qu.1.topic=Section 1.1.1: Basic Rules@

qu.1.1.mode=Plain Number@
qu.1.1.name=1.1.7 Functions@
qu.1.1.comment=<p class="noindent">Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${c}</mn>
 </math>
into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <mo>+</mo>
  <mn>${b}</mn>
  <mi>x</mi>
 </math>
gives
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${c}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${a}</mn>
 <mo>+</mo>
 <mn>${b}</mn>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${c}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${ans}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=int(rint(5)+1);
$b=int(rint(5)+2);
$c=int(rint(5)+1);
$ans=int($a+($b)*($c));@
qu.1.1.uid=5aa57827-2c0b-493a-a4d2-1bc6a92bc555@
qu.1.1.question=<p class="noindent">Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <mo>+</mo>
  <mn>${b}</mn>
  <mi>x</mi>
 </math>
find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.1.1.answer=
${ans} @

qu.1.2.mode=Blanks@
qu.1.2.name=1.1.2 Non-linear@
qu.1.2.comment=<p class="noindent">On the interval from 0 to 1 the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
increases by ${diffh1}. On the interval from 1 to 2 the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
increases by ${diffh2}. And on the interval from 2 to 3 the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
increases by ${diffh3}. Thus, the function does not have a constant rate of change and it could therefore
cannot be a linear function.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$c1=int(20+rint(5));
$f=int(2+rint(4));
$f1=int($c1);
$f2=int(-$f+$c1);
$f3=int(-$f*2+$c1);
$f4=int(-$f*3+$c1);
$diff=$f;
$b2=int(20+rint(5));
$g=int(2+rint(4));
$g1=int($g*15+$b2);
$g2=int($g*20+$b2);
$g3=int($g*25+$b2);
$g4=int($g*30+$b2);
$diffg=int(5*$g);
$h1=int(10);
$diffh1=int(rint(5)+1 );
$h2=int($h1+$diffh1);
$diffh2=int(rint(5)+1 );
$h3=int($h2+$diffh2);
$diffh3=$diffh2+1;
$h4=int($h3+$diffh3);
$anse=$g t + $b2@
qu.1.2.uid=f6786205-2d31-436a-9911-2ea3db260fe0@
qu.1.2.info=  difficulty=easy;
@
qu.1.2.question=<p class="noindent">Does the following table represent a linear function?
</p>
<div class="tabular">
 <table class="tabular" cellspacing="0" cellpadding="0" rules="groups" frame="border">
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
   <col/>
   <col/>
   <col/>
  </colgroup>
  <tr valign="baseline">
   <td align="center" style="white-space:nowrap;" class="td11">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>t</mi>
    </math>
   </td>
   <td align="center" style="white-space:nowrap;" class="td11">15</td>
   <td align="center" style="white-space:nowrap;" class="td11">20</td>
   <td align="center" style="white-space:nowrap;" class="td11">25</td>
   <td align="center" style="white-space:nowrap;" class="td11">30</td>
  </tr>
  <tr class="hline">
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
  </tr>
  <tr valign="baseline">
   <td align="center" style="white-space:nowrap;" class="td11">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>s</mi>
    </math>
   </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${h1}  </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${h2}  </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${h3}  </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${h4}  </td>
  </tr>
 </table>
</div>
<br class="newline"/>
<br class="newline"/><1>@
qu.1.2.blank.1=non-linear, linear@
qu.1.2.grader.1=menu@
qu.1.2.extra=@

qu.1.3.mode=Multipart@
qu.1.3.name=1.1.3 Functions@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$v1=int(rint(3)+1);
$f1=int($v1*$v1);
$g1=int(3*$v1-1);
$a1=int($f1+$g1);
$v2=int($v1+1);
$f2=int($v2*$v2);
$g2=int(3*$v2-1);
$a2=int($f2*$g2);
$v3=int($v1+2);
$g3=int(3*$v3-1);
$a3=int($g3*$g3);
$f3=int($v3*$v3);
$a4=int(3*$f3-1);@
qu.1.3.uid=f13afc9b-a047-4726-a5a2-9803c08d2d21@
qu.1.3.question=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>3</mn>
  <mi>x</mi>
  <mo>&minus;</mo>
  <mn>1</mn>
 </math>.
Find the following:</p>@
qu.1.3.weighting=1,1,1,1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>+</mo>
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>@
qu.1.3.part.1.answer=${a1}@
qu.1.3.part.1.mode=Plain Number@
qu.1.3.part.1.comment=<p class="noindent">Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${f1}</mn>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${g1}</mn>
  <mo>.</mo>
 </math>
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>+</mo>
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${f1}</mn>
  <mo>+</mo>
  <mn>${g1}</mn>
  <mo>=</mo>
  <mn>${a1}</mn>
 </math>
</p>@
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.question=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v2}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>&sdot;</mo>
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v2}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>@
qu.1.3.part.2.answer=${a2}@
qu.1.3.part.2.mode=Plain Number@
qu.1.3.part.2.comment=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v2}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>&sdot;</mo>
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v2}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${f2}</mn>
  <mo>&sdot;</mo>
  <mn>${g2}</mn>
  <mo>=</mo>
  <mn>${a2}</mn>
 </math>
</p>@
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.question=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>g</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${v3}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>@
qu.1.3.part.3.answer=${a3}@
qu.1.3.part.3.mode=Plain Number@
qu.1.3.part.3.comment=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v3}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${g3}</mn>
 </math> so
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>g</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${v3}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${g3}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a3}</mn>
 </math>
</p>@
qu.1.3.part.4.editing=useHTML@
qu.1.3.part.4.question=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${v3}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
</p>@
qu.1.3.part.4.answer=${a4}@
qu.1.3.part.4.mode=Plain Number@
qu.1.3.part.4.comment=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${v3}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${f3}</mn>
 </math> so
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>f</mi>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${v3}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mi>g</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${f3}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a4}</mn>
 </math>
</p>@

