ࡱ> y ~bjbj |{{R+44444HHH8,H2'^&&&&&&&&&&&$)+J&!4^&44&&g&iiiF4&4&&i&iir$T%&pI1R$&'02'%,|,%%&,4%i&&j2', : Center for Academic Success Library-Lower Level h (678) 466-4070 http://adminservices.clayton.edu/cas Mathematics Success Tips Series: Asymptotes  *Warning* The information given here is not given in technical mathematical definitions and is not the complete knowledge that you need. Instead it attempts to give a basic understanding of the concept using non-rigorous terms. An asymptote is a line that a graph gets closer and closer to, but never touches or crosses it. There are three main types of asymptotes that functions may have: horizontal, vertical, or oblique. Using the conditions set forth, you may be able to determine where the asymptotes exist.  Horizontal Asymptotes If the denominator has a higher degree than the numerator, then y=0 is your horizontal asymptote.  EMBED Equation.3  If the degrees of both the numerator and denominator are the same, then the coefficients of the two highest degrees will be where the horizontal asymptote exists.  EMBED Equation.3  If the numerator has a higher degree than the denominator, then there is an undefined asymptote.  EMBED Equation.3  Vertical Asymptotes Set the denominator of your function equal to zero and solve. The value(s) you get will be where the vertical asymptote(s) exist.  EMBED Equation.3  Oblique or Slant Asymptotes Slant asymptotes only exist when the numerator has a higher degree than the denominator. Divide the numerator with the denominator with either long division or synthetic division. The quotient will be your slant asymptote (you can ignore the remainder).  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Therefore, the horizontal asymptote is y=0.  FMicrosoft Equation 3.0 DS Equation Equation.39q  f(x)=3x 2 +x+4x 2 "6x+9The highest degree of the nuObjInfo Equation Native _1390803454FpIpIOle $merator is to the power of 2. The coefficient is 3. The highest degree of the denominator is to the power of 2. The coefficient is 1. Therefore, the horizontal asymptote is y=31 or y=3. FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj%fObjInfo'Equation Native (&_1390802532FpIpI  f(x)=3x 2 +2x"1x"7The highest degree of the numerator is to the power of 3. The highest degree of the denominator is to the power of 1. Therefore, the horizontal asymptote is undefined.   FMicrosoft Equation 3.0 DS Equation Equation.39qOle 5CompObj6fObjInfo8Equation Native 91 f(x)=3x(x-2)(x+5)If we set the denominator to zero, then we would get:(x-2)(x+5) = 0If x-2=0 then x=2If x+5=0 then x=-5Therefore, the vertical asymptotes are at x=2 and x=-5 FMicrosoft Equation 3.0 DS Equation Equation.39q_1390804808FpIpIOle ECompObjFfObjInfoH f(x)=3x 2 +2x"1x"2The highest degree of the numerator is to the power of 2. The highest degree of tEquation Native I1Table,,SummaryInformation(]DocumentSummaryInformation8dhe denominator is to the power of 1.Using synthetic division, find the quotient. 232"1!6163815The quotient is y=3x+8 with a remainder of 15. Therefore, the slant asymptote is y=3x+8. 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