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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 42 "EXPLORING POLYNOMIAL FUNCTIONS
 GRAPHICALLY" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 
259 0 "" }{TEXT 260 20 "Precalculus Project\n" }}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 1 " " }{TEXT 271 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT 
-1 125 "To find patterns in the graphs of polynomial functions.\nTo co
nstruct a polynomial function using its end behavior and zeros.\n" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 272 17 "Solved Example 1
:" }}{PARA 0 "" 0 "" {TEXT -1 48 "A polynomial function is a function \+
of the form " }{XPPEDIT 18 0 "f(x) = a[n];" "6#/-%\"fG6#%\"xG&%\"aG6#%
\"nG" }{XPPEDIT 18 0 "x^n;" "6#)%\"xG%\"nG" }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "a[n-1];" "6#&%\"aG6#,&%\"nG\"\"\"F(!\"\"" }{XPPEDIT 18 
0 "x^(n-1);" "6#)%\"xG,&%\"nG\"\"\"F'!\"\"" }{TEXT -1 10 " +  ... + " 
}{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{XPPEDIT 18 0 "x;" "6#%\"xG
" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT 
-1 37 ",  where n is a nonnegative integer, " }{XPPEDIT 18 0 "a[0];" "
6#&%\"aG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"
\"\"" }{TEXT -1 8 ", ... , " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" 
}{TEXT -1 23 " are real numbers, and " }{XPPEDIT 18 0 "a[n] <> 0;" "6#
0&%\"aG6#%\"nG\"\"!" }{TEXT -1 435 ".   The graphs of polynomial funct
ions are smooth, continuous curves.   Patterns can be found in familie
s of polynomial functions that are useful in sketching polynomial func
tions.\n\nUsing a graphing utility (calculator or graphing program), s
ketch the following polynomial functions on one set of axes.  Discuss \+
their similarities and their differences, including their intercepts, \+
end behavior, domain and range.\n                     " }{XPPEDIT 18 
0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 46 "            \+
                                  " }{XPPEDIT 18 0 "g(x) = x^4;" "6#/-
%\"gG6#%\"xG*$F'\"\"%" }{TEXT -1 32 "                                \+
" }{XPPEDIT 18 0 "h(x) = x^6;" "6#/-%\"hG6#%\"xG*$F'\"\"'" }{TEXT -1 
6 "  \011\011\011 " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 
273 1 " " }{TEXT 274 9 "Solution:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "plot([x^2, x^4, x^6], x = -2..2, y = -5..10, title =
 \"Polynomials\", legend = [\"x^2\", \"x^4\", \"x^6\"]);      " }}
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{PARA 0 "" 0 "" {TEXT -1 140 "Each curve has an x-intercept at (0,0). \+
 The domain for all three curves is the set of all real numbers.  The \+
range for all three curves is " }}{PARA 0 "" 0 "" {TEXT -1 48 "       \+
                                         " }{XPPEDIT 18 0 "0 <= y;" "6
#1\"\"!%\"yG" }}{PARA 0 "" 0 "" {TEXT -1 167 " For all three polynomia
l functions as x increases without bounds, the graphs rise to the righ
t, and as x decreases without bounds, the graphs rise to the left.  Wh
en " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "1 < abs(x);" "6#2\"\"\"-%$absG6#
%\"xG" }{TEXT -1 84 "     and the power of the function increases, the
 curves move closer to the y-axis. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 0 "" }{TEXT 287 18 " Solved Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 
212 "Write an equation for a third degree polynomial function with zer
os (x-intercepts) of -2, 0, and 1, and whose end behavior indicates th
at the curve rises on the left and falls on the right.  Sketch the fun