qu.1.4.mode=Equation@
qu.1.4.name=1.1.1 Slope of a line@
qu.1.4.comment=<p class="noindent">The slope is
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${y1}</mn>
    <mo>&minus;</mo>
    <mn>${y0}</mn>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mn>${slope}</mn>
</math>
<p class="nopar"> so the equation of the line is
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>y</mi>
 <mo>=</mo>
 <mn>${slope}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${intercept}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$slope=int(rint(3)+3);
$intercept=int(rint(3)+3);
$x0=int(rint(5)+1);
$y0=int($intercept+$slope*$x0);
$x1=int(rint(5)+$x0+1);
$y1=int($intercept+$slope*$x1);@
qu.1.4.uid=fd19645f-074f-4037-aede-e50b55202461@
qu.1.4.question=<p class="noindent">Find the equation of the line that passes through the points
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.4.answer=y=${slope} x
+ ${intercept}@

qu.1.5.mode=Multipart@
qu.1.5.name=1.1.2 Linear@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$slope=int(-(rint(3)+3));
$intercept=int(rint(3)+35);
$x0=int(rint(5)+1);
$dx=int(rint(3)+1);
$y0=int($intercept+$slope*$x0);
$x1=int($x0+$dx);
$y1=int($intercept+$slope*$x1);
$x2=int($x1+$dx);
$y2=int($intercept+$slope*$x2);
$x3=int($x2+$dx);
$y3=int($intercept+$slope*$x3);
$diff=abs($y1-$y0);@
qu.1.5.uid=97c15d94-e258-4fc4-8662-f899e5a0543e@
qu.1.5.info=  difficulty=easy;
@
qu.1.5.question=@
qu.1.5.weighting=1,1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.extra=@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=<p class="noindent">Does the following table represent a linear function?
</p>
<div class="tabular">
 <table class="tabular" cellspacing="0" cellpadding="0" rules="groups" frame="border">
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
   <col/>
   <col/>
   <col/>
  </colgroup>
  <tr valign="baseline">
   <td align="center" style="white-space:nowrap;" class="td11">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>x</mi>
    </math>
   </td>
   <td align="center" style="white-space:nowrap;" class="td11">${x0}</td>
   <td align="center" style="white-space:nowrap;" class="td11">${x1}</td>
   <td align="center" style="white-space:nowrap;" class="td11">${x2}</td>
   <td align="center" style="white-space:nowrap;" class="td11">${x3}</td>
  </tr>
  <tr class="hline">
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
  </tr>
  <tr valign="baseline">
   <td align="center" style="white-space:nowrap;" class="td11">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>y</mi>
    </math>
   </td>
   <td align="center" style="white-space:nowrap;" class="td11">${y0}</td>
   <td align="center" style="white-space:nowrap;" class="td11">${y1}</td>
   <td align="center" style="white-space:nowrap;" class="td11">${y2}</td>
   <td align="center" style="white-space:nowrap;" class="td11">${y3}</td>
  </tr>
 </table>
</div>
<br class="newline"/>
<br class="newline"/><1>@
qu.1.5.part.1.blank.1=linear, non-linear@
qu.1.5.part.1.grader.1=menu@
qu.1.5.part.1.info=  difficulty=easy;
@
qu.1.5.part.1.mode=Blanks@
qu.1.5.part.1.comment=<p class="noindent">On the interval from ${x1} to ${x2} the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
decreases by ${diff}. On the interval from ${x2} to ${x3} the value of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math> decreases by ${diff}. And on the
interval from ${x2} to ${x3} the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
decreases by ${diff}. Thus, the function has a constant rate of change and it could therefore be
linear.</p>@
qu.1.5.part.2.editing=useHTML@
qu.1.5.part.2.question=<p class="noindent">Find the equation of the line that passes through the points in the table in part (a). </p>@
qu.1.5.part.2.answer=y=${slope} x +
${intercept}@
qu.1.5.part.2.mode=Equation@
qu.1.5.part.2.comment=<p class="noindent">The slope is
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${y1}</mn>
    <mo>&minus;</mo>
    <mn>${y0}</mn>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mn>${slope}</mn>
</math>
<p class="nopar"> so the equation of the line is
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>y</mi>
 <mo>=</mo>
 <mn>${slope}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${intercept}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@

qu.1.6.mode=Multipart@
qu.1.6.name=1.1.9 Slope and intercept@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$a=int(rint(5)+2);
$k=int(rint(5)+1);
$l=int(rint(5)+2);
$b=int($a*$k);
$c=int($a*$l);
$slope=-$k;
$intercept=$l;@
qu.1.6.uid=c37ccec8-2e65-4af6-8ebc-bf16956615e9@
qu.1.6.question=<p class="noindent">Determine the slope and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
of the line whose equation is given by
</p>
<p class="noindent"/>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mn>${a}</mn>
 <mi>y</mi>
 <mo>+</mo>
 <mn>${b}</mn>
 <mi>x</mi>
 <mo>=</mo>
 <mn>${c}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.6.weighting=1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=<p class="noindent">The slope is</p>@
qu.1.6.part.1.answer=${slope}@
qu.1.6.part.1.mode=Formula@
qu.1.6.part.1.comment=<p class="noindent">Rewriting the equation as
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mn>${a}</mn>
 <mi>y</mi>
 <mo>=</mo>
 <mo>&minus;</mo>
 <mn>${b}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${c}</mn>
</math>
<p class="nopar"> shows that the line has slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mo>&minus;</mo>
  <mn>${b}</mn>
  <mo>&#8725;</mo>
  <mn>${a}</mn>
 </math>.
</p>@
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.question=<p class="noindent">The intercept is</p>@
qu.1.6.part.2.answer=${intercept}@
qu.1.6.part.2.mode=Formula@
qu.1.6.part.2.comment=<p class="noindent">Rewriting the equation as
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mn>${a}</mn>
 <mi>y</mi>
 <mo>=</mo>
 <mo>&minus;</mo>
 <mn>${b}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${c}</mn>
</math>
<p class="nopar"> shows that the vertical intercept is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${c}</mn>
  <mo>&#8725;</mo>
  <mn>${a}</mn>
 </math>.</p>@

