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

    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')
   20 ## Author('Adam Spiegler')
   21 ## Institution('Loyola University Chicago')
   22 
   23 DOCUMENT();
   24 
   25 loadMacros("PG.pl",
   26            "PGbasicmacros.pl",
   27            "PGchoicemacros.pl",
   28            "PGanswermacros.pl",
   29            "PGgraphmacros.pl",
   30            "PGauxiliaryFunctions.pl",
   31            "extraAnswerEvaluators.pl",
   32            "MathObjects.pl",
   33            "PGcourse.pl",
   34      "AnswerFormatHelp.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();