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    <name>
      <text>17.3.1.3 Parametric Curves Information</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>Sketch the curve \(\quad\displaystyle x=t+1, \qquad y=2t^2-3\qquad, 0\le t\le 1\)</p>
<h4 class="HELM_solution">Solution</h4>
<p>Similar to a previous Task, we first construct the corresponding table of \((x,y)\) values as \(t\) ranges from \(0 \to 1.\): </p>
<table class="HELM_table">
  <thead>
    <tr>
      <th>\(t\)</th>
      <th>\(x\)</th>
       <th>\(y\)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\(0\)</td>
      <td>\(1\)</td>
      <td>\(-3\)</td>
    </tr>
    <tr>
      <td>\(\dfrac{1}{4}\)</td>
      <td>\(\dfrac{5}{4}\)</td>
      <td>\(\dfrac{-23}{8}\)</td>
    </tr>
    <tr>
      <td>\(\dfrac{1}{2}\)</td>
      <td>\(\dfrac{3}{2}\)</td>
      <td>\(\dfrac{-5}{2}\)</td>
    </tr>
     <tr> 
      <td>\(\dfrac{3}{4}\)</td>
      <td>\(\dfrac{7}{4}\)</td>
      <td>\(\dfrac{-15}{8}\)</td>
    </tr>
    <tr>
      <td>\(1\)</td>
      <td>\(2\)</td>
      <td>\(-1\)</td>
    </tr>
  </tbody>
</table>
<p> And the curve can be plotted as </p>
<div style="text-align: center;">
<img src="@@PLUGINFILE@@/17-3-fig17-parametric.svg" alt="The described curve plotted on the xy plane for t ranging from 0 to 1." width="208" height="173" class="img-fluid atto_image_button_text-bottom">
</div>
<br><p> And eliminating the \(t\)-parameter from the expressions defining \(x,y\), we have that \(t = x-1 \implies y = 2(x-1)^2 - 3.\)

</p><hr>

<h4 class="HELM_example">Example</h4>
<p>Sketch the curve \(\quad\displaystyle x=t^2+1, \qquad y=2t^4-3\qquad, 0\le t\le 1\)</p>
<h4 class="HELM_solution">Solution</h4>
<p>This is very similar to the previous example (except for \(t^4\) replacing \(t^2\) in the expression for \(y\) and \(t^2\) replacing \(t\) in the expression for \(x\)).  The corresponding table of values is
</p><table class="HELM_table">
  <thead>
    <tr>
      <th>\(t\)</th>
      <th>\(x\)</th>
       <th>\(y\)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\(0\)</td>
      <td>\(1\)</td>
      <td>\(-3\)</td>
    </tr>
    <tr>
      <td>\(\dfrac{1}{4}\)</td>
      <td>\(\dfrac{17}{16}\)</td>
      <td>\(\dfrac{-383}{128}\)</td>
    </tr>
    <tr>
      <td>\(\dfrac{1}{2}\)</td>
      <td>\(\dfrac{5}{4}\)</td>
      <td>\(\dfrac{-23}{8}\)</td>
    </tr>
     <tr> 
      <td>\(\dfrac{3}{4}\)</td>
      <td>\(\dfrac{25}{16}\)</td>
      <td>\(\dfrac{-303}{128}\)</td>
    </tr>
    <tr>
      <td>\(1\)</td>
      <td>\(2\)</td>
      <td>\(-1\)</td>
    </tr>
  </tbody>
</table>
<p></p><div style="text-align: center;"><img src="@@PLUGINFILE@@/17-3-fig17-parametric2.svg?time=1753798509001" alt="The same curve as above" width="209" height="179" class="img-fluid atto_image_button_text-bottom"><br></div>

<br><p>We see that this is <strong>identical</strong> to the curve drawn previously. This is confirmed by eliminating the \(t\)-parameter from the expressions defining \(x,y\). Here \(t^2=x-1\) so \(y=2(x-1)^2-3\) which is the same as obtained in the last example. The main difference is that particular values of \(t\) locate (in general) different \((x,y)\) points on the curve for the two parametric representations. We conclude that a given curve in the \(xy\) plane can have many (in fact infinitely many) parametric descriptions.</p>]]></text>
<file name="17-3-fig17-parametric.svg" path="/" encoding="base64">[encoded data, 14508 characters]</file>
<file name="17-3-fig17-parametric2.svg" path="/" encoding="base64">[encoded data, 15964 characters]</file>
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