qu.1.7.mode=Equation@
qu.1.7.name=1.1.11 Equation of a line@
qu.1.7.comment=<p class="noindent">The slope is
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${y1}</mn>
    <mo>&minus;</mo>
    <mn>${y0}</mn>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mn>${slope}</mn>
</math>
<p class="nopar"> so the equation of the line is

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>y</mi>
 <mo>=</mo>
 <mn>${slope}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${intercept}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$slope=int(rint(3)+3);
$intercept=int(rint(3)+3);
$x0=int(rint(5)+1);
$y0=int($intercept+$slope*$x0);
$x1=int(rint(5)+$x0+1);
$y1=int($intercept+$slope*$x1);@
qu.1.7.uid=964d7da5-0722-4f66-bf22-6b47f4d5bcc3@
qu.1.7.question=<p class="noindent">Find the equation of the line that passes through the points
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.7.answer=y=${slope} x
+ ${intercept}@

qu.1.8.mode=Multipart@
qu.1.8.name=1.1.15 Application@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$b=int(10+rint(4));
$m=int(rint(3)+1)/10;
$m1=int(100*$m);
$c1=int($b+100*$m);
$c2=int($b+180*$m);@
qu.1.8.uid=0244793c-0ad8-481b-adea-4c566f55207c@
qu.1.8.info=  difficulty=easy;
@
qu.1.8.question=<p class="noindent">The monthly charge for a waste collection service is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${c1}</mn>
 </math> dollars for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>100</mn>
 </math> kg of waste
and is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${c2}</mn>
 </math>
dollars for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>180</mn>
 </math>
kg of waste. </p>@
qu.1.8.weighting=1,1,1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.question=<p class="noindent">Find a linear formula for the cost, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>C</mi>
 </math>,
of waste collection as a function of the number of kilograms of waste,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>w</mi>
 </math>.
<br class="newline"/>
 <br class="newline"/>(Give any coefficients correct to 1 decimal place.) </p>@
qu.1.8.part.1.answer=
C=${b}+${m}w@
qu.1.8.part.1.info=  difficulty=medium;
@
qu.1.8.part.1.mode=Equation@
qu.1.8.part.1.comment=<p class="noindent">We find the slope, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math>,
and intercept, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
 </math>, in
the linear equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>C</mi>
  <mo>=</mo>
  <mi>b</mi>
  <mo>+</mo>
  <mi>m</mi>
  <mi>w</mi>
 </math>.
To find the slope we use

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>m</mi>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mi>&Delta;</mi>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>&Delta;</mi>
    <mi>w</mi>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>${c2}</mn>
    <mo>&minus;</mo>
    <mn>${c1}</mn>
   </mrow>
   <mrow>
    <mn>180</mn>
    <mo>&minus;</mo>
    <mn>100</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mn>${m}</mn>
</math>
<p class="nopar"> We substitute to find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
 </math>:
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtable columnspacing="0" columnalign="right center left">
  <mtr>
   <mtd>
    <mi>C</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mi>b</mi>
    <mo>+</mo>
    <mi>m</mi>
    <mi>w</mi>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mn>32</mn>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mi>b</mi>
    <mo>+</mo>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${m}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>100</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mi>b</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${b}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
 </mtable>
</math>
<p class="nopar">
The linear formula is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>C</mi>
  <mo>=</mo>
  <mn>${b}</mn>
  <mo>+</mo>
  <mn>${m}</mn>
  <mi>w</mi>
 </math>.
</p>@
qu.1.8.part.2.blank.4=kilogram, dollar@
qu.1.8.part.2.blank.3=kilogram, dollar@
qu.1.8.part.2.blank.2=dollars, kilograms@
qu.1.8.part.2.blank.1=%24%7bm%7d@
qu.1.8.part.2.info=  difficulty=easy;
@
qu.1.8.part.2.extra=@
qu.1.8.part.2.editing=useHTML@
qu.1.8.part.2.question=<p class="noindent">The slope of the line found in part (a) is <1><2> per <3><br class="newline"/>This means that for each additional <4> the additional cost is <5> cents.</p>@
qu.1.8.part.2.comment=<p class="noindent">The slope of the line found in part (a) is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${m}</mn>
 </math>
dollars per kilogram. Each additional kilogram of waste costs an additional
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${m1}</mn>
 </math> cents. </p>@
qu.1.8.part.2.mode=Blanks@
qu.1.8.part.2.grader.5=formula@
qu.1.8.part.2.grader.4=menu@
qu.1.8.part.2.grader.3=menu@
qu.1.8.part.2.grader.2=menu@
qu.1.8.part.2.grader.1=formula@
qu.1.8.part.2.blank.5=%24%7bm1%7d@
qu.1.8.part.3.blank.3=no waste, average amount of waste, overfill the trash can@
qu.1.8.part.3.blank.2=dollars, kilograms@
qu.1.8.part.3.blank.1=%24%7bb%7d@
qu.1.8.part.3.info=  difficulty=easy;
@
qu.1.8.part.3.extra=@
qu.1.8.part.3.editing=useHTML@
qu.1.8.part.3.question=<p class="noindent">The vertical intercept of the line found in part (a) is <1>. The units of this intercept is <2>. This is the
amount charges if you have <3>. </p>@
qu.1.8.part.3.comment=<p class="noindent">The flat monthly fee to subscribe to the waste collection service is
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${b}</mn>
 </math>
dollars. This is the amount charged even if there is no waste.</p>@
qu.1.8.part.3.mode=Blanks@
qu.1.8.part.3.grader.3=menu@
qu.1.8.part.3.grader.2=menu@
qu.1.8.part.3.grader.1=formula@

