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<title>Quadratic Functions and Their Graphs</title>
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<div class="section" id="fwk-redden-ch06_s04" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.4</span> Quadratic Functions and Their Graphs</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch06_s04_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch06_s04_o01" numeration="arabic">
<li>Graph a parabola.</li>
<li>Find the intercepts and vertex of a parabola.</li>
<li>Find the maximum and minimum <em class="emphasis">y</em>-value.</li>
<li>Find the vertex of a parabola by completing the square.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch06_s04_s01" version="5.0" lang="en">
<h2 class="title editable block">The Graph of a Quadratic Function</h2>
<p class="para editable block" id="fwk-redden-ch06_s04_s01_p01">A <strong class="emphasis bold">quadratic function</strong> is a polynomial function of degree 2 which can be written in the general form,</p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0886" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p03">Here <em class="emphasis">a</em>, <em class="emphasis">b</em> and <em class="emphasis">c</em> represent real numbers where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0887" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span> The squaring function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0888" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> is a quadratic function whose graph follows.</p>
<div class="informalfigure large block">
<img src="section_09/f7c27c7d79c9141d0731362a4554caa7.png">
</div>
<p class="para block" id="fwk-redden-ch06_s04_s01_p05">This general curved shape is called a <span class="margin_term"><a class="glossterm">parabola</a><span class="glossdef">The U-shaped graph of any quadratic function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0889" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>, where <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> are real numbers and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0890" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> and is shared by the graphs of all quadratic functions. Note that the graph is indeed a function as it passes the vertical line test. Furthermore, the domain of this function consists of the set of all real numbers <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0891" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span> and the range consists of the set of nonnegative numbers <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0892" display="inline"><mrow><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p06">When graphing parabolas, we want to include certain special points in the graph. The <em class="emphasis">y</em>-intercept is the point where the graph intersects the <em class="emphasis">y</em>-axis. The <em class="emphasis">x</em>-intercepts are the points where the graph intersects the <em class="emphasis">x</em>-axis. The <span class="margin_term"><a class="glossterm">vertex</a><span class="glossdef">The point that defines the minimum or maximum of a parabola.</span></span> is the point that defines the minimum or maximum of the graph. Lastly, the <span class="margin_term"><a class="glossterm">line of symmetry</a><span class="glossdef">The vertical line through the vertex, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0893" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></math></span>, about which the parabola is symmetric.</span></span> (also called the <span class="margin_term"><a class="glossterm">axis of symmetry</a><span class="glossdef">A term used when referencing the line of symmetry.</span></span>) is the vertical line through the vertex, about which the parabola is symmetric.</p>
<div class="informalfigure large block">
<img src="section_09/9d42e34d8294dfb85c5491c48840be45.png">
</div>
<p class="para editable block" id="fwk-redden-ch06_s04_s01_p08">For any parabola, we will find the vertex and <em class="emphasis">y</em>-intercept. In addition, if the <em class="emphasis">x</em>-intercepts exist, then we will want to determine those as well. Guessing at the <em class="emphasis">x</em>-values of these special points is not practical; therefore, we will develop techniques that will facilitate finding them. Many of these techniques will be used extensively as we progress in our study of algebra.</p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p09">Given a quadratic function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0894" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>, find the <em class="emphasis">y</em>-intercept by evaluating the function where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0895" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> In general, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0896" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mo>=</mo><mi>c</mi></mrow></math></span>, and we have</p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0897" display="block"><mtable columnalign="center" columnspacing="0.1em"><mtr columnalign="center"><mtd columnalign="center"><mstyle color="#007fbf"><mrow><mi>y</mi><mi>-</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi></mrow></mstyle></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>c</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p11">Next, recall that the <em class="emphasis">x</em>-intercepts, if they exist, can be found by setting <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0898" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> Doing this, we have <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0899" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span>, which has general solutions given by the quadratic formula, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0900" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></math></span> Therefore, the <em class="emphasis">x</em>-intercepts have this general form:</p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0901" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="center"><mstyle color="#007fbf"><mrow><mi>x</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>−</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>+</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mtext> </mtext></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s04_s01_p13">Using the fact that a parabola is symmetric, we can determine the vertical line of symmetry using the <em class="emphasis">x</em>-intercepts. To do this, we find the <em class="emphasis">x</em>-value midway between the <em class="emphasis">x</em>-intercepts by taking an average as follows:</p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0902" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>−</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mo>+</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mtext> </mtext><mo>÷</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>−</mo><menclose notation="updiagonalstrike"><mrow><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow></menclose><mo>−</mo><mi>b</mi><mo>+</mo><menclose notation="updiagonalstrike"><mrow><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow></menclose></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo>÷</mo><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>1</mn></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>2</mn><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>⋅</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p15">Therefore, the line of symmetry is the vertical line <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0903" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></math></span> We can use the line of symmetry to find the the vertex.</p>