ction.\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 288 0 "" }
{TEXT 289 10 " Solution:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT -1 
152 " -2 is a zero means  (x + 2) is a factor of the polynomial, 0 is \+
a zero means x is a factor and 1 is a zero means (x - 1) is a factor o
f the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 190 "   Therefore the re
quired equation is f(x) = a(x + 2)x (x - 1).  Since we want a third de
gree polynomial, a is a constant.  We choose a negative constant to sa
tisfy the required condition.  " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " f:= x-> -2*(x + 2)*x*(x - 1
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&arrowGF(,$**\"\"#\"\"\",&9$F/F.F/F/F1F/,&F1F/F/!\"\"F/F3F(F(F(" }}}
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39++v3'>$[FF0$!3OH9vt\"zIc%F37$$\"37++D6EjpGF0$!3bM\"*)fOq_A&F37$$\"\"
$F*$!#gF*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fc[l-%&TITLEG6#%,Graph~of~f~
G-%+AXESLABELSG6$Q\"x6\"Q\"yF\\\\l-%%VIEWG6$;F(Fhz;$FasF*$\"#?F*" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 240 " The graph of the third d
egree polynomial function has x-intercepts -2, 0 and 1.  As x decrease
s without bounds,  f(x), the polynomial with a negative coefficient, r
ises on the left and as x increases without bounds f(x) falls on the r
ight." }}}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 275 91 "_____________
______________________________________________________________________
________" }}{PARA 257 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }{TEXT 257 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
0 "" }{TEXT 276 11 " Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 245 "Using
 a graphing utility (calculator or graphing program), sketch the follo
wing polynomial functions on one set of axes.  Discuss their similarit
ies and differences including their intercepts, end behavior, domain a
nd range.\n                    " }{XPPEDIT 18 0 "f(x) = -x^2;" "6#/-%
\"fG6#%\"xG,$*$F'\"\"#!\"\"" }{TEXT -1 28 "                           \+
 " }{XPPEDIT 18 0 "g(x) = -x^4;" "6#/-%\"gG6#%\"xG,$*$F'\"\"%!\"\"" }
{TEXT -1 20 "                    " }{XPPEDIT 18 0 "h(x) = -x^6;" "6#/-
%\"hG6#%\"xG,$*$F'\"\"'!\"\"" }{TEXT -1 17 "                 " }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 277 11 " Problem 2:" }}
{PARA 0 "" 0 "" {TEXT -1 117 "Compare and contrast the graphs of the f
unctions in the solved example and the graphs of the functions in Prob
lem 1.\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 278 11 " Pro
blem 3:" }}{PARA 0 "" 0 "" {TEXT -1 193 "Using a graphing utility, ske
tch the following polynomial functions on one set of axes.  Discuss th
eir similarities and differences including their intercepts, end behav
ior, domain and range. \011" }}{PARA 0 "" 0 "" {TEXT -1 19 "          \+
         " }{XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\"fG6#%\"xG*$F'\"\"$" }
{TEXT -1 30 "                              " }{XPPEDIT 18 0 "g(x) = x^
5;" "6#/-%\"gG6#%\"xG*$F'\"\"&" }{TEXT -1 22 "                      " 
}{XPPEDIT 18 0 "h(x) = x^7;" "6#/-%\"hG6#%\"xG*$F'\"\"(" }{TEXT -1 10 
"        \011 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 0 "" }{TEXT 279 11 " Problem 4:" }}{PARA 0 "" 0 "" 
{TEXT -1 212 "Using a graphing utility, sketch the following polynomia
l functions on one set of axes.  Discuss their similarities and differ
ences including their intercepts, end behavior, domain and range.\011 \+
                   " }{XPPEDIT 18 0 "f(x) = -x^3;" "6#/-%\"fG6#%\"xG,$