qu.1.9.mode=Blanks@
qu.1.9.name=1.1.4 Odd/Even functions@
qu.1.9.comment=<p class="noindent">The function is ${correcta} because<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
${sign} <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mo>&minus;</mo>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$type=rint(3);
$n1=switch($type,2,1,2);
$n2=switch($type,4,3,1);
$a=switch($type,rint(3),-rint(3),rint(3));
$b0=switch($type,-rint(3),-rint(3),rint(3));
$b=if(eq($a,$b0),$b0+1,$b0);
$correct=switch($type,"Even","Odd","Neither");
$correcta=switch($type," even "," odd "," neither odd nor even ");
$incorrect1=switch($type,"Odd","Neither","Even");
$incorrect2=switch($type,"Neither","Even","Odd");
$sign=switch($type," is the same as "," is the same as -"," is not related to ");@
qu.1.9.uid=5a2abb6f-70b3-4ad7-b4c1-30dbe8c098ba@
qu.1.9.question=<p class="noindent">Is the function shown below odd, even or neither?

</p>
<applet code="applets.grapher.Graph" width="250" height="250" archive="graphing.jar">
 <param name="y1" value="${a}*x^(${n1})+${b}*x^(${n2})"/>
 <param name="gridLines" value="1"/>
 <param name="xMin" value="-4"/>
 <param name="xMax" value="4"/>
 <param name="yMin" value="-4"/>
 <param name="yMax" value="4"/>
</applet><1>@
qu.1.9.blank.1=%24%7bcorrect%7d, %24%7bincorrect1%7d, %24%7bincorrect2%7d@
qu.1.9.grader.1=menu@
qu.1.9.extra=@

qu.1.10.mode=Equation@
qu.1.10.name=1.1.13 Application@
qu.1.10.comment=<p class="noindent">This is a linear function with vertical intercept
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${a}</mn>
 </math> and
slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${b}</mn>
 </math>.</p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$a=int(20+rint(6));
$b=int(rint(8)+1)/100;
$b1=int(100*$b);@
qu.1.10.uid=51211d74-922f-4115-bfcb-d32ae461c1a9@
qu.1.10.info=  difficulty=easy;
@
qu.1.10.question=<p class="noindent">A cell phone company charges a monthly fee of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${a}</mn>
 </math> dollars
plus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${b1}</mn>
 </math>
cents per minute. Find a formula for the monthly charge,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>C</mi>
 </math>, in dollars, as a function
of the number of minutes, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math>,
the phone is used during the month.</p>@
qu.1.10.answer=
C=${a}+${b} m@

qu.1.11.mode=Plain Number@
qu.1.11.name=1.1.8 Functions@
qu.1.11.comment=<p class="noindent">Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${c}</mn>
 </math>
into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <mi>x</mi>
  <mo>&minus;</mo>
  <mn>${b}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math>
gives
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${c}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${a}</mn>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${c}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>&minus;</mo>
 <mn>${b}</mn>
 <msup>
  <mrow>
   <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
     <mn>${c}</mn>
    </mrow>
    <mo stretchy="false">)</mo>
   </mrow>
  </mrow>
  <mrow>
   <mn>2</mn>
  </mrow>
 </msup>
 <mo>=</mo>
 <mn>${ans}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$a=int(rint(5)+2);
$b=int(rint(5)+2);
$c=int(rint(5)+1);
$ans=int($a*$c-$b*$c*$c);@
qu.1.11.uid=87e5bf7f-ad0b-4875-92d0-a31b11d91083@
qu.1.11.question=<p class="noindent">Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <mi>x</mi>
  <mo>&minus;</mo>
  <mn>${b}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math>
find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${c}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.</p>@
qu.1.11.answer=${ans} ?
0.1@

