Vertical and Horizontal Asymptotes - Chandler-Gilbert Community ...

Horizontal Asymptotes: These horizontal lines are written in the form: y = k , where k is a constant.Because functions approach horizontal asymptotes for very large positive or negative input values,ONLY the terms with the highest degree (largest exponent) in both the numerator and denominatorneed to be considered when finding the horizontal asymptote. The resulting ratio (fraction) shouldthen be reduced by “cancelling” common factors.Case Example GraphCase 1: If the result has novariables in the denominator,the function has nohorizontal asymptote.2x + 1ax ( ) =x − 32 2x + 1 x xax ( ) = ⇒ =2x−6 2x2No horizontal asymptoteax ( )No HorizontalAsymptoteCase 2: If the result isa number, thehorizontal asymptote is1−6xbx ( ) =3x+ 71−6x−6x⇒ =−2 ⇒ y =−23x+9 3xy = that number. The horizontal asymptote is y = − 2bx ( )Function approachingthe line y = −2Horizontal Asymptotey = −2Case 3: If the result has novariables in the numerator,the horizontal asymptote is1+2xcx ( ) =233x− 42 21+2x2x2cx ( ) = ⇒ = ⇒ y=03 33x − 4 3x 3xy = 0 . The horizontal asymptote is y = 0cx ( )Function approachingthe line y = 0Horizontal Asymptotey = 0Final Note:There are other types of functions that have vertical and horizontal asymptotes not discussed in thishandout.There are other types of straight-line asymptotes called oblique or slant asymptotes.There are other asymptotes that are not straight lines.Example graphs of other functions with asymptotes:© Chandler-Gilbert Community College