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

Vertical and Horizontal Asymptotes(This handout is specific to rational functions Px ( )Qx ( )where Px ( ) and Qx ( ) are polynomial functions.)What is an asymptote? An asymptote is a line that the graph of a function approaches.IMPORTANT: The graph of a function may cross a horizontal asymptote any number of times, but thegraph continues to approach the asymptote as the input increases and/or decreases without bound.Example:IMPORTANT NOTE ON HOLES: In order to find asymptotes, functions must FIRST bereduced. Factor both the numerator and denominator to see if they have any factors incommon that can be “cancelled” out. Any factor that “cancels” produces a hole in thegraph. A hole exists at a value of x if that value makes both the numerator anddenominator equal to zero (0).x−2 x−2 1f( x)= ⇒ ⇒2x −4 ( x− 2)( x+ 2) x+2This means that the graph of f( x ) has a hole when x = 2x − 2 1The graphs of f( x)= and y=are exactly the same except for the hole when x = 2 on the2x − 4 x + 2first graph. Graphing calculators will not show the hole in the first graph, but the table will show anerror when x = 2 .Vertical Asymptotes: These vertical lines are written in the form: x= k , where k is a constant.Once a rational function is reduced, vertical asymptotes may be found by setting the denominatorequal to zero (0) and solving for the input variable.Function approachingf( x)2x+ 1Example: f( x)=3x− 6denominator = 0 ⇒ 3x− 6 = 0 ⇒ 3x= 6 ⇒ x=2the line x = 2The graph of f( x ) has a the vertical asymptote x = 2Vertical Asymptotex = 21Example: gx ( ) =2x − 42 2denominator = 0 ⇒ x − 4 = 0 ⇒ x = 4 ⇒ x=±2The graph of gx ( ) has thevertical asymptotes x = − 2 and x = 2gx ( )Function approachingthe line x = 2Vertical Asymptotesx=− 2 & x=2© Chandler-Gilbert Community College

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