qu.1.12.mode=Multipart@
qu.1.12.name=1.1.14 Application@
qu.1.12.comment=@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$a=int(40+rint(9)+1);
$b=int($a+10);@
qu.1.12.uid=3f86541d-e500-4f22-a56f-f049ff58a0ac@
qu.1.12.info=  difficulty=medium;
@
qu.1.12.question=<p class="noindent">Company A rents cars at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${a}</mn>
 </math>
dollars a day and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>15</mn>
 </math>
cents a mile. Its competitor's, Company B, cars are
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${b}</mn>
 </math> dollars
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>10</mn>
 </math>
cents per mile. </p>@
qu.1.12.weighting=1,1,1,1@
qu.1.12.numbering=alpha@
qu.1.12.part.1.editing=useHTML@
qu.1.12.part.1.question=<p class="noindent">Give a formula for, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>A</mi>
   </mrow>
  </msub>
 </math>,
the cost of renting a car from Company A for a day as a function of,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math>, the
distance travelled</p>@
qu.1.12.part.1.answer=
${a}+0.15 m@
qu.1.12.part.1.info=  difficulty=easy;
@
qu.1.12.part.1.mode=Formula@
qu.1.12.part.1.comment=<p class="noindent">Company A's price for a day's rental with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math>
miles is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>A</mi>
   </mrow>
  </msub>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>m</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <mo>+</mo>
  <mn>0.15</mn>
  <mi>m</mi>
 </math>.</p>@
qu.1.12.part.2.editing=useHTML@
qu.1.12.part.2.question=<p class="noindent">Give a formula for, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>B</mi>
   </mrow>
  </msub>
 </math>
the cost of renting a car from the competitor company for a day as a function of,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math>, the
distance travelled</p>@
qu.1.12.part.2.answer=
${b}+0.10 m@
qu.1.12.part.2.info=  difficulty=easy;
@
qu.1.12.part.2.mode=Formula@
qu.1.12.part.2.comment=<p class="noindent">Company B's price for a day's rental with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>m</mi>
 </math>
miles is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>B</mi>
   </mrow>
  </msub>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>m</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${b}</mn>
  <mo>+</mo>
  <mn>0.10</mn>
  <mi>m</mi>
 </math>.</p>@
qu.1.12.part.3.blank.2=Company A, Company B@
qu.1.12.part.3.blank.1=Company B, Company A@
qu.1.12.part.3.extra=@
qu.1.12.part.3.editing=useHTML@
qu.1.12.part.3.question=<p class="noindent">If you intend to travel less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>100</mn>
 </math>
miles, which company should you select?
<br class="newline"/><1><br class="newline"/>What about if you intended to travel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>250</mn>
 </math>
miles?
<br class="newline"/><2></p>@
qu.1.12.part.3.comment=<p class="noindent">Plotting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>A</mi>
   </mrow>
  </msub>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>B</mi>
   </mrow>
  </msub>
 </math>
shows that it is cheaper to use the first company if you intend to travel only
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>100</mn>
 </math> miles. If you
intended to travel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>250</mn>
 </math>
miles then the competitor is cheaper.
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" archive="graphing.jar">
 <param name="y1" value="(${a}+0.15*x)"/>
 <param name="y2" value="(${b}+0.10*x)"/>
 <param name="gridLines" value="1"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="500"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="200"/>
</applet>
<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>A</mi>
   </mrow>
  </msub>
 </math> is shown
in blue, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>B</mi>
   </mrow>
  </msub>
 </math> in
red. The horizontal axis shows the miles traveled.)
<br class="newline"/>
</p>@
qu.1.12.part.3.mode=Blanks@
qu.1.12.part.3.grader.2=menu@
qu.1.12.part.3.grader.1=menu@
qu.1.12.part.4.editing=useHTML@
qu.1.12.part.4.question=<p class="noindent">At what mileage should you switch from the first company to its competitor?</p>@
qu.1.12.part.4.answer=
200 ? 1@
qu.1.12.part.4.info=  difficulty=medium;
@
qu.1.12.part.4.mode=Plain Number@
qu.1.12.part.4.comment=<p class="noindent">We need to determine where he two lines intersect. We let
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>A</mi>
   </mrow>
  </msub>
  <mo>=</mo>
  <msub>
   <mrow>
    <mi>C</mi>
   </mrow>
   <mrow>
    <mi>B</mi>
   </mrow>
  </msub>
 </math>,
thus
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtable columnspacing="0" columnalign="right center left">
  <mtr>
   <mtd>
    <mn>${a}</mn>
    <mo>+</mo>
    <mn>0.15</mn>
    <mi>m</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${b}</mn>
    <mo>+</mo>
    <mn>0.10</mn>
    <mi>m</mi>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mn>0.05</mn>
    <mi>m</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>10</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mi>m</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>200</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
 </mtable>
</math>
<p class="nopar">
If you are going more that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>200</mn>
 </math>
miles a day, then Company B is cheaper. If you are going less than
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>200</mn>
 </math> miles a
day, the Company A is cheaper. </p>@

qu.1.13.mode=Multipart@
qu.1.13.name=1.1.5 Functions@
qu.1.13.comment=@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$a=int(rint(2)+2);
$b=int(rint(3)+1);
$x=int(rint(3)+1);
$ansb=int($a*$x*($x)+ $b);
$xc=int($x+1);
$xcm=int(-$xc);
$c1=int($xc*$xc);
$ansc=int($a*($xc)*($xc) + $b);
$qd=int($b-1);
$xmax=int(2*$xc);
$ymax=int($a*$xmax*$xmax + $b + 2);@
qu.1.13.uid=0767dc7b-5b64-4d9d-93a3-126308df23a1@
qu.1.13.question=<p class="noindent">Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>${b}</mn>
 </math>.
</p>@
qu.1.13.weighting=1,1,1,1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.question=<p class="noindent">Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
is zero. </p>@
qu.1.13.part.1.answer=
${b}? 0.05@
qu.1.13.part.1.mode=Plain Number@
qu.1.13.part.1.comment=<p class="noindent">We are asked for the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math> is zero. That
is, we are asked for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>0</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>.
Plugging in we get

