x-axis)

i.e.f(x)=ax3+......bx5+......i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......} → horizontal asymptote: y=0y = 0

2) Case 2:

if: degree of numerator = degree of denominator

then: horizontal asymptote: y = leading  coefficient  of  numeratorleading  coefficient  of  denominator\frac{leading\; coefficient \;of\; numerator}{leading\; coefficient\; of\; denominator}

i.e.f(x)=ax5+......bx5+......i.e. f(x) = \frac{ax^{5}+......}{bx^{5}+......} → horizontal asymptote: y=aby = \frac{a}{b}

3) Case 3:

if: degree of numerator > degree of denominator

then: horizontal asymptote: NONE

i.e.f(x)=ax5+......bx3+......i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......} NO  horizontal  asymptote NO\; horizontal\; asymptote