Division of Polynomials and Slant Asymptotes
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1 Division o Polynomials and Slant Asymptotes Here is more detail about lon run behavior o rational unctions when the deree o the numerator is reater than the deree o the denominator. As the book points out (see p. in the tetbook), the lon run behavior o rational unctions mimics the ratio o the leadin terms o the numerator and denominator unction. Eample: -- the lon run behavior o this rational unction is similar to. That is, it would look like a cubic. But which cubic? To ind the eact answer, you need to do a little lon division. 7 Reminder: Write the improper raction as a mied number. 6 Do you remember what to do? In the United States, you probably use the azinta method divide 7 by 6: Narration: 6 azinta 7 times, 8 6 =, subtract, 6 7 Brin down the, 6 azinta 8 times, 8 6 = 8, subtract, the remainder is, so We ll use the same technique to ind the speciic lon run behavior o rational unctions. I you learned a dierent alorithm or lon division, the same ideas should apply. p Deinition: The rational unction is said to be asymptotic to a unction q i or suiciently ar away rom in either direction, the distance between p p and tends to zero. Inormally, the raph o hus the raph q q o is called an asymptote.. I the raph o happens to be a line,
2 To ind the speciic unction division. Eample: Consider to. that p q 6 is asymptotic to, we will perorm lon. Find so that is asymptotic Solution: Divide, just as we would with numbers. It s helpul to line up powers o : Narration: azinta times, is, subtract (careully!), Brin down the 6, azinta 6 6 times, 6 is 6 8, subtract, The remainder is, so Then we can write 6. What happens to this unction in the lon run? The rational bit that we built out o the remainder has deree o the numerator less than the deree o the denominator, so it tends to zero. As ets ar rom zero in either direction, will look like 6. So is asymptotic to 6. Since this is a linear unction, we can say that y 6 is an asymptote or (). This asymptote is neither horizontal nor vertical; it is a slant asymptote or oblique asymptote. You can check that this is correct by raphin both unctions in your calculator and then zoomin out. I can see a convincin picture or.
3 Eample: Find the equation o the curve that represents the lon run behavior o h. Solution: Divide. I ind it easier to keep thins straiht i I write zeros in place o missin powers o : This means that h. (Note that you could also have perormed this division by simply cancelin powers o.) The rational part oes to zero, so h has a slant asymptote o y. You can see this lon run behavior starts to dominate or very near zero. Eample: -- the lon run behavior o this rational unction is similar to. That is, it would look like a cubic. But which cubic?
4 Solution: Divide, just as we would with numbers is asymptotic to y. So y is the cubic asked or in the question. We typically don t call this unction an asymptote, because it is not a line. But you can see the same behavior, where hus the curve. In this eample, the lon run behavior is visible very close to the vertical asymptotes. Eercises: For problems throuh 6, use division to ind the equation o the curve that represents the lon run behavior o each o the ollowin rational unctions. Does the unction have a horizontal or slant asymptote? Conirm your answer by raphin with your calculator...
5 For problems 7 throuh, ind the intercepts, the vertical intercept, the vertical asymptotes, and ind the equation o the curve that represents the lon run behavior o the rational unction. Use that inormation to sketch a raph o the unction. Conirm your answer by raphin with your calculator
6 Brie answers to selected eercises:. The deree o the numerator is equal to the deree o the denominator, so the book s shortcut applies: this unction has a horizontal asymptote, y. But you would et the same answer i you divided:. You may use either technique to et the riht answer.. The unction has a slant asymptote, y 6.. The unction is asymptotic to y ; this is a parabola, so it is neither a horizontal nor a slant asymptote. (In order to conirm this with my raphin device, I used a window o and y.) 7. The -intercepts are at =, =, and =. There is one vertical asymptote, =, and so there is no vertical intercept. Usin division (multiply out the numerator irst), you ind that is asymptotic to y. This is a slant asymptote.. The -intercept is at =, so that is also the vertical intercept. There is one vertical asymptote, =. Usin lon division, you ind that is asymptotic to y. This is neither a horizontal nor a slant asymptote.
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