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>0</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${a}</mn>
 <msup>
  <mrow>
   <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
     <mn>0</mn>
    </mrow>
    <mo stretchy="false">)</mo>
   </mrow>
  </mrow>
  <mrow>
   <mn>2</mn>
  </mrow>
 </msup>
 <mo>+</mo>
 <mn>${b}</mn>
 <mo>=</mo>
 <mn>0</mn>
 <mo>+</mo>
 <mn>${b}</mn>
 <mo>=</mo>
 <mn>${b}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.13.part.2.extra=@
qu.1.13.part.2.editing=useHTML@
qu.1.13.part.2.question=<p class="noindent">
 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
 </math><1></p>@
qu.1.13.part.2.blank.1=%24%7bansb%7d %3f%0a0.05@
qu.1.13.part.2.grader.1=formula@
qu.1.13.part.2.mode=Blanks@
qu.1.13.part.2.comment=<p class="noindent">Substituting we get
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>f</mi>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${x}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${a}</mn>
 <msup>
  <mrow>
   <mrow>
    <mo stretchy="false">(</mo>
    <mrow>
     <mn>${x}</mn>
    </mrow>
    <mo stretchy="false">)</mo>
   </mrow>
  </mrow>
  <mrow>
   <mn>2</mn>
  </mrow>
 </msup>
 <mo>+</mo>
 <mn>${b}</mn>
 <mo>=</mo>
 <mn>${ansb}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.13.part.3.editing=useHTML@
qu.1.13.part.3.question=<p class="noindent">Which values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math> a
value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${ansc}</mn>
 </math>?
</p>@
qu.1.13.part.3.answer=
${xc};${xcm} @
qu.1.13.part.3.mode=Multi Formula@
qu.1.13.part.3.comment=<p class="noindent">Asking what values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
give a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-value

of 11 is the same as solving
</p>
<p class="noindent"/>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtable columnspacing="0" columnalign="right center left">
  <mtr>
   <mtd>
    <mi>y</mi>
    <mo>=</mo>
    <mn>${ansc}</mn>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${a}</mn>
    <msup>
     <mrow>
      <mi>x</mi>
     </mrow>
     <mrow>
      <mn>2</mn>
     </mrow>
    </msup>
    <mo>+</mo>
    <mn>${b}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <msup>
     <mrow>
      <mi>x</mi>
     </mrow>
     <mrow>
      <mn>2</mn>
     </mrow>
    </msup>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mn>${c1}</mn>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mi>x</mi>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mo>&plusmn;</mo>
    <mn>${xc}</mn>
    <mo>.</mo>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
 </mtable>
</math>
<p class="nopar">
We can also solve this problem graphically. Looking at the graph of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${a}</mn>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>${b}</mn>
 </math>
below
<br class="newline"/>
</p>
<applet code="applets.grapher.Graph" width="250" height="250" archive="graphing.jar">
 <param name="y1" value="${a}*x*x+${b}"/>
 <param name="gridLines" value="1"/>
 <param name="xMin" value="0"/>
 <param name="xMax" value="${xmax}"/>
 <param name="yMin" value="0"/>
 <param name="yMax" value="${ymax}"/>
</applet>
<p class="noindent">We see that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
intersects the line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${ansc}</mn>
 </math>
at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mn>${xc}</mn>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
  <mo>=</mo>
  <mo>&minus;</mo>
  <mn>${xc}</mn>
 </math>. Thus,
when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${xc}</mn>
 </math> or

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math> equals
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mo>&minus;</mo>
  <mn>${xc}</mn>
 </math> we
have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>f</mi>
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>=</mo>
  <mn>${ansc}</mn>
 </math>.</p>@
qu.1.13.part.4.extra=@
qu.1.13.part.4.editing=useHTML@
qu.1.13.part.4.question=<p class="noindent">Are there any values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>x</mi>
 </math>
that give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math> a
value of ${qd}?
<br class="newline"/><1></p>@
qu.1.13.part.4.blank.1=No, yes@
qu.1.13.part.4.grader.1=menu@
qu.1.13.part.4.mode=Blanks@
qu.1.13.part.4.comment=<p class="noindent">No. No matter what, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
 </math> is
greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
 </math>,
so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <msup>
   <mrow>
    <mi>x</mi>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </msup>
  <mo>+</mo>
  <mn>2</mn>
 </math> is greater
than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>2</mn>
 </math>.</p>@

