Lecture 4-5A: An Introduction to Rational Functions A Rational Function f () is epressed as a fraction with a functiong() in the numerator and a function h() in the denominator. f () = g() h() Eamples f () = 6 f () = 5 f () = sin + tan! f () = e ln() This section will focus on rational functions with a polnomial function in the numerator and the denominator. The terms of the polnomials with have powers of that are positive integers and coefficients that are real numbers. The terms will be written in decreasing powers of. f () = a n n ± a n n ± a n n ±...a b m m ± b m m ± b m m ±...b where n z + and a n R f () = 6 Eamples f () = 5 4 To analze the behavior of rational functions we prefer the polnomials in the numerator and denominator to be epressed in factored form. Man of the polnomials we stud will have have higher orders for the powers of and factoring all of them would require a great deal of time. it is common to epress man of the rational functions with the polnomials alread factored to help speed up the analsis of the function and the sketch associated with the function. ( )( +) f () = ( 4) Eamples f () = ( )( + ) 4 ( + ) M-7 4-5A Lecture! Page of! 8 Eitel
Rational functions are represented b man more tpes of graphs than power function we learned to graph in the last section. This means that the material in this section will be broken down into more specific cases and more theorems will be needed to analze the behavior of the graphs. The Graphs of Rational Functions Rational functions can be represented b are several different tpes of graphs.. The right and left right and left ends of the graph of rational functions ma have values that increase to positive infinit or decrease to negative infinit like power functions but the ma also approach horizontal asmptotes. The end behavior of the graph is determined b the term with the highest power in the numerator and the term with the highest power in the denominator.. The graph of a rational function ma bounce off a root on the ais and turn upward or downward at that point. The graph of a rational function ma go through a point on the ais at that point. These behaviors are determined b the roots of the rational functions. 4. Ration functions ma have holes in the graphs depending on common factors in the numerator and dominator. Some holes are on the ais and act like roots. Other holes are above or below the ais. All holes create a gap in the graph. Rational functions ma have vertical asmptotes depending on the factors in the denominator. The graph of a rational function turns upwards and approaches positive infinit or turns downward and approaches negative on either side of a vertical asmptote Sketch Graphs We use sketch graphs to help us get a general feel about the graph of a function. A sketch graph displas onl the basic behaviors of the graphs of rational functions. The graphs are different from an actual graph obtained from plotting millions of (, ) points that are solutions to the function. A sketch graph displas the behavior of the graph around gaps in the graph. The sketch graphs also displa the basic behavior of the graph at the ends of the graph where as the values for approach negative or positive infinit. Sketch do not show all the possible wiggles in the graph that do not involve behavior of the graph around gaps in the graph. The cannot be used to predict etra turns that are not found using the critical numbers of the function. Think of a sketch graph as a cartoon rendering of an idea. Do not be fooled b the cartoon. It contains most of the important concepts and is used to get a feel for where problems ma be found. It does leave out some important details. After the cartoon is approved an animator creates the actual graphic in much more detail and ou get a final product. Pre calculus starts ou off drawing the cartoon version. Your Calculus course will introduce the concept of derivatives. You will use that tool to refine the sketch graph we make here into a final product. M-7 4-5A Lecture! Page of! 8 Eitel
Before we go further lets be clear about the limitations of sketch graphs The sketch graphs shown in these lectures are used to displa onl the basic behaviors of roots, intervals where the graph is increasing and decreasing and the behavior of the graph around gaps in the graph. The sketch graphs also displa the basic behavior of the graph at the ends of the graph where as the values for approach negative or positive infinit. Sketch graphs do not displa a scale on the or ais so the graphs cannot displa the eact rate of increase do decrease. The do not show the eact and values for man of the turns the graph ma take. Sketch graphs use smooth curves to connect the roots. The do not show all the possible wiggles in the graph that do not involve behavior of the graph around gaps in the graph. The cannot be used to predict etra turns that are not found using the critical numbers of the function. Sketch graphs are used to analze the limits at critical points that produce roots, holes and asmptotes in the graph. You will discover in our calculus class how the derivative of the function can be used to refine the graph, The derivative will help ou find all the wiggles in the graph and add accurate values where the occur. The two sketches shown below describe the eact same behavior in terms of the behavior of the graph around gaps in the graph and the basic behavior of the graph at the ends. Both of the sketch graphs displaed below show the same limits for the values as the values approach the vertical asmptote at = 4. Both graphs show the graph going through the odd root at =. Both graphs show that as approaches + or the values approach. The sketch graph cannot be used to predict etra turns that are not found using the critical numbers of the function. The two graphs have different scales and this distorts the rate of change in the increase or decrease around the turn in the graphs. The different scales for each graph also means that the height of the turn is also not shown as being the same. None of these differences change the answers we can obtain about the limits of the values as approach the critical numbers that are associated with roots, holes and asmptotes in the graph which are the focus for this section. M-7 4-5A Lecture! Page of! 8 Eitel
The basic behaviors of Rational Functions that we can displa with a sketch graph are shown below. both ends approach the ais. As the values for approach positive infinit + or negative infinit the values ma approach the ais as a horizontal asmptote. As the values for approach positive infinit + or negative infinit all of the function below have the horizontal line = as a limit for the values 4 M-7 4-5A Lecture! Page 4 of! 8 Eitel
both ends approach a non zero horizontal asmptote As the values for approach positive infinit + or negative infinit the values ma approach a non zero constant value 4 4 5 4 / / M-7 4-5A Lecture! Page 5 of! 8 Eitel
BOTH ends of the graph approach positive infinit + As the values for approach positive infinit + or negative infinit the values approach positive infinit + BOTH ends of the graph approach negative infinit As the values for approach positive infinit + or negative infinit the values approach negative infinit M-7 4-5A Lecture! Page 6 of! 8 Eitel
the right end approach positive infinit + the left end of the graph approaches negative infinit As the values for approach + the values approach + as the values for approach negative infinit the values approach negative infinit the right end approach negative infinit the left end approaches positive infinit + As the values for approach + the values approach the values for approach the values approach positive infinit + M-7 4-5A Lecture! Page 7 of! 8 Eitel
The graphs of rational functions have even and odd roots. The roots of rational functions act just like the even and odd roots of power functions. The graph goes through odd roots The graph bounces off even roots. The graphs of rational functions can have or more holes in the graph. The hole in the graph ma be on the ais. If so the act like roots. The graph goes through odd roots. The graph bounces off even roots. The holes in the graph ma be above or below the ais. All holes in the graph create a gap in the graph. M-7 4-5A Lecture! Page 8 of! 8 Eitel
The graphs of rational functions have even and odd vertical asmptotes. the graph of the right side of the vertical asmptote approaches positive infinit + the graph on the left side of the vertical asmptote approaches negative infinit odd asmptote odd asmptotes the graph on the right side of the vertical asmptote approaches negative infinit the graph on the left side of the vertical asmptote approaches positive infinit + odd asmptote odd asmptotes M-7 4-5A Lecture! Page 9 of! 8 Eitel
the graph on BOTH sides of the vertical asmptote approach positive infinit + even asmptote 4 even asmptotes the graph on BOTH sides of the vertical asmptote approach negative infinit even asmptote even asmptotes M-7 4-5A Lecture! Page of! 8 Eitel