/ Interactive Algebra & Tri…f Guided Study Companion / Interactive Algebra and Trigonometry - A Self-Guided Study Companion.iso / tutor / chap_4 / 4-2-1.tut

< prev

next >

MacOS 8.1

Win98

DOS

view JSON data

view as text

+--------+-------------------------+-------------------------+--------+--------+
|00000000| 54 55 54 4f 52 20 30 36 | 21 0b 00 00 4f 00 00 00 |TUTOR 06|!...O...|
|00000010| 53 65 63 74 69 6f 6e 20 | 34 2e 32 20 20 47 72 61 |Section |4.2  Gra|
|00000020| 70 68 73 20 6f 66 20 52 | 61 74 69 6f 6e 61 6c 20 |phs of R|ational |
|00000030| 46 75 6e 63 74 69 6f 6e | 73 0d 0b 00 46 6f 72 20 |Function|s...For |
|00000040| 6d 6f 72 65 20 70 72 61 | 63 74 69 63 65 3a 0d 0a |more pra|ctice:..|
|00000050| 00 0d 0a 00 20 20 20 20 | 20 10 34 2d 32 2d 33 0e |....    | .4-2-3.|
|00000060| 78 34 2d 32 0e 45 78 65 | 72 63 69 73 65 73 0f 0d |x4-2.Exe|rcises..|
|00000070| 0a 00 20 20 20 20 20 10 | 34 2d 32 2d 32 0e 65 34 |..     .|4-2-2.e4|
|00000080| 2d 32 0e 47 75 69 64 65 | 64 20 45 78 65 72 63 69 |-2.Guide|d Exerci|
|00000090| 73 65 73 0f 0d 0a 00 0d | 0a 00 54 6f 70 69 63 73 |ses.....|..Topics|
|000000a0| 20 66 6f 72 20 65 78 70 | 6c 6f 72 61 74 69 6f 6e | for exp|loration|
|000000b0| 3a 0d 0a 00 0d 0a 00 20 | 20 20 20 20 0e 73 34 2d |:...... |    .s4-|
|000000c0| 32 2d 31 0e 47 75 69 64 | 65 6c 69 6e 65 73 20 66 |2-1.Guid|elines f|
|000000d0| 6f 72 20 47 72 61 70 68 | 69 6e 67 20 52 61 74 69 |or Graph|ing Rati|
|000000e0| 6f 6e 61 6c 20 46 75 6e | 63 74 69 6f 6e 73 0f 0d |onal Fun|ctions..|
|000000f0| 0a 00 20 20 20 20 20 0e | 73 34 2d 32 2d 32 0e 53 |..     .|s4-2-2.S|
|00000100| 6c 61 6e 74 20 41 73 79 | 6d 70 74 6f 74 65 73 0f |lant Asy|mptotes.|
|00000110| 0d 0a 00 53 65 63 74 69 | 6f 6e 20 34 2e 32 20 20 |...Secti|on 4.2  |
|00000120| 47 72 61 70 68 73 20 6f | 66 20 52 61 74 69 6f 6e |Graphs o|f Ration|
|00000130| 61 6c 20 46 75 6e 63 74 | 69 6f 6e 73 0d 0b 00 54 |al Funct|ions...T|
|00000140| 68 65 72 65 20 61 72 65 | 20 61 20 66 65 77 20 67 |here are| a few g|
|00000150| 65 6e 65 72 61 6c 20 12 | 31 67 75 69 64 65 6c 69 |eneral .|1guideli|
|00000160| 6e 65 73 20 66 6f 72 20 | 67 72 61 70 68 69 6e 67 |nes for |graphing|
|00000170| 20 72 61 74 69 6f 6e 61 | 6c 20 66 75 6e 63 74 69 | rationa|l functi|
|00000180| 6f 6e 73 12 30 20 6c 69 | 73 74 65 64 20 0d 0a 00 |ons.0 li|sted ...|
|00000190| 62 65 6c 6f 77 2e 20 20 | 4c 65 74 20 11 33 66 28 |below.  |Let .3f(|
|000001a0| 78 29 20 3d 20 70 28 78 | 29 2f 71 28 78 29 11 31 |x) = p(x|)/q(x).1|
|000001b0| 2c 20 77 68 65 72 65 20 | 11 33 70 28 78 29 20 11 |, where |.3p(x) .|
|000001c0| 31 61 6e 64 20 11 33 71 | 28 78 29 20 11 31 61 72 |1and .3q|(x) .1ar|
|000001d0| 65 20 70 6f 6c 79 6e 6f | 6d 69 61 6c 73 20 77 69 |e polyno|mials wi|
|000001e0| 74 68 20 6e 6f 20 0d 0a | 00 63 6f 6d 6d 6f 6e 20 |th no ..|.common |
|000001f0| 66 61 63 74 6f 72 73 2e | 0d 0a 00 0d 0b 00 31 2e |factors.|......1.|
|00000200| 20 20 46 69 6e 64 20 61 | 6e 64 20 70 6c 6f 74 20 |  Find a|nd plot |
|00000210| 74 68 65 20 11 33 79 11 | 31 2d 69 6e 74 65 72 63 |the .3y.|1-interc|
|00000220| 65 70 74 20 28 69 66 20 | 61 6e 79 29 20 62 79 20 |ept (if |any) by |
|00000230| 65 76 61 6c 75 61 74 69 | 6e 67 20 11 33 66 28 30 |evaluati|ng .3f(0|
|00000240| 29 11 31 2e 0d 0a 00 32 | 2e 20 20 46 69 6e 64 20 |).1....2|.  Find |
|00000250| 74 68 65 20 7a 65 72 6f | 73 20 6f 66 20 74 68 65 |the zero|s of the|
|00000260| 20 6e 75 6d 65 72 61 74 | 6f 72 20 28 69 66 20 61 | numerat|or (if a|
|00000270| 6e 79 29 20 62 79 20 73 | 6f 6c 76 69 6e 67 20 74 |ny) by s|olving t|
|00000280| 68 65 20 65 71 75 61 74 | 69 6f 6e 20 11 33 70 28 |he equat|ion .3p(|
|00000290| 78 29 20 3d 20 30 11 31 | 2e 20 20 0d 0a 00 20 20 |x) = 0.1|.  ...  |
|000002a0| 20 20 54 68 65 6e 20 70 | 6c 6f 74 20 74 68 65 20 |  Then p|lot the |
|000002b0| 63 6f 72 72 65 73 70 6f | 6e 64 69 6e 67 20 11 33 |correspo|nding .3|
|000002c0| 78 11 31 2d 69 6e 74 65 | 72 63 65 70 74 73 2e 0d |x.1-inte|rcepts..|
|000002d0| 0a 00 33 2e 20 20 46 69 | 6e 64 20 74 68 65 20 7a |..3.  Fi|nd the z|
|000002e0| 65 72 6f 73 20 6f 66 20 | 74 68 65 20 64 65 6e 6f |eros of |the deno|
|000002f0| 6d 69 6e 61 74 6f 72 20 | 28 69 66 20 61 6e 79 29 |minator |(if any)|
|00000300| 20 62 79 20 73 6f 6c 76 | 69 6e 67 20 74 68 65 20 | by solv|ing the |
|00000310| 65 71 75 61 74 69 6f 6e | 20 0d 0a 00 20 20 20 20 |equation| ...    |
|00000320| 11 33 71 28 78 29 20 3d | 20 30 2e 20 20 11 31 54 |.3q(x) =| 0.  .1T|
|00000330| 68 65 6e 20 73 6b 65 74 | 63 68 20 74 68 65 20 63 |hen sket|ch the c|
|00000340| 6f 72 72 65 73 70 6f 6e | 64 69 6e 67 20 76 65 72 |orrespon|ding ver|
|00000350| 74 69 63 61 6c 20 61 73 | 79 6d 70 74 6f 74 65 73 |tical as|ymptotes|
|00000360| 2e 0d 0a 00 34 2e 20 20 | 46 69 6e 64 20 61 6e 64 |....4.  |Find and|
|00000370| 20 73 6b 65 74 63 68 20 | 74 68 65 20 68 6f 72 69 | sketch |the hori|
|00000380| 7a 6f 6e 74 61 6c 20 61 | 73 79 6d 70 74 6f 74 65 |zontal a|symptote|
|00000390| 20 28 69 66 20 61 6e 79 | 29 20 62 79 20 75 73 69 | (if any|) by usi|
|000003a0| 6e 67 20 74 68 65 20 72 | 75 6c 65 20 66 6f 72 20 |ng the r|ule for |
|000003b0| 0d 0a 00 20 20 20 20 66 | 69 6e 64 69 6e 67 20 74 |...    f|inding t|
|000003c0| 68 65 20 68 6f 72 69 7a | 6f 6e 74 61 6c 20 61 73 |he horiz|ontal as|
|000003d0| 79 6d 70 74 6f 74 65 20 | 6f 66 20 61 20 72 61 74 |ymptote |of a rat|
|000003e0| 69 6f 6e 61 6c 20 66 75 | 6e 63 74 69 6f 6e 2e 0d |ional fu|nction..|
|000003f0| 0a 00 35 2e 20 20 50 6c | 6f 74 20 61 74 20 6c 65 |..5.  Pl|ot at le|
|00000400| 61 73 74 20 6f 6e 65 20 | 70 6f 69 6e 74 20 62 6f |ast one |point bo|
|00000410| 74 68 20 11 33 62 65 74 | 77 65 65 6e 20 61 6e 64 |th .3bet|ween and|
|00000420| 20 62 65 79 6f 6e 64 20 | 11 31 65 61 63 68 20 11 | beyond |.1each .|
|00000430| 33 78 11 31 2d 69 6e 74 | 65 72 63 65 70 74 20 61 |3x.1-int|ercept a|
|00000440| 6e 64 20 0d 0a 00 20 20 | 20 20 76 65 72 74 69 63 |nd ...  |  vertic|
|00000450| 61 6c 20 61 73 79 6d 70 | 74 6f 74 65 2e 0d 0a 00 |al asymp|tote....|
|00000460| 36 2e 20 20 55 73 65 20 | 73 6d 6f 6f 74 68 20 63 |6.  Use |smooth c|
|00000470| 75 72 76 65 73 20 74 6f | 20 63 6f 6d 70 6c 65 74 |urves to| complet|
|00000480| 65 20 74 68 65 20 67 72 | 61 70 68 20 62 65 74 77 |e the gr|aph betw|
|00000490| 65 65 6e 20 61 6e 64 20 | 62 65 79 6f 6e 64 20 74 |een and |beyond t|
|000004a0| 68 65 20 76 65 72 74 69 | 63 61 6c 20 0d 0a 00 20 |he verti|cal ... |
|000004b0| 20 20 20 61 73 79 6d 70 | 74 6f 74 65 73 2e 0d 0a |   asymp|totes...|
|000004c0| 00 0d 0a 00 4e 6f 74 65 | 20 74 68 61 74 2c 20 69 |....Note| that, i|
|000004d0| 6e 20 61 64 64 69 74 69 | 6f 6e 20 74 6f 20 74 68 |n additi|on to th|
|000004e0| 65 20 73 74 65 70 73 20 | 61 62 6f 76 65 2c 20 79 |e steps |above, y|
|000004f0| 6f 75 20 6d 61 79 20 77 | 69 73 68 20 74 6f 20 74 |ou may w|ish to t|
|00000500| 65 73 74 20 66 6f 72 20 | 73 79 6d 6d 65 74 72 79 |est for |symmetry|
|00000510| 20 0d 0a 00 74 6f 20 72 | 65 64 75 63 65 20 74 68 | ...to r|educe th|
|00000520| 65 20 6e 75 6d 62 65 72 | 20 6f 66 20 11 33 78 11 |e number| of .3x.|
|00000530| 31 2d 76 61 6c 75 65 73 | 20 79 6f 75 20 6e 65 65 |1-values| you nee|
|00000540| 64 20 74 6f 20 65 76 61 | 6c 75 61 74 65 2e 0d 0a |d to eva|luate...|
|00000550| 00 53 65 63 74 69 6f 6e | 20 34 2e 32 20 20 47 72 |.Section| 4.2  Gr|
|00000560| 61 70 68 73 20 6f 66 20 | 52 61 74 69 6f 6e 61 6c |aphs of |Rational|
|00000570| 20 46 75 6e 63 74 69 6f | 6e 73 0d 0b 00 4c 65 74 | Functio|ns...Let|
|00000580| 20 11 33 66 20 11 31 62 | 65 20 74 68 65 20 72 61 | .3f .1b|e the ra|
|00000590| 74 69 6f 6e 61 6c 20 66 | 75 6e 63 74 69 6f 6e 20 |tional f|unction |
|000005a0| 67 69 76 65 6e 20 62 79 | 0d 0a 00 20 20 20 20 20 |given by|...     |
|000005b0| 20 20 20 20 20 20 20 20 | 20 20 11 32 6e 20 20 20 |        |  .2n   |
|000005c0| 20 20 20 20 20 6e 2d 31 | 0d 0b 00 20 20 20 20 20 |     n-1|...     |
|000005d0| 20 20 20 20 20 20 20 11 | 33 61 20 78 20 20 2b 20 |       .|3a x  + |
|000005e0| 61 20 20 20 78 20 20 20 | 20 2b 20 11 34 2a 2a 2a |a   x   | + .4***|
|000005f0| 20 11 33 2b 20 61 20 78 | 20 2b 20 61 0d 0b 00 20 | .3+ a x| + a... |
|00000600| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 6e 20 |        |    .2n |
|00000610| 20 20 20 20 20 6e 2d 31 | 20 20 20 20 20 20 20 20 |     n-1|        |
|00000620| 20 20 20 20 20 20 31 20 | 20 20 20 20 30 0d 0b 00 |      1 |    0...|
|00000630| 20 20 20 20 20 11 33 66 | 28 78 29 20 3d 20 11 34 |     .3f|(x) = .4|
|00000640| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|00000650| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|00000660| 11 31 2e 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |.1....  |        |
|00000670| 20 20 20 20 20 11 32 6d | 20 20 20 20 20 20 20 20 |     .2m|        |
|00000680| 6d 2d 31 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |m-1...  |        |
|00000690| 20 20 11 33 62 20 78 20 | 20 2b 20 62 20 20 20 78 |  .3b x | + b   x|
|000006a0| 20 20 20 20 2b 20 11 34 | 2a 2a 2a 20 11 33 2b 20 |    + .4|*** .3+ |
|000006b0| 62 20 78 20 2b 20 62 0d | 0b 00 20 20 20 20 20 20 |b x + b.|..      |
|000006c0| 20 20 20 20 20 20 20 11 | 32 6d 20 20 20 20 20 20 |       .|2m      |
|000006d0| 6d 2d 31 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |m-1     |        |
|000006e0| 20 31 20 20 20 20 20 30 | 0d 0a 00 0d 0b 00 11 31 | 1     0|.......1|
|000006f0| 49 66 20 11 33 6e 20 3d | 20 6d 20 2b 20 31 20 11 |If .3n =| m + 1 .|
|00000700| 31 28 74 68 61 74 20 69 | 73 2c 20 74 68 65 20 64 |1(that i|s, the d|
|00000710| 65 67 72 65 65 20 6f 66 | 20 74 68 65 20 6e 75 6d |egree of| the num|
|00000720| 65 72 61 74 6f 72 20 69 | 73 20 65 78 61 63 74 6c |erator i|s exactl|
|00000730| 79 20 6f 6e 65 20 67 72 | 65 61 74 65 72 20 0d 0a |y one gr|eater ..|
|00000740| 00 74 68 61 6e 20 74 68 | 65 20 64 65 67 72 65 65 |.than th|e degree|
|00000750| 20 6f 66 20 74 68 65 20 | 64 65 6e 6f 6d 69 6e 61 | of the |denomina|
|00000760| 74 6f 72 29 2c 20 74 68 | 65 6e 20 74 68 65 20 67 |tor), th|en the g|
|00000770| 72 61 70 68 20 6f 66 20 | 11 33 66 20 11 31 68 61 |raph of |.3f .1ha|