qu.1.14.mode=Blanks@
qu.1.14.name=1.1.6 Average velocity@
qu.1.14.comment=<p class="noindent"/>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtable columnspacing="0" columnalign="right center left">
  <mtr>
   <mtd>
    <mtext>Average&nbsp;velocity&nbsp;over&nbsp;</mtext>
    <mn>0</mn>
    <mo>&lt;</mo>
    <mi>t</mi>
    <mo>&lt;</mo>
    <mn>0.2</mn>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mstyle displaystyle="true">
     <mfrac>
      <mrow>
       <mi>s</mi>
       <mrow>
        <mo stretchy="false">(</mo>
        <mrow>
         <mn>0.2</mn>
        </mrow>
        <mo stretchy="false">)</mo>
       </mrow>
       <mo>&minus;</mo>
       <mi>s</mi>
       <mrow>
        <mo stretchy="false">(</mo>
        <mrow>
         <mn>0</mn>
        </mrow>
        <mo stretchy="false">)</mo>
       </mrow>
      </mrow>
      <mrow>
       <mn>0.2</mn>
       <mo>&minus;</mo>
       <mn>0</mn>
      </mrow>
     </mfrac>
    </mstyle>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd/>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mstyle displaystyle="true">
     <mfrac>
      <mrow>
       <mn>${s2}</mn>
       <mo>&minus;</mo>
       <mn>${s1}</mn>
      </mrow>
      <mrow>
       <mn>0.2</mn>
      </mrow>
     </mfrac>
    </mstyle>
    <mo>=</mo>
    <mn>${v1}</mn>
    <mtext>&nbsp;ft/sec.</mtext>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd>
    <mtext>&nbsp;Average&nbsp;velocity&nbsp;over&nbsp;</mtext>
    <mn>0.2</mn>
    <mo>&lt;</mo>
    <mi>t</mi>
    <mo>&lt;</mo>
    <mn>0.4</mn>
   </mtd>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mstyle displaystyle="true">
     <mfrac>
      <mrow>
       <mi>s</mi>
       <mrow>
        <mo stretchy="false">(</mo>
        <mrow>
         <mn>0.4</mn>
        </mrow>
        <mo stretchy="false">)</mo>
       </mrow>
       <mo>&minus;</mo>
       <mi>s</mi>
       <mrow>
        <mo stretchy="false">(</mo>
        <mrow>
         <mn>0.2</mn>
        </mrow>
        <mo stretchy="false">)</mo>
       </mrow>
      </mrow>
      <mrow>
       <mn>0.2</mn>
       <mo>&minus;</mo>
       <mn>0</mn>
      </mrow>
     </mfrac>
    </mstyle>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd/>
   <mtd>
    <mo>=</mo>
   </mtd>
   <mtd>
    <mstyle displaystyle="true">
     <mfrac>
      <mrow>
       <mn>${s3}</mn>
       <mo>&minus;</mo>
       <mn>${2}</mn>
      </mrow>
      <mrow>
       <mn>0.2</mn>
      </mrow>
     </mfrac>
    </mstyle>
    <mo>=</mo>
    <mn>${v2}</mn>
    <mtext>&nbsp;ft/sec.</mtext>
   </mtd>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
  <mtr>
   <mtd/>
   <mtd/>
   <mtd/>
   <mtd>
    <mspace width="1em"/>
   </mtd>
  </mtr>
 </mtable>
</math>
<p class="nopar">
A reasonable estimate of the velocity at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0.2</mn>
 </math>
is the average:
<br class="newline"/>
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mn>1</mn>
   </mrow>
   <mrow>
    <mn>2</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mrow>
  <mo stretchy="false">(</mo>
  <mrow>
   <mn>${v1}</mn>
   <mo>+</mo>
   <mn>${v2}</mn>
  </mrow>
  <mo stretchy="false">)</mo>
 </mrow>
 <mo>=</mo>
 <mn>${v}</mn>
 <mtext>&nbsp;ft/sec</mtext>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$s1=0.1+rint(3)*0.05;
$s2=0.5+rint(3)*0.05;
$s3=2.1+rint(3)*0.05;
$s4=3.1+rint(3)*0.05;
$v1=decimal(4,($s2-$s1)/0.2);
$v2=decimal(4,($s3-$s2)/0.2);
$v=decimal(4,0.5*($v1+$v2));@
qu.1.14.uid=6d829b3d-f077-4d92-a013-1bd211c4559d@
qu.1.14.question=<p class="noindent">Find  the  average  velocity  over  the  interval
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>0</mn>
  <mo>&#8804;</mo>
  <mi>t</mi>
  <mo>&#8804;</mo>
  <mn>0.8</mn>
 </math>, and estimate the velocity
at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>t</mi>
  <mo>=</mo>
  <mn>0.2</mn>
 </math> of a car whose position,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>s</mi>
 </math>, is given by the following table.
</p>
<div class="tabular">
 <table class="tabular" cellspacing="0" cellpadding="0" rules="groups" frame="border">
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
  </colgroup>
  <colgroup>
   <col/>
  </colgroup>
  <tr valign="baseline">
   <td align="center" style="white-space:nowrap;" class="td11">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>t</mi>
    </math> (sec)  </td>
   <td align="center" style="white-space:nowrap;" class="td11">0</td>
   <td align="center" style="white-space:nowrap;" class="td11">0.2</td>
   <td align="center" style="white-space:nowrap;" class="td11">0.4</td>
   <td align="center" style="white-space:nowrap;" class="td11">0.6</td>
   <td align="center" style="white-space:nowrap;" class="td11">0.8</td>
   <td align="center" style="white-space:nowrap;" class="td11">1.0</td>
  </tr>
  <tr class="hline">
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
   <td>
    <hr/>
   </td>
  </tr>
  <tr valign="baseline">
   <td align="center" style="white-space:nowrap;" class="td11">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
     <mi>s</mi>
    </math> (metres)</td>
   <td align="center" style="white-space:nowrap;" class="td11"> ${s1} </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${s2}  </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${s3}  </td>
   <td align="center" style="white-space:nowrap;" class="td11">  ${s4}  </td>
   <td align="center" style="white-space:nowrap;" class="td11">6.5</td>
   <td align="center" style="white-space:nowrap;" class="td11">9.6</td>
  </tr>
 </table>
</div>
<br class="newline"/>
<br class="newline"/>(a) Using the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
 <mn>0</mn>
 <mo>&lt;</mo>
 <mi>t</mi>
 <mo>&lt;</mo>
 <mn>0.2</mn>
</math><1> to 2 decimal places.
<br class="newline"/>
<br class="newline"/>(b) Using the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
 <mn>0.2</mn>
 <mo>&lt;</mo>
 <mi>t</mi>
 <mo>&lt;</mo>
 <mn>0.4</mn>
</math><2> to 2 decimal places.
<br class="newline"/>
<br class="newline"/>(c) Using the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
 <mn>0</mn>
 <mo>&lt;</mo>
 <mi>t</mi>
 <mo>&lt;</mo>
 <mn>0.4</mn>
</math><3> to 2 decimal places. @
qu.1.14.blank.1=%24%7bv1%7d%3f0.005@
qu.1.14.blank.2=%24%7bv2%7d%3f0.005@
qu.1.14.blank.3=%24%7bv%7d%3f0.005@
qu.1.14.grader.1=formula@
qu.1.14.grader.2=formula@
qu.1.14.grader.3=formula@
qu.1.14.extra=@