*$F'\"\"$!\"\"" }{TEXT -1 28 "                            " }{XPPEDIT 
18 0 "g(x) = -x^5;" "6#/-%\"gG6#%\"xG,$*$F'\"\"&!\"\"" }{TEXT -1 20 " \+
                   " }{XPPEDIT 18 0 "h(x) = -x^7;" "6#/-%\"hG6#%\"xG,$
*$F'\"\"(!\"\"" }{TEXT -1 8 "        " }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 280 11 " Problem 5:" }}{PARA 0 "" 0 "" {TEXT -1 
108 "Compare and contrast the graphs of the functions in Problem 3 and
 the graphs of the functions in Problem 4.\n" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 0 "" }{TEXT 281 11 " Problem 6:" }}{PARA 0 "" 0 "" {TEXT 
-1 198 "Using a graphing utility, sketch the following polynomial func
tions on one set of axes.  Discuss their similarities and differences \+
including their intercepts, end behavior, domain and range.\011  \011 \+
  " }}{PARA 0 "" 0 "" {TEXT -1 19 "                   " }{XPPEDIT 18 
0 "f(x) = x^2-x;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F'!\"\"" }{TEXT 
-1 24 "                        " }{XPPEDIT 18 0 "g(x) = x^4-x^2;" "6#/
-%\"gG6#%\"xG,&*$F'\"\"%\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 15 "          \+
     " }{XPPEDIT 18 0 "h(x) = x^4+x^3-2*x^2;" "6#/-%\"hG6#%\"xG,(*$F'
\"\"%\"\"\"*$F'\"\"$F+*&\"\"#F+*$F'F/F+!\"\"" }{TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 282 11 " Problem 7:" }}{PARA 0 "" 0 "" {TEXT -1 199 "Using a gra
phing utility, sketch the following polynomial functions on one set of
 axes.  Discuss their similarities and differences including their int
ercepts, end behavior, domain and range.\011  \011   \011" }}{PARA 0 "
" 0 "" {TEXT -1 20 "                    " }{XPPEDIT 18 0 "f(x) = -x^2-
x;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#!\"\"F'F+" }{TEXT -1 22 "             \+
         " }{XPPEDIT 18 0 "g(x) = -x^4+4*x^2;" "6#/-%\"gG6#%\"xG,&*$F'
\"\"%!\"\"*&F*\"\"\"*$F'\"\"#F-F-" }{TEXT -1 10 "          " }
{XPPEDIT 18 0 "h(x) = -x^4+x^3+2*x^2;" "6#/-%\"hG6#%\"xG,(*$F'\"\"%!\"
\"*$F'\"\"$\"\"\"*&\"\"#F.*$F'F0F.F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 283 
11 " Problem 8:" }}{PARA 0 "" 0 "" {TEXT -1 195 "Using a graphing util
ity, sketch the following polynomial functions on one set of axes.  Di
scuss their similarities and differences including their intercepts, e
nd behavior, domain and range.\011  \011" }}{PARA 0 "" 0 "" {TEXT -1 
20 "                    " }{XPPEDIT 18 0 "f(x) = x^3-x;" "6#/-%\"fG6#%
\"xG,&*$F'\"\"$\"\"\"F'!\"\"" }{TEXT -1 25 "                         \+
" }{XPPEDIT 18 0 "g(x) = x^5-4*x^3;" "6#/-%\"gG6#%\"xG,&*$F'\"\"&\"\"
\"*&\"\"%F+*$F'\"\"$F+!\"\"" }{TEXT -1 12 "            " }{XPPEDIT 18 
0 "h(x) = -x^3+2*x^2;" "6#/-%\"hG6#%\"xG,&*$F'\"\"$!\"\"*&\"\"#\"\"\"*
$F'F-F.F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 284 11 " Problem 9:" }}{PARA 0 "
" 0 "" {TEXT -1 74 "Compare and contrast the graphs of the functions i
n Problems 6, 7, and 8.\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 285 12 " Problem 10:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 286 12 " Problem 11:" }}{PARA 
0 "" 0 "" {TEXT -1 195 "Write an equation for a fourth degree polynomi
al function with zeros of -3, 0, and 3, and whose end behavior indicat
es that the curve falls on both the left and right side.  Sketch the f
unction.\n" }}}{PARA 0 "" 0 "" {TEXT -1 166 " ________________________
________________________________________________________\nMSEIP Grant \+
#  P120A010031      \"Four Colleges: Calculus + Enhancements\", 2001-2
004\n" }}}{MARK "1 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }