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# Class Note for MATH 1330 with Professor Flagg at UH

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Date Created: 02/06/15

PreCalSection23Page 1 of9 Math 1330 Section 23 Rational Functions A rational function is a function of the form rx Px Qx where Px and Qx are polynomials The domain of a rational functions in the set of all real numbers EXCEPT the values of x for which the bottom is zero ie the domain is all real numbers except x such that Qx0 Finding the range of a rational function is more complicated and we will really only do this in conjunction with drawing the graph To do all the math necessary to draw the graph of a rational function it is necessary to factor the polynomial in the numerator and the polynomial in the denominator if they are not already factored for you Write out the whole factored fraction first do not cancel common terms until after you have written the full factored form down the factors we cancel will be important to the graph Example Let rx x2 x 6 3 2x2 3x Then in factored form rx x 2x 3 xx lx 3 Once I have the full factored form I know that when x at 3 the x3 factors in the top and bottom cancel However the number 73 is NOT in the domain of rx because the bottom is zero there Therefore when we draw the graph there can be no point on the graph at x3 Before listing the graphing strategy let s explore the different features a graph of a rational function might have one at a time HOLES The graph of the rational function rx Px Qx has a HOLE at the point xa if x a is a factor of Qx that is cancelled when rx is reduced to lowest form ie common factors are cancelled and xa is not a factor of the denominator of the reduced fraction PreCalSection23Page 2 of9 Example rx W has a hole at X3 since the factor x3 is cancelled from xx 7 lx 3 the bottom When reducing the fraction to lowest terms Example 1 Determine if the rational function x2 7 x 7 2 110C 7 W has any holes in its graph Example 2 Determine the values of x Where the graph of the following function has a hole x752x 3Jc7l4 fm x753x7l2x6 VERTICAL ASYMP TOTES A vertical line Fa is called a vertical asymptote of the rational function yrx if y7gtooory7gt7ooasx7gtzf orx7gta39 On the graph this looks like PreCalSection23Page 3 of9 How to nd a vertical asymptote A Factor the polynomials in numerator and denominator B Cancel the common terms C The values of x for which the denominator is zero in this reduced fraction are the x values at which the graph has a vertical asymptote D The equations of the lines of the vertical asymptotes are xc for c a value such that xc is a factor of the denominator in the reduced fraction W reduces to the fraction rx xx lx3 xx l asymptotes are the lines x0 and xl Example rx and the vertical Example 3 Determine the vertical asymptotes of the function x 52x3x 1 fw x 53x l2x6 Example 4 Determine the values of x for which the following function has a hole and find the equations of the vertical asymptotes x3x2 2x x f x22x 3 PreCalSection23Page 4 of9 HORIZONTAL ASYMPTOTES A horizontal line yb is called a horizontal asymptote of the rational function rX if y gtb asx gtoo orx gt oo Consider the rational function V x Px anxquot 61Hxquot391 alxa0 x b xmb xm39lbxb m m 1 1 0 Notice that the degree of PX is n and the degree of QX is In How to tell if rX has a horizontal asymptote A If n gt m then rX does NOT have a horizontal asymptote B If m gt n then the line y 0 is a horizontal asymptote n C Ifmnthentheline y bm lead1ng coefficient of QX a leading coefficient of PX is a horizontal asymptote One thing to remember if the line y b is a horizontal asymptote that does not necessarily mean that b is not in the range of the function rX What this means is that it is still possible in some cases to find an X such that rx b even if y b is an asymptote The reason this is true is that horizontal asymptotes come from the behavior as X gets very large either positive or negative not what happens in the middle of the graph Example 5 Determine the horizontal asymptote if it exists for the functions x4 A I m m B gee Zizfjx s C x x S x3x l x 53 x l2 x 62 PreCalSection23Page 5 of9 SLANT ASYMPTOTE This is probably a new concept for you so read carefully P be a rational function where the degree of the numerator is one more x than the degree of the denominator From the previous page we know that rX does not have a horizontal asymptote However the graphs of these functions do have special shapes that include an asymptote line The asymptote is a line that is not horizontal hence it slants and that is why it is called a slant asymptote Let rx Where a slant asymptote comes from By the division algorithm the rational function rX can be written in the form b rx mx Qx where m at 0 and RX is a polynomial of degree less than the degree of the polynomial QX R As x gt 00 the value of the rational function Q5 goes to zero Since it has a horizontal x asymptote at y0 So the original function rX has values at large X that are very close to mx b Therefore the line y mx b is an asymptote for the function rX How to find a slant asymptote 1 Check and see that the degree of the top is one more than the degree of the bottom 2 If so it has a slant asymptote 3 Divide the top polynomial by the bottom polynomial See dividing polynomials in the appendices of the online textbook if you don t know how to do this 4 The answer to the division problem is a polynomial mxb with a remainder RX 5 The slant asymptote is y mx b 2 2 Example 6 Find the slant asymptote 1f1t eX1sts for the function rx x PreCalSection23Page 6 of 9 This is a picture of the graph of this function Do you see Why we say it has a slant asymptote PreCalSection23Page 7 of9 OVERALL STRATEGY FOR GRAPHING RATIONAL FUNCTIONS 1 Factor the numerator and denominator and check for common factors if not in factored form Write out the FULL factored form then cancel common factors and write the reduced form 2 Identify HOLES If x c is a factor of the denominator in the FULL factored form but it cancels and is NOT a factor of the denominator in the reduced fraction the graph has a hole at x c 3 Identify Vertical Asymptotes If x c is a factor of the denominator in the reduced fraction then the line x c is a vertical asymptote of the graph 4 Identify Horizontal Asymptotes by comparing the degrees of the polynomials in the numerator and denominator 5 If the degree of the numerator is one more than the degree of the denominator find the slant asymptote 6 Find the yintercept 7 Find the Xintercepts x a is an Xintercept if x a is a factor of the numerator of the reduced fraction In other words these are the X s where the top is 0 and the bottom is not 8 Find points on the graph including at least one between each vertical asymptote if there are more than one Only a few points are necessary we only need a sketch 9 If there are no horizontal or slant asymptotes what happens at the end ie as x gt 00 is the function positive or negative and as x gt w is the function positive or negative 10 Determine the behavior of the function near the asymptotes A For a horizontal asymptote which side of the line is the function on as x gtoo andas x gt oo B For a vertical asymptote for X values close to the asymptote what is the sign of the function is it positive or negative Check points on BOTH SIDES of the asymptote C For a slant asymptote which side of the asymptote line is the function on Finally sketch the graph This is complicated but does not have to be that bad So don t panic and follow the steps A helpful way to remember the important parts is the following description of what places on the graph are described by the above steps A The X values not in the domain Vertical Asymptotes and Holes B The places where the graph crosses an aXis X and yintercepts C The end behavior of the function horizontal asymptote slant asymptote goes up to infinity or goes down toward negative infinity PreCalSecti0n23Page 8 of9 Example 7 Use the graphing strategies to sketch a graph of the function x5 fx x2 PreCalSecti0n23Page 9 of9 Example 8 Use the graphing strategy to sketch the graph of the function x3 4x x p x2 2x 8

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