qu.1.15.mode=Equation@
qu.1.15.name=1.1.12 Equation of a line@
qu.1.15.comment=<p class="noindent">Using the points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>,
we have
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mtext>Slope</mtext>
 <mo>=</mo>
 <mstyle displaystyle="true">
  <mfrac>
   <mrow>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${y1}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
    <mo>&minus;</mo>
    <mrow>
     <mo stretchy="false">(</mo>
     <mrow>
      <mn>${y0}</mn>
     </mrow>
     <mo stretchy="false">)</mo>
    </mrow>
   </mrow>
   <mrow>
    <mn>${x1}</mn>
    <mo>&minus;</mo>
    <mn>${x0}</mn>
   </mrow>
  </mfrac>
 </mstyle>
 <mo>=</mo>
 <mn>${slope}</mn>
 <mo>.</mo>
</math>
<p class="nopar"> Now we know that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
  <mo>=</mo>
  <mn>${slope}</mn>
  <mi>x</mi>
  <mo>+</mo>
  <mi>b</mi>
 </math>.
Using the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math>,
we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${y0}</mn>
  <mo>=</mo>
  <mn>${slope}</mn>
  <mn>${x0}</mn>
  <mo>+</mo>
  <mi>b</mi>
 </math>,
which yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>b</mi>
  <mo>=</mo>
  <mn>${intercept}</mn>
 </math>.
Thus the equation of the line is

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mi>y</mi>
 <mo>=</mo>
 <mn>${slope}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${intercept}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$slope=int(-rint(3)-3);
$intercept=int(rint(3)+3);
$x0=int(-(rint(10)+1));
$y0=int($intercept+$slope*$x0);
$x1=int(rint(5)+$x0+1);
$y1=int($intercept+$slope*$x1);@
qu.1.15.uid=c40ad2ea-d505-4918-93d0-d78eb91263ef@
qu.1.15.question=<p class="noindent">Find the equation of the line that passes through the points
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x0}</mn>
    <mo>,</mo>
    <mn>${y0}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
 </math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mrow>
   <mo stretchy="false">(</mo>
   <mrow>
    <mn>${x1}</mn>
    <mo>,</mo>
    <mn>${y1}</mn>
   </mrow>
   <mo stretchy="false">)</mo>
  </mrow>
  <mo>.</mo>
 </math>
</p>@
qu.1.15.answer=y=${slope} x
+ ${intercept}@

qu.1.16.mode=Multipart@
qu.1.16.name=1.1.10 Slope and intercept@
qu.1.16.comment=@
qu.1.16.editing=useHTML@
qu.1.16.solution=@
qu.1.16.algorithm=$a=int(rint(5)+2);
$k=int(rint(5)+1);
$l=int(rint(5)+2);
$b=int($k*$a);
$c=int($l*$a);
$slope=$b/$a;
$intercept=-$c/$a;@
qu.1.16.uid=29dacb64-3764-42c3-a7a5-b238c86bd9e7@
qu.1.16.question=<p class="noindent">Determine the slope and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mi>y</mi>
 </math>-intercept
of the line whose equation is given by
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mn>${b}</mn>
 <mi>x</mi>
 <mo>=</mo>
 <mn>${a}</mn>
 <mi>y</mi>
 <mo>+</mo>
 <mn>${c}</mn>
 <mo>.</mo>
</math>
<p class="nopar"/>@
qu.1.16.weighting=1,1@
qu.1.16.numbering=alpha@
qu.1.16.part.1.extra=@
qu.1.16.part.1.editing=useHTML@
qu.1.16.part.1.question=<p class="noindent">The slope is <1></p>@
qu.1.16.part.1.blank.1=%24%7bslope%7d@
qu.1.16.part.1.grader.1=formula@
qu.1.16.part.1.mode=Blanks@
qu.1.16.part.1.comment=<p class="noindent">Rewriting the equation as
</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mo>&minus;</mo>
 <mn>${a}</mn>
 <mi>y</mi>
 <mo>=</mo>
 <mo>&minus;</mo>
 <mn>${b}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${c}</mn>
</math>
<p class="nopar"> shows that the line has slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mn>${b}</mn>
  <mo>&#8725;</mo>
  <mn>${a}</mn>
 </math>.
</p>@
qu.1.16.part.2.extra=@
qu.1.16.part.2.editing=useHTML@
qu.1.16.part.2.question=<p class="noindent">The intercept is <1></p>@
qu.1.16.part.2.blank.1=%24%7bintercept%7d@
qu.1.16.part.2.grader.1=formula@
qu.1.16.part.2.mode=Blanks@
qu.1.16.part.2.comment=<p class="noindent">Rewriting the equation as

</p>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
 <mo>&minus;</mo>
 <mn>${a}</mn>
 <mi>y</mi>
 <mo>=</mo>
 <mo>&minus;</mo>
 <mn>${b}</mn>
 <mi>x</mi>
 <mo>+</mo>
 <mn>${c}</mn>
</math>
<p class="nopar"> shows that the vertical intercept is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
  <mo>&minus;</mo>
  <mn>${c}</mn>
  <mo>&#8725;</mo>
  <mn>${a}</mn>
 </math>.</p>@

