[npl] / trunk / NationalProblemLibrary / LoyolaChicago / Precalc / Chap9Sec5 / Q24.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# View of /trunk/NationalProblemLibrary/LoyolaChicago/Precalc/Chap9Sec5/Q24.pg

Fri Oct 7 16:42:13 2011 UTC (19 months, 2 weeks ago) by glarose
File size: 4581 byte(s)
LoyolaChicago 9.5: 4e tagging, updates.

    1 # DESCRIPTION
2 # Problem from Functions Modeling Change, Connally et al., 3rd ed.
3 # WeBWorK problem written by Adam Spiegler, <aspiegler@luc.edu>
4 # ENDDESCRIPTION
5
6 ## DBsubject('Precalculus')
7 ## DBchapter('Polynomial And Rational Functions')
8 ## DBsection('The Short-Run Behavior Of Rational Functions')
9 ## KEYWORDS('rational','fraction','polynomial,'asymptote','intercept')
10 ## TitleText1('Functions Modeling Change')
11 ## EditionText1('3')
12 ## AuthorText1('Connally')
13 ## Section1('9.5)
14 ## Problem1('24')
15 ## TitleText2('Functions Modeling Change');
16 ## EditionText2('4')
17 ## AuthorText2('Connally')
18 ## Section2('11.5')
19 ## Problem2('22')
21 ## Institution('Loyola University Chicago')
22
23 DOCUMENT();
24
26            "PGbasicmacros.pl",
27            "PGchoicemacros.pl",
29            "PGgraphmacros.pl",
30            "PGauxiliaryFunctions.pl",
32            "MathObjects.pl",
33            "PGcourse.pl",
35 );
36
37 TEXT(beginproblem());
38
39 Context("Point");
40
41 $showPartialCorrectAnswers = 1; 42 43$h = non_zero_random(-5,5,2);
44 $k = non_zero_random(-6,6,2); 45$top = $h+.01; 46$bot = $h - .01; 47 48 if ($h < 0) {$hor = "left";$disp_h = -$h} 49 else {$hor = "right"; $disp_h =$h};
50 if ($k < 0) {$ver = "down"; $disp_k = -$k}
51        else {$ver = "up";$disp_k = $k}; 52 53 54$f[0] = "1/(x-$h)+$k for x in <-10,$bot> using color:blue and weight:2"; 55$f[1] = "1/(x-$h)+$k for x in <$top,10> using color:blue and weight:2"; 56 57$graph = init_graph(-10,-10,10,10,'axes'=>[0,0],'ticks'=>[20,20]);
58 $graph->lb('reset'); 59 for ($i = 1; $i <= 4;$i++) {
60    $graph->lb(new Label(2*$i,-.1,2*$i,'black','center','top')); 61$graph->lb(new Label(-2*$i,-.1,-2*$i,'black','center','top'));
62    $graph->lb(new Label(-.1,2*$i,2*$i,'black','right','middle')); 63$graph->lb(new Label(-.1,-2*$i,-2*$i,'black','right','middle'))};
64 $graph->lb(new Label(9.8,-0.1,"x",'black','right','top')); 65$graph->lb(new Label(-.1,9.8,"y",'black','right','top'));
66 $graph->lb(new Label(8.7,-.05,"x",'black','left','bottom')); 67$graph->moveTo($h,$10);
68 $graph->lineTo($h,-10,'red');
69 $graph->moveTo(10,$k);
70 $graph->lineTo(-10,$k,'green');
71 plot_functions( $graph,$f[0],$f[1] ); 72$fig = image(insertGraph($graph), width => 300, height => 300, tex_size => 500); 73 74$A = -$h; 75$B = $k; 76$M = $k; 77$C = 1-$k*$h;
78 $D = -$h;
79
80 $xint = "-$C/$M"; 81$yint = "$C/$D";
82
83 Context()->texStrings;
84 BEGIN_TEXT
85
86 The graph below is a vertical and/or horizontal shift of $$y=1/x$$
87 (assume no reflections or compression/expansions have been applied).
88 $BR 89$BCENTER
90 $fig 91$ECENTER
92 $PAR 93 (a) The graph's equation can be written in the form 94 $f(x) = \frac{1}{x+A} + B$ 95 for constants $$A$$ and $$B$$. Based on the graph above, 96 find the values for $$A$$ and $$B$$. 97$BR
98 $$A =$$ \{ ans_rule(8) \} and $$B =$$ \{ ans_rule(8) \}
99 $PAR 100 (b) Now take your formula from part (a) and write it as the ratio 101 of two linear polynomials of the form, 102 $f(x) = \frac{M x + C}{x+D}$ 103 for constants $$M$$ , $$C$$, and $$D$$. What are the values 104 of $$M$$ , $$C$$, and $$D$$? 105$BR
106 $$M =$$ \{ ans_rule(8) \} , $$C =$$ \{ ans_rule(8) \},
107 and $$D =$$ \{ ans_rule(8) \}
108 $PAR 109 (c) Complete the exact values of the coordinates of the intercepts 110 of the graph. 111$BR
112 $$x$$-intercept: \{ ans_rule(10) \} \{AnswerFormatHelp('points')\}
113 $BR 114 $$y$$-intercept: \{ ans_rule(10) \} \{AnswerFormatHelp('points')\} 115 116 END_TEXT 117 Context()->normalStrings; 118 119 ANS( Compute($A)->cmp() );
120 ANS( Compute($B)->cmp() ); 121 ANS( Compute($M)->cmp() );
122 ANS( Compute($C)->cmp() ); 123 ANS( Compute($D)->cmp() );
124 ANS( Point( "($xint,0)" )->cmp() ); 125 ANS( Point( "(0,$yint)" )->cmp() );
126
127 $s = ($C/$M > 0 ) ? '-' : ''; 128 ($n,$d) = reduce(abs($C),abs($M)); 129$xfr = ( $d == 1 ) ? "$s$n" : "$s\frac{$n}{$d}";
130
131 $s = ($C/$D > 0 ) ? '' : '-'; 132 ($n,$d) = reduce(abs($C),abs($D)); 133$yfr = ( $d == 1 ) ? "$s$n" : "$s\frac{$n}{$d}";
134
135 Context()->texStrings;
136 SOLUTION(EV3(<<'END_SOLUTION'));
137 $PAR 138$BBOLD  SOLUTION $EBOLD 139$PAR
140 (a) The graph shows $$y= 1/x$$ shifted to
141 the $hor$disp_h and $ver$disp_k units. Thus,
142 $y = \frac{1}{x-h}+k$
143 is a choice for a formula.
144 $PAR 145 146 (b) The equation 147 $y = \frac{1}{x-h}+k = \frac{ M x + C}{x+D}.$ 148$PAR
149
150 (c) We see that the graph has both an $$x$$-and
151 $$y$$-intercept.
152 If $$y=0$$ then $$M x+C=0$$, so $$x= xfr$$.
153 The $$x$$-intercept is $$\left( xfr, 0 \right)$$.
154 \$BR
155 When $$x=0$$,
156 $y= f(0) = \frac{ M (0) + C}{0+D} = \frac{C}{D},$
157 so the $$y$$-intercept
158 is $$\left( 0, yfr \right)$$.
159
160 END_SOLUTION
161 Context()->normalStrings;
162
163
164 ENDDOCUMENT();