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{SECT 0 {PARA 256 "" 0 "" {TEXT 259 40 "EXPLORING RATIONAL FUNCTIONS G
RAPHICALLY" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
261 20 "Precalculus Project\n" }{TEXT 256 6 "      " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 263 
11 "Objectives:" }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "To fi
nd patterns in the graphs of rational functions.\nTo construct a ratio
nal function using its properties.\n" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 277 22 " Required Information:" }}{PARA 0 "" 0 "
" {TEXT -1 47 "A rational function is a function of the form  " }
{XPPEDIT 18 0 "f(x)/g(x);" "6#*&-%\"fG6#%\"xG\"\"\"-%\"gG6#F'!\"\"" }
{TEXT -1 3 " , " }{XPPEDIT 18 0 "g(x) <> 0;" "6#0-%\"gG6#%\"xG\"\"!" }
{TEXT -1 568 " where f(x) and g(x) are polynomials.\n\nThe graphs of r
ational functions may have horizontal asymptotes, vertical asymptotes,
 slant asymptotes or \"holes\".   Just as patterns can be found in fam
ilies of polynomial functions that are useful in sketching the functio
ns, patterns can be found in families of rational functions.\nThe hori
zontal asymptotes are found by observing the end behavior i.e. behavio
r as x increases or decreases without bounds.  The horizontal asymptot
es exist whenever the degree of the numerator is less than or equal to
 the degree of denominator." }}{PARA 0 "" 0 "" {TEXT -1 76 "The vertic
al asymptotes are the zeroes of the polynomial in the denominator." }}
{PARA 0 "" 0 "" {TEXT -1 140 "Slant asymptotes exist when the degree o
f the numerator is one more than the degree of the denominator and fou
nd by computing the quotient.." }}{PARA 0 "" 0 "" {TEXT -1 146 "A \"ho
le\" in any function is created at a point when the limit of the funct
ion exists at that point but is not equal to the value of the function
. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 18 " Solved Ex
ample 1:" }}{PARA 0 "" 0 "" {TEXT -1 220 "Using a graphing utility (ca
lculator or graphing program), sketch the following rational functions
 on one set of axes.  Discuss their similarities and their differences
. Include asymptotes, intercepts, domain and range.\n" }{XPPEDIT 18 0 
"f(x) = 1/(x^2);" "6#/-%\"fG6#%\"xG*&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT 
-1 51 "                                                   " }{XPPEDIT 
18 0 "g(x) = 1/(x^4);" "6#/-%\"gG6#%\"xG*&\"\"\"F)*$F'\"\"%!\"\"" }
{TEXT -1 42 "                                          " }{XPPEDIT 18 
0 "h(x) = 1/(x^6);" "6#/-%\"hG6#%\"xG*&\"\"\"F)*$F'\"\"'!\"\"" }{TEXT 
-1 9 "        \n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 
1 " " }{TEXT 266 9 "Solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "restart:with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, th
e name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "f:= 1/x^2:  g:= 1/x^4:  h:= 1/x^6:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 119 "plot([f,g,h], x = -4..4, y = -2..12, tit
le = \"graphs of 1/x^n, n = 2, 4, and 6\", legend = [\"1/x^2\", \"1/x^
4\", \"1/x^6\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 534 171 171 {PLOTDATA 
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\"#7F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "1/x^2
" "1/x^4" "1/x^6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 467 "The domain \+
of all three curves is the set of all real numbers except x = 0. \nThe
 range of all three curves is y > 0.  \nx = 0 is the vertical asymptot
e for all three curves. \ny = 0 is the horizontal asymptote for all th
ree curves.\nAll three curves are in the first and second quadrants.\n
As the exponent increases, the graph is further from the y-axis for -1
 < x < 1 and closer to the x-axis for x < -1 and x > 1.\nAll graphs pa
ss through the points (-1, 1) and ( 1,1).\n" }}}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 0 "" }{TEXT 278 18 " Solved Example 2:" }}{PARA 0 "" 0 "" 
{TEXT -1 106 "Define and sketch a rational function which has vertical
 asymptotes at x = -2 and 1 and a slant asymptote." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 279 1 " \+
" }{TEXT 280 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 260 "A rational \+
function has vertical asymptotes at x = -2 and x = 1 if the denominato
r has factors (x + 2) and (x - 1).  It has a slant asymptote if the de
gree of the numerator is one more than the degree of the denominator a
nd is obtained by finding the quotient." }}{PARA 0 "" 0 "" {TEXT -1 
19 "We define:  f(x) = " }{XPPEDIT 19 1 "x^3/((x+2)*(x-1));" "6#*&)%\"
xG\"\"$\"\"\"*&,&%\"xGF'\"\"#F'F',&F*F'F'!\"\"F'F-" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "f:= x ->x^3/((x+2)*(x - 1));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*(9$\"\"$
,&F-\"\"\"\"\"#F0!\"\",&F-F0F0F2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "quo(x^3,(x+2)*(x - 1),x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&\"\"\"!\"\"%\"xGF$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "implicitplot(\{y=f(x),x = -2, x = 1, y = -1 +x\}, x =
 -10..10, y = -10..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 411 209 209 
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{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 274 18 " Solved Example 3:" }}
{PARA 0 "" 0 "" {TEXT -1 226 "Write equations for two different ration
al functions with the given properties:\n1.  The horizontal asymptote \+
is y = 1.\n2.  \011The vertical asymptote is x = -3.\n3.   There is a \+
hole (discontinuity) at x = 1.\nSketch the function.\n" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 275 1 " " }{TEXT 276 9 "Solution:
" }}{PARA 0 "" 0 "" {TEXT -1 140 " From the solved example above the h
orizontal asymptote is y = 1 if the function has a limit 1 as x increa
ses or decreases without bounds.  " }}{PARA 0 "" 0 "" {TEXT -1 105 "If
 a function is undefined at x = -3 then the rational function has a ve
rtical asymptote at that point.  " }}{PARA 0 "" 0 "" {TEXT -1 207 "The
 hole in any function is created when the limit exists at x = 1 but is
 not equal to the value of the function.  Many functions can be define
d to satisfy these conditions.  We define two in the following. " }}
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m$\"35.++++++NFco7$7$F]fmFgclFg\\n7$F[]n7$F]fm$\"3W/++++++vFco7$7$F]fm
F+F]]n7$Fa]n7$F]fm$\"3O++++++]6F-7$7$F]fmFjtFc]n7$Fg]n7$F]fm$\"3[+++++
+]:F-7$7$F]fmF]yFi]n7$F]^n7$F]fm$\"3S++++++]>F-7$7$F]fmFczF_^n7$Fc^n7$
F]fm$\"33++++++]BF-7$7$F]fmFi[lFe^n7$Fi^n7$F]fm$\"3+++++++]FF-7$7$F]fm
Fb]lF[_n7$F__n7$F]fm$\"3O++++++]JF-7$7$F]fmFd^lFa_n7$Fe_n7$F]fm$\"3#)*
***********\\NF-7$7$F]fmF__lFg_n7$F[`n7$F]fm$\"3=++++++]RF-7$7$F]fmF[`
lF]`n7$Fa`n7$F]fm$\"3a++++++]VF-7$7$F]fmFg`lFc`n7$Fg`n7$F]fm$\"3!4++++
++v%F-7$7$F]fmF]blFi`nFfs-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }
}}}{PARA 0 "" 0 "" {TEXT -1 140 " These two functions satisfy the give
n conditions. Maple does not plot the point (1, 2).  The functions hav
e a hole discontinuity at x = 1, " }}{PARA 0 "" 0 "" {TEXT -1 9 "becau
se  " }{XPPEDIT 19 1 "limit(f(x),x = 1);" "6#-%&limitG6$-%\"fG6#%\"xG/
F)\"\"\"" }{TEXT -1 28 " = 1/4, while f(1) = 2  and " }{XPPEDIT 19 1 "
limit(g(x),x = 1);" "6#-%&limitG6$-%\"gG6#%\"xG/F)\"\"\"" }{TEXT -1 
54 " = 1/16 while g(1) = 2.  This can be seen as follows: " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "limi
t (f(x), x = 1); f(1);limit(g(x), x = 1); g(1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 257 "" 0 "" 
{TEXT -1 102 "\n______________________________________________________
_______________________________________________" }{TEXT 257 1 "\n" }
{MPLTEXT 1 0 0 "" }{TEXT -1 1 "\n" }{TEXT 260 10 "ASSIGNMENT" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 267 13 " Problem 1:  " }}{PARA 0 "" 0 "" {TEXT -1 222 "Using a g
raphing utility (calculator or graphing program), sketch the following
 rational functions on one set of axes.  Discuss their similarities an
d their differences.  Include asymptotes, intercepts, domain, and rang
e.\n" }{XPPEDIT 18 0 "f(x) = 1/x;" "6#/-%\"fG6#%\"xG*&\"\"\"F)F'!\"\"
" }{TEXT -1 47 "                                               " }
{XPPEDIT 18 0 "g(x) = 1/(x^3);" "6#/-%\"gG6#%\"xG*&\"\"\"F)*$F'\"\"$!
\"\"" }{TEXT -1 41 "                                         " }
{XPPEDIT 18 0 "h(x) = 1/(x^5);" "6#/-%\"hG6#%\"xG*&\"\"\"F)*$F'\"\"&!
\"\"" }{TEXT -1 2 " \n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }
{TEXT 268 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 143 "Compare and \+
contrast the graphs of the functions in the solved example (even power
s) and the graphs of the functions in Problem 1 (odd powers)." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" }{TEXT 270 11 " \+
Problem 3:" }}{PARA 0 "" 0 "" {TEXT -1 229 "Using a graphing utility (
calculator or graphing program), sketch the following rational functio
ns on one set of axes.  Discuss their similarities and their differenc
es.  Include asymptotes, holes, intercepts, domain, and range.\n" }
{XPPEDIT 18 0 "f(x) = 1/(x-5);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F)\"\"&
!\"\"F," }{TEXT -1 55 "                                               \+
        " }{XPPEDIT 18 0 "g(x) = (x+2)/(x^2-3*x-10);" "6#/-%\"gG6#%\"x
G*&,&F'\"\"\"\"\"#F*F*,(*$F'F+F**&\"\"$F*F'F*!\"\"\"#5F0F0" }{TEXT -1 
0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 271 11 " Problem \+
4:" }}{PARA 0 "" 0 "" {TEXT -1 76 "Find all vertical and horizontal as
ymptotes for each of the given functions." }}{PARA 0 "" 0 "" {XPPEDIT 
18 0 "f(x) = 2*x^2/(3*x^2-12);" "6#/-%\"fG6#%\"xG*(\"\"#\"\"\"*$F'F)F*
,&*&\"\"$F**$F'F)F*F*\"#7!\"\"F1" }{TEXT -1 48 "                      \+
                          " }{XPPEDIT 18 0 "g(x) = 4/(x^2-5*x+6);" "6#
/-%\"gG6#%\"xG*&\"\"%\"\"\",(*$F'\"\"#F**&\"\"&F*F'F*!\"\"\"\"'F*F0" }
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 281 11 "
 Problem 5:" }}{PARA 0 "" 0 "" {TEXT -1 114 "Find a rational function \+
with vertical asymptotes at x = -2 and x = 4 and a slant asymptote.  S
ketch the function." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 
273 11 " Problem 6:" }}{PARA 0 "" 0 "" {TEXT -1 112 "Write equations f
or two different rational functions with the given properties:\n1.  Th
e horizontal asymptote is " }{XPPEDIT 18 0 "y = 4/5;" "6#/%\"yG*&\"\"%
\"\"\"\"\"&!\"\"" }{TEXT -1 82 "\n2. \011The vertical asymptotes occur
 at x = -2 and x = 4.     \nSketch the function.\n" }}}{PARA 0 "" 0 "
" {TEXT 258 1 " " }{TEXT -1 153 "_____________________________________
__________________________________\nMSEIP Grant #P120A010031:    \"Fou
r Colleges: Calculus + Enhancements\", 2001-2004\n" }}}{MARK "5 4 0" 
26 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }