12 Rational Functions

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Kelley Griffith
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1 Funtions Conepts: The Definition of a Funtion Identifing Funtions Finding the Domain of a Funtion The Big-Little Priniple Vertial and Horizontal Asmptotes The Graphs of Funtions (Setion.) Definition. A rational funtion is a funtion that is equivalent to a funtion of the following form. r() = polnomial polnomial Eample. Whih of the following are rational funtions? If the funtion is rational, find its domain. f() = f() is defined if 3, so the domain of f() is (, 3) (3, ) g() = + + Not rational sine the numerator is not a polnomial. h() = +3 h() is defined for all eept 0, so the domain of h() is (, 0) (0, ). j() = + + = + ( ) + = ++ = + j() is defined if, so the domain of j() is (, ) (, ). k() = + += + + k() is defined for all so its domain is (, ). Note.3 Polnomials are a speial ase of rational funtions.

2 . Graphs of Some Simple Funtions Let f() =. Below is a hart of the basi shapes that the graph of f() antakeon. n n odd n even Eamples: f() =, g() = 7 Eample: f() =, h() = 0 Propert. (The Big-Little Priniple) If is a number that is lose to 0 on the number line, then is a number that is far from 0 on the number line. If is a number that is far from 0 on the number line, then is a number that is lose to 0 on the number line. How should ou remember the Big-Little Priniple? If is big then is little and if is big then is little. Can ou use the Big-Little Priniple to eplain the shape of the graph of = f() =? n How do ou desribe the end behavior of the graph of = f() =? n If is big (in either the positive or negative sense) then is little so is also little. As n gets bigger, gets littler - that is, it gets loser to zero. So, the end behavior of f() is n 0as 0as

3 Man (though not all) of the graphs of rational funtions have asmptotes. Intuitivel, if a graph approahes another graph and eventuall gets as lose to that other graph as anone ould possibl hope without neessaril touhing the other graph, then the other graph is alled an asmptote for the original graph. Two tpes of asmptotes that often our in the graphs of rational funtions are horizontal asmptotes (asmptotes that are horizontal lines) and vertial asmptotes (asmptotes that are vertial lines). You an determine a rational funtion s horizontal asmptotes b onsidering the leading terms of the numerator and denominator. (This is onept is ver similar to the end behavior of a polnomial.) You an determine a rational funtion s vertial asmptotes b finding the values whih are zeros of the denominator but are not zeros of the numerator. Eample. Desribe the horizontal asmptotes of the graph of = g() = The leading term of the numerator is 3 and the leading term of the denominator is 3. The ratio of the leading terms is 3 3 = 3.Thus,g() hasthesameendbehavioras whih is 0as and 0as So, =0isahorizontalasmptote. Eample. Desribe the horizontal and vertial asmptotes of the graph of = f() = + 3. The leading term of the numerator is and the leading term of the denominator is. Theratio of the leading terms is =. Thus,f() hasthesameendbehavioras =whihis as and as So, =isahorizontalasmptote. =3isazeroofthedenominatorbutnotthenumerator. So =3isavertialasmptoteoff(). Eample.7 Desribe the horizontal and vertial asmptotes of the graph of = g() = The leading term of the numerator is and the leading term of the denominator is. Theratio of the leading terms is =. Thus,g() hasthesameendbehavioras = whih is as and as So, g()doesnothaveahorizontalasmptote. Notiethatthenumerator ++ = (+3)(+). So, = isazeroofthedenominatorbutnotthenumerator.so = isavertialasmptote of g(). 3

4 . The Graphs of Funtions - Some Eamples We have used a graphing alulator (TI-3 Plus) to approimate the graphs of a few rational funtions. Graphing alulators do not do a ver good job of skething asmptotes. Nevertheless, we an use them to better understand the graphs of rational funtions. For eah graph, look at the algebrai desription of the funtion and the approimate graph to better understand its asmptotes. Show algebraiall how ou would find the asmptotes of eah graph. Draw a better sketh of the graph that inludes all asmptotes. Make sure that asmptotes are drawn with dotted lines. Note: Eah graph is in a [, 0] [, 0] viewing window. The graph of = f() =. The leading term of the numerator is and the leading term of the denominator is. The ratio of the leading terms is =3. Thus,f() has the same end behavior as = 3 whih is 3as and as So, = 3 is a horizontal asmptote of f(). Notie that = is a zero of the denominator but not the numerator. So = + The graph of = f() = + +. is a vertial asmptote of f(). The leading term of the numerator is and the leading term of the denominator is. The ratio of the leading terms is =. Thus, f() has the same end behavior as = whih is 0 as and 0as So, = 0 is a horizontal asmptote of g(). The denominator + +=( +3)( + ) is zero when = 3 andwhen =. Sine neither of these make the numerator zero then = 3 and = are vertial asmptotes of f().

5 Eample. (Graphs of Funtions) Let h() = +.Skeththegraphofh() without using our alulator. Be sure 0 to label all asmptotes and interepts of the graph. The leading term of the numerator is and the leading term of the denominator is. The ratio of the leading terms is =.Thus,h() has the same end behavior as = whih is as and as So, = is a horizontal asmptote. Notie that the numerator + =( +)( ) and the denominator 0 = ( +)( ). So, = and =arezerosof the denominator but not the numerator. So = and = are vertial asmptotes of h() The -interept of the graph is = (0) (0) 0 = =. The -interepts are found b setting h() = 0. The-interepts are where the numerator of h() are 0. Sine we ve alread fatored the numerator above, its eas to see that the -interepts are = and = = = = 0.

Week 3. Topic 5 Asymptotes

Week 3. Topic 5 Asymptotes Week 3 Topic 5 Asmptotes Week 3 Topic 5 Asmptotes Introduction One of the strangest features of a graph is an asmptote. The come in three flavors: vertical, horizontal, and slant (also called oblique).