Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials ad if q( ) is ot the q( ) zero polyomial. The domai of f ( ) cosists of all real umbers for which q( ) 0 is udefied., sice divisio by zero Cosider the ratioal fuctio f ( ). Its graph is below. Notice how the graph looks like it has bee ripped apart ear 0. Not all graphs of ratioal fuctios separate like this. Whe they do we call the lie of separatio a asymptote. http://www.sit.ac.ae/hd/ide.html
Asymptotes There are three major types of asymptotes. Vertical asymptotes which are also called poles Horizotal asymptotes, ad Oblique asymptotes We eed to determie what asymptotes there are for a give ratioal fuctio i order to graph it. Vertical asymptotes of a ratioal fuctio occur at ay values of that make the deomiator equal to zero, provided that all commo factors, other tha costats have bee elimiated. Eample : Fid the vertical asymptotes of the followig fuctios. a. f ( ) There are o factors, other tha, commo to both the umerator ad the deomiator. So, set the deomiator equal to zero ad solve for. So the vertical asymptote is the vertical lie 0. + 6 b. f ( ). To fid out if there are commo factors, I eed to completely factor the ( )( + ) umerator ad deomiator. This produces the followig. f ( ) +. So, there is ot a vertical asymptote for this ratioal fuctio. c. f ( ). Agai, I eed to completely factor both the umerator ad deomiator. 4 This produces the followig. f ( ). So, the umerator ad deomiator have ( )( + ) o commo factors other tha. Hece there are two vertical asymptotes ad they are the vertical lies ±. Horizotal asymptotes of a ratioal fuctio occur as follows. If the degree of the umerator is less tha the degree of the deomiator, the horizotal asymptote is always y 0. If the degree of the umerator ad the deomiator are the same, the the horizotal asymptote is determied by the ratio of the coefficiets of the leadig terms. For a + L eample, if f ( ), where a ad b are the leadig terms of the b + L umerator ad deomiator respectively, the the horizotal asymptote occurs at a y. b If the degree of the umerator is greater tha the degree of the deomiator, o horizotal asymptote occurs. http://www.sit.ac.ae/hd/ide.html
Eample : Fid the horizotal asymptotes of the followig fuctios. a. f ( ). The degree of the umerator is 0 ad the degree of the deomiator is. Hece the horizotal asymptote is y 0. + 6 b. f ( ). The degree of the umerator is ad the degree of the deomiator is. Hece there is o horizotal asymptote. c. f ( ). The degree of the umerator ad deomiator is. Hece the horizotal 4 asymptote is y 7 + 9 d. f ( ). The degree of the umerator ad deomiator is. Hece the + + 0 7 horizotal asymptote is y. 5. A oblique asymptote occurs oly whe the degree of the umerator is greater tha the degree of the deomiator. If the differece of the degrees is oly oe, the oblique asymptote is a lie (ot vertical or horizotal). This is the oly type of oblique asymptote we will discuss. However, please otice that a ratioal fuctio caot have both a horizotal ad oblique asymptote. The oblique asymptote is foud by dividig the ratioal epressio through the process of log divisio. The quotiet obtaied is the lie which makes the oblique asymptote. Eample : Determie the oblique asymptotes of the followig fuctios. a. f ( ). Performig the log divisio we obtai the followig. http://www.sit.ac.ae/hd/ide.html
( 4) + ( ) So the oblique asymptote is y + b. 8 f ( ) Performig the log divisio we obtai the followig. + 4 + 0 + 4 8 + 0 8 + 0 + 0 ( 8 + 0 + ) + 0 So the oblique asymptote is y 8. c. + f ( ) Performig the log divisio we obtai the followig. 5 + 0 5 + + + 0 ( + 0 5) + + 0 So the oblique asymptote is y + ( + 0 50) + 50 The Graph of a Ratioal Fuctio p( ) To graph a ratioal fuctio f ( ), where p ( ) ad q ( ) have o commo factors: q( ). Fid the vertical asymptotes ad sketch them ito the graph with a dotted lie.. Fid the horizotal or the oblique asymptote, if there is oe, ad sketch it ito the graph with a dotted lie.. Fid ad plot the zeros of the fuctio. The zeros are the values of that make the umerator equal to zero. I other words, the zeros of f () are the zeros of p (). 4. Fid ad plot f ( 0). This is the y-itercept of the fuctio. 5. Fid other fuctio values to determie the geeral shape of the fuctio. The draw the graph. http://www.sit.ac.ae/hd/ide.html 4
Note: The graph of a ratioal fuctio ever crosses a vertical asymptote. However, it may or may ot cross a horizotal or oblique asymptote. I additio, the ed behavior of the graph will approach the horizotal or oblique asymptote. Eamples: Graph each of the followig fuctios. a. 8 f ( ) + 4 b. f ( ) http://www.sit.ac.ae/hd/ide.html 5
c. + f ( ) + d. f ( ) + 5 5 4 http://www.sit.ac.ae/hd/ide.html 6