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    <name>
      <text>1.3.18 (Intersections between parabolas and lines)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h1>Intersections with lines</h1>
<br>
The graph of the quadratic \(ax^2+bx+c\) consists of all points \((x,y)\) which satisfy
\(y=ax^2+bx+c\). So:<p></p>

<ul type="disc">
 <li><p>if a point \((x,y)\) is on the
     parabola then we know \(y=ax^2+bx+c\)</p></li>
 <li><p>for points not on the parabola, we know \(y\neq ax^2+bx+c\).</p></li>
</ul>

<p>If we want to find the points of intersection of the line \(y=mx+k\) with the parabola&nbsp;\(y=ax^2+bx+c\), we solve
\[
 mx+k = ax^2+bx+c. \quad\quad(\ast) 
\]
This is an example of solving <i>simultaneous equations</i> which we will discuss in more detail in week 7.</p>

<p>Another way to think
about this is to measure the vertical distance between the line and parabola at a given \(x\) value, which is given by</p><table style="text-align: center; margin-left:auto; margin-right:auto;" cellpadding="10">
<tbody><tr>
<td>\[&nbsp;\underbrace{\left(ax^2+bx+c\right)}_{\substack{\text{height of point} \\ \text{on parabola}}} - \underbrace{\left(mx+k\right)}_{\substack{\text{height of point} \\ \text{on line}}} \]</td>
<td><img src="@@PLUGINFILE@@/3_2_1_distance.svg" alt="Distance between line and parabola" width="141" height="103" class="img-responsive atto_image_button_text-top" <="" td="">
</td></tr>
</tbody></table>
<br><p>This shows that finding roots of \((\ast)\) is equivalent to
finding \(x\)-coordinates where there is no distance between the points on
the line and parabola.</p>

<p>Now,&nbsp;\((\ast)\) is a quadratic (we can write it in standard form if we wished by rearranging) so its discriminant, \(\Delta\), tells us about its roots:</p><table style="text-align: center; margin-left:auto; margin-right:auto;" cellpadding="10" border="1" bordercolor="#3a87ad">
<thead>
<tr style="background: #c1ddea">
<th scope="col">Discriminant</th>
<th scope="col">Interpretation</th>
<th scope="col">Sketch</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(\Delta&gt;0\)</td>
<td>there are two distinct
roots, so the line and curve intersect twice</td>
<td><img src="@@PLUGINFILE@@/3_2_1_pos.svg" alt="Parabola and line with two intersections" width="64" height="52" class="img-responsive atto_image_button_middle"></td>
</tr>
<tr>
<td>\(\Delta=0\)</td>
<td><p align="left">there is one real root, so the line and curve intersect
once</p>

(i.e. the line is a tangent to the curve)</td>
<td><img src="@@PLUGINFILE@@/3_2_1_zero.svg" alt="Parabola and line with one intersection" width="64" height="62" class="img-responsive atto_image_button_middle"></td>
</tr>
<tr>
<td>\(\Delta&lt;0\)</td>
<td>there are no real roots, so
the line and the parabola do not intersect</td>
<td><img src="@@PLUGINFILE@@/3_2_1_neg.svg" alt="Parabola and line with no intersections" width="64" height="68" class="img-responsive atto_image_button_middle"></td>
</tr>
</tbody>
</table>]]></text>
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<file name="3_2_1_zero.svg" path="/" encoding="base64">[encoded data, 5508 characters]</file>
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