Finding Asymptotes KEY
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1 Unit: 0 Lesson: 0 Discontinuities Rational functions of the form f ( are undefined at values of that make 0. Wherever a rational function is undefined, a break occurs in its graph. Each such break is called a discontinuity. For rational functions, there are two main types: A function of the form f ( will have A function of the form f ( will have a vertical asymptote at a if a removable discontinuity at b if a is a zero of (but NOT ). b is a zero of both and. For the rational functions given, Factor (if possible) to identify more easily the zeros of the numerator and the denominator. Then, list the values of at which the function is undefined, and use these to describe the function s domain. Finally, decide which of these values provide locations of vertical asymptotes (VA), and which denote removable discontinuities (RD). ) 5 f ( ) f ( + 5 Undefined at: -5 Undefined at: ½ Domain: (, -5) (-5, ) Domain: (,½) (½, ) VA: -5 VA: ½ RD: None RD: None ) f ( 4) 6 ( 6) Factored: f ( ( 6) f ( Factored: ( 6)( 4) f ( ( )( ) Undefined at: 0, 6 Undefined at: -, Domain: (, 0) (0, 6) (6, ) Domain: (, -) (-, ) (, ) VA: 0 VA: -, RD: 6 RD: None 0, TESCCC 0/09/ page of 6
2 Unit: 0 Lesson: 0 5) 6 f ( 6) + ( ) Factored: f ( ( 4)( ) 9 f ( Factored: f ( ( )( ) ( 4)( ) Undefined at: -4, Undefined at:, 4 Domain: (, -4) (-4, ) (, ) Domain: (, ) (, 4) (4, ) VA: -4 VA: 4 RD: RD: Limits A rational function f ( will have a horizontal asymptote if f ( approaches a limit as -values approach infinity ( ) or negative infinity ( ). One method for approimating a limit is to create an end behavior table for a function, using etreme positive and negative values for. ) f ( ) f ( 9) + f ( f( f( f( Does a limit seem to eist? Does a limit seem to eist? Does a limit seem to eist? Yes Yes No If so, name the horizontal y 0.5 If so, name the horizontal y 0 If so, name the horizontal None 0, TESCCC 0/09/ page of 6
3 Unit: 0 Lesson: 0 There are other shortcut methods for determining if (and where) rational functions have horizontal asymptotes. Here, for functions of the form f (, consider the degrees of the numerator,, and the denominator,. If the degree of is If the degree of is If the degree of is Less than Equal to Greater than the degree of, then the degree of, then the degree of, then the function will have a the function will have a the function will have horizontal asymptote at y 0 horizontal asymptote given by The ratio of the leading coefficients of and No horizontal asymptote Compare the degrees of the polynomials ( and ) in each numerator and denominator using <,, or >. Then use the rules above quickly to determine the horizontal asymptote of each function (if one eists). 0) 4 f ( ) + 4 f ( ) + f 4 + ( of of of < of of < of y 4/ y 0 y 0 ) 5 9 f ( 4) f ( 5) 5 6 f + ( of > of of of of > of None y 5/- -5 None 0, TESCCC 0/09/ page of 6
4 Unit: 0 Lesson: 0 Slant For functions of the form f (, if the degree of is one more than the degree of, the function can have a slant asymptote (sometimes called oblique asymptotes). A slant asymptote is a linear asymptote that is neither vertical nor horizontal. The best method for determining the linear equation for a slant asymptote is to find the quotient using polynomial long division. When a slant asymptote is present, then r ( f ( can be written as q ( + (where q( is the quotient and r( is the remainder), and the quotient, y q(, is the slant asymptote. Eample: f ( NOTE: one degree higher ( 4 + ( + 6) Quotient r ( Use polynomial long division to write each rational function in the form f ( q( +. Then identify the slant asymptote ) f ( ) f ( ) f ( 6 f( (½) + + f( 5 +, or 0 f( + + f( (½) Slant y 5 Slant y (½) + Slant y + 0, TESCCC 0/09/ page 4 of 6
5 Unit: 0 Lesson: 0 End Behavior f ( Consider the rational functions where the numerator is or more degrees greater than that of the denominator. Sometimes, these rational functions look almost identical to other functions (ecept near the middle of the graphs, where the vertical asymptotes occur). f ( [-6, 6] by [-0, 0] For eample, the rational function shown appears to be almost parabolic (ecept, of course, for the vertical asymptote at -). In these cases, we say the rational function has an end behavior asymptote. [-0, 0] by [-40, 0] One method for determining the equation for an end behavior asymptote is to find the quotient from polynomial long division. When an end behavior asymptote is present, then r ( f ( can be written as q ( + (where q( is the quotient and r( is the remainder), and the quotient, y q(, is the end behavior asymptote. Eample: f ( ( + ) ( + ( + ) NOTE: Deg Deg Quotient two degrees higher f ( Quotient + + End behavior asymptote is y Quotient coefficients - - NOTE: Sometimes, you can use synthetic division! 9) Under what circumstances can a synthetic process be used in place of long division? Synthetic substitution/division can only be used when the divisor (denominator) is of the form ( a) 0, TESCCC 0/09/ page 5 of 6
6 Unit: 0 Lesson: 0 Use synthetic substitution to rewrite each rational r ( function in the form f ( q( + (where q( is the quotient and r( is the remainder). 0) f ( Then, with the aid of a calculator, sketch the graphs of both the rational function and the end behavior asymptote in the space provided Rewrite: f( + Graph (sketch): ) f ( ) f ( Rewrite: Rewrite: f( f( Graph (sketch): Graph (sketch): 0, TESCCC 0/09/ page 6 of 6
Domain: The domain of f is all real numbers except those values for which Q(x) =0.
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