<p class="para block" id="fwk-redden-ch06_s04_s01_p16"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0904" display="block"><mrow><mtable columnspacing="0.1em" columnalign="center"><mtr columnalign="center"><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mrow><mi>L</mi><mi>i</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>s</mi><mi>y</mi><mi>m</mi><mi>m</mi><mi>e</mi><mi>t</mi><mi>r</mi><mi>y</mi></mrow></mstyle></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007fbf"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="center"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>,</mo><mi>f</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s04_s01_p17">Generally three points determine a parabola. However, in this section we will find five points so that we can get a better approximation of the general shape. The steps for graphing a parabola are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch06_s04_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch06_s04_s01_p18">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0905" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p19"><strong class="emphasis bold">Step 1</strong>: Determine the <em class="emphasis">y</em>-intercept. To do this, set <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0906" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span> and find <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0907" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0908" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><msup><mrow><mo>(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p21">The <em class="emphasis">y</em>-intercept is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0909" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p22"><strong class="emphasis bold">Step 2</strong>: Determine the <em class="emphasis">x</em>-intercepts if any. To do this, set <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0910" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0911" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>M</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mo>−</mo><mn>1</mn><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>e</mi><mi>a</mi><mi>c</mi><mi>h</mi><mtext> </mtext><mi>f</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>z</mi><mi>e</mi><mi>r</mi><mi>o</mi><mo>.</mo></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p24">Here where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0912" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span>, we obtain two solutions. Hence, there are two <em class="emphasis">x</em>-intercepts, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0913" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0914" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p25"><strong class="emphasis bold">Step 3</strong>: Determine the vertex. One way to do this is to first use <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0915" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></math></span> to find the <em class="emphasis">x</em>-value of the vertex and then substitute this value in the function to find the corresponding <em class="emphasis">y</em>-value. In this example, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0916" display="inline"><mrow><mi>a</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0917" display="inline"><mrow><mi>b</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p26"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0918" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>2</mn><mrow><mo>−</mo><mn>2</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p27">Substitute −1 into the original function to find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0919" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><msup><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p29">The vertex is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0920" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p30"><strong class="emphasis bold">Step 4</strong>: Determine extra points so that we have at least five points to plot. Ensure a good sampling on either side of the line of symmetry. In this example, one other point will suffice. Choose <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0921" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span> and find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0922" display="block"><mrow><mtable columnspacing="0.3em" columnalign="left" columnlines="solid none" rowlines="solid none"><mtr columnalign="left"><mtd columnalign="right" style="border-bottom:1pt solid black"><mi>x</mi><mi> </mi></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mi> </mi><mi>y</mi></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>Point</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>2</mn><mi> </mi></mrow></mtd><mtd columnalign="left"><mi> </mi><mn>3</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>4</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>3</mn><mo>=</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p32">Our fifth point is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0923" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p33"><strong class="emphasis bold">Step 5</strong>: Plot the points and sketch the graph. To recap, the points that we have found are</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p34"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0924" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>y</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>s</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mtext> and </mtext><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi>a</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p35">Answer:</p>
<div class="informalfigure large">
<img src="section_09/c4011fd29ced185631240cc25889d3c5.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch06_s04_s01_p37">The parabola opens downward. In general, use the leading coefficient to determine if the parabola opens upward or downward. If the leading coefficient is negative, as in the previous example, then the parabola opens downward. If the leading coefficient is positive, then the parabola opens upward.</p>
<div class="informalfigure large block">
<img src="section_09/4c56e02218b986fe483396687512f1ff.png">
</div>
<p class="para block" id="fwk-redden-ch06_s04_s01_p39">All quadratic functions of the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0925" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> have parabolic graphs with <em class="emphasis">y</em>-intercept <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0926" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>c</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> However, not all parabolas have <em class="emphasis">x</em>-intercepts.</p>
<div class="callout block" id="fwk-redden-ch06_s04_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch06_s04_s01_p40">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0927" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p41">Because the leading coefficient 2 is positive, we note that the parabola opens upward. Here <em class="emphasis">c</em> = 5 and the <em class="emphasis">y</em>-intercept is (0, 5). To find the <em class="emphasis">x</em>-intercepts, set <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0928" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0929" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p43">In this case, <em class="emphasis">a</em> = 2, <em class="emphasis">b</em> = 4, and <em class="emphasis">c</em> = 5. Use the discriminant to determine the number and type of solutions.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p44"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0930" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>16</mn><mo>−</mo><mn>40</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>24</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p45">Since the discriminant is negative, we conclude that there are no real solutions. Because there are no real solutions, there are no <em class="emphasis">x</em>-intercepts. Next, we determine the <em class="emphasis">x</em>-value of the vertex.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p46"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0931" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mrow><mo>(</mo><mstyle color="#007f3f"><mn>4</mn></mstyle><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>2</mn></mstyle><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>4</mn></mrow><mn>4</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p47">Given that the <em class="emphasis">x</em>-value of the vertex is −1, substitute −1 into the original equation to find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p48"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0932" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><msup><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mo>−</mo><mn>4</mn><mo>+</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p49">The vertex is (−1, 3). So far, we have only two points. To determine three more, choose some <em class="emphasis">x</em>-values on either side of the line of symmetry, <em class="emphasis">x</em> = −1. Here we choose <em class="emphasis">x</em>-values −3, −2, and 1.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p50"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0933" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left" columnlines="solid none" rowlines="solid none"><mtr columnalign="left"><mtd columnalign="right" style="border-bottom:1pt solid black"><mi>x</mi><mi> </mi></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mi> </mi><mi>y</mi></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>Points</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>3</mn><mi> </mi></mrow></mtd><mtd columnalign="left"><mrow><mi> </mi><mn>11</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>3</mn></mrow></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>3</mn></mrow></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>3</mn></mrow></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn><mo>=</mo><mn>18</mn><