|00000780| 73 20 61 20 12 31 73 6c | 61 6e 74 20 61 73 79 6d |s a .1sl|ant asym|
|00000790| 70 2d 0d 0a 00 74 6f 74 | 65 12 30 2e 20 20 54 6f |p-...tot|e.0.  To|
|000007a0| 20 66 69 6e 64 20 74 68 | 65 20 73 6c 61 6e 74 20 | find th|e slant |
|000007b0| 61 73 79 6d 70 74 6f 74 | 65 2c 20 77 65 20 75 73 |asymptot|e, we us|
|000007c0| 65 20 6c 6f 6e 67 20 64 | 69 76 69 73 69 6f 6e 20 |e long d|ivision |
|000007d0| 74 6f 20 72 65 64 75 63 | 65 20 11 33 66 20 11 31 |to reduc|e .3f .1|
|000007e0| 74 6f 20 0d 0a 00 74 68 | 65 20 66 6f 72 6d 0d 0a |to ...th|e form..|
|000007f0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00000800| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00000810| 32 6e 2d 32 20 20 20 20 | 20 20 20 20 6e 2d 33 0d |2n-2    |    n-3.|
|00000820| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |..      |      .3|
|00000830| 61 20 20 20 20 20 20 20 | 20 20 20 20 20 63 20 20 |a       |     c  |
|00000840| 20 78 20 20 20 20 2b 20 | 63 20 20 20 78 20 20 20 | x    + |c   x   |
|00000850| 20 2b 20 11 34 2a 2a 2a | 20 11 33 2b 20 63 20 78 | + .4***| .3+ c x|
|00000860| 20 2b 20 63 0d 0b 00 20 | 20 20 20 20 20 20 20 20 | + c... |        |
|00000870| 20 20 20 20 11 32 6e 20 | 20 20 20 20 20 20 20 20 |    .2n |        |
|00000880| 20 20 20 6e 2d 32 20 20 | 20 20 20 20 20 20 6e 2d |   n-2  |      n-|
|00000890| 33 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 31 |3       |       1|
|000008a0| 20 20 20 20 20 30 0d 0b | 00 20 20 20 20 20 11 33 |     0..|.     .3|
|000008b0| 66 28 78 29 20 3d 20 11 | 34 32 32 11 33 78 20 2b |f(x) = .|422.3x +|
|000008c0| 20 63 20 20 20 20 2b 20 | 11 34 32 32 32 32 32 32 | c    + |.4222222|
|000008d0| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|000008e0| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 0d 0b |22222222|222222..|
|000008f0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 62 |.       |     .3b|
|00000900| 20 20 20 20 20 20 11 32 | 6e 2d 31 20 20 20 20 20 |      .2|n-1     |
|00000910| 20 20 20 6d 20 20 20 20 | 20 20 20 20 6d 2d 31 0d |   m    |    m-1.|
|00000920| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 6d |..      |       m|
|00000930| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 62 |        |     .3b|
|00000940| 20 78 20 20 2b 20 62 20 | 20 20 78 20 20 20 20 2b | x  + b |  x    +|
|00000950| 20 11 34 2a 2a 2a 20 11 | 33 2b 20 62 20 78 20 2b | .4*** .|3+ b x +|
|00000960| 20 62 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 | b...   |        |
