################################################################
#
#  Example showing how to use the Parser to create functions you
#  can call from perl, to substitute values into a formula, and to
#  convert a formula to a form that can be used in graphics generated
#  on the fly.
#

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
  "PG.pl",
  "PGbasicmacros.pl",
  "PGanswermacros.pl",
  "PGgraphmacros.pl",
  "Parser.pl",
  "parserUtils.pl",
);

TEXT(&beginproblem);

##############################################
#
#  The setup
#

Context('Vector');
Context()->variables->add(a => 'Real', b => 'Real');
$a = non_zero_random(-4,-1,1);
$b = non_zero_random(-3,3,1);

#
#  The function to plot
#
$f = Formula("ax^2 + by");     #  the function to display

#
#  Traces to show
#
$x = non_zero_random(-1,1,1);
$y = non_zero_random(-1,1,1);

#
#  Graph domain and size
#
($xm,$xM) = (-2,2);
($ym,$yM) = (-2,2);
($zm,$zM) = (-5,5);
$size = [200,300];
$tex_size = 350;

##############################################

#
#  The plot defaults
#
@Goptions = (
  $ym,$zm,$yM,$zM,                            # dimensions of graph
  axes => [0,0], grid => [$yM-$ym,$zM-$zm],   # number of grid lines
  size => $size                               # pixel dimension
);
@imageoptions = (size=>$size, tex_size=>$tex_size);

$xdomain  = "x in <$xm,$xM>";
#$ydomain  = "y in <$ym,$yM>";  # plot_functions only handles variable x
$ydomain  = "x in <$ym,$yM>";
$plotoptions = "using color:red and weight:2";

#
#  Make the traces
#
$fx = $f->substitute(x=>$x, a=>$a, b=>$b, y=>'x')->reduce; # must have variable x
$Gx = init_graph(@Goptions);
plot_functions($Gx,"$fx for $ydomain $plotoptions");
$Xtrace = Image($Gx,@imageoptions);

$fy = $f->substitute(y=>$y, a=>$a, b=>$b)->reduce;
$Gy = init_graph(@Goptions);
plot_functions($Gy,"$fy for $xdomain $plotoptions");
$Ytrace = Image($Gy,@imageoptions);

#
#  Make the table of images
#
@rowopts = (indent=>0, separation=>30);
$Images = 
  BeginTable().
    AlignedRow([$Xtrace,$Ytrace], @rowopts).
    AlignedRow(["Trace for \(x=$x\)","Trace for \(y=$y\)"], @rowopts).
  EndTable();

##############################################

BEGIN_TEXT

The graphs below are traces for a function \(f(x,y)\) at \(x=$x\) and 
\(y=$y\).
$PAR

$Images
$PAR

If \(f(x,y) = \{$f->TeX\}\) then
\(a\) = \{ans_rule(6)\} and \(b\) = \{ans_rule(6)\}.

END_TEXT

##################################################

ANS(std_num_cmp($a));
ANS(std_num_cmp($b));

##################################################

ENDDOCUMENT();        # This should be the last executable line in the problem.