|00000970| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000980| 20 11 32 6d 20 20 20 20 | 20 20 6d 2d 31 20 20 20 | .2m    |  m-1   |
|00000990| 20 20 20 20 20 20 20 20 | 20 20 20 31 20 20 20 20 |        |   1    |
|000009a0| 20 30 0d 0a 00 11 31 4e | 6f 77 20 77 65 20 63 61 | 0....1N|ow we ca|
|000009b0| 6e 20 73 65 65 20 74 68 | 61 74 20 11 33 66 20 11 |n see th|at .3f .|
|000009c0| 31 69 73 20 67 69 76 65 | 6e 20 61 73 20 74 68 65 |1is give|n as the|
|000009d0| 20 73 75 6d 20 6f 66 20 | 61 20 6c 69 6e 65 61 72 | sum of |a linear|
|000009e0| 20 66 75 6e 63 74 69 6f | 6e 20 61 6e 64 20 61 20 | functio|n and a |
|000009f0| 72 61 74 69 6f 6e 61 6c | 20 0d 0a 00 66 75 6e 63 |rational| ...func|
|00000a00| 74 69 6f 6e 2e 20 20 53 | 69 6e 63 65 20 74 68 65 |tion.  S|ince the|
|00000a10| 20 6c 65 66 74 6f 76 65 | 72 20 72 61 74 69 6f 6e | leftove|r ration|
|00000a20| 61 6c 20 70 61 72 74 20 | 6f 66 20 11 33 66 20 11 |al part |of .3f .|
|00000a30| 31 61 70 70 72 6f 61 63 | 68 65 73 20 7a 65 72 6f |1approac|hes zero|
|00000a40| 20 61 73 20 11 33 78 20 | 11 34 32 33 20 2b 38 11 | as .3x |.423 +8.|
|00000a50| 31 2c 20 0d 0a 00 77 65 | 20 63 61 6e 20 73 65 65 |1, ...we| can see|
|00000a60| 20 74 68 61 74 20 11 33 | 66 20 11 31 61 70 70 72 | that .3|f .1appr|
|00000a70| 6f 61 63 68 65 73 20 74 | 68 65 20 6c 69 6e 65 20 |oaches t|he line |
|00000a80| 11 33 79 20 3d 20 61 20 | 78 2f 62 20 20 2b 20 63 |.3y = a |x/b  + c|
|00000a90| 20 20 20 2e 20 20 20 20 | 20 20 20 20 20 20 20 20 |   .    |        |
|00000aa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000ab0| 20 20 20 20 20 0d 0b 00 | 20 20 20 20 20 20 20 20 |     ...|        |
|00000ac0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000ae0| 20 20 20 11 32 6e 20 20 | 20 6d 20 20 20 20 6e 2d |   .2n  | m    n-|
|00000af0| 31 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |1       |        |
|00000b00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000b10| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 2e 0d 0a |        |   .1...|
|00000b20| 00 39 00 00 00 da 00 00 | 00 4d 29 00 00 10 00 00 |.9......|.M).....|
|00000b30| 00 00 00 00 00 73 34 2d | 32 00 3c 01 00 00 15 04 |.....s4-|2.<.....|
|00000b40| 00 00 4d 29 00 00 13 01 | 00 00 00 00 00 00 73 34 |..M)....|......s4|
|00000b50| 2d 32 2d 31 00 7a 05 00 | 00 a7 05 00 00 4d 29 00 |-2-1.z..|.....M).|
|00000b60| 00 51 05 00 00 00 00 00 | 00 73 34 2d 32 2d 32 00 |.Q......|.s4-2-2.|
+--------+-------------------------+-------------------------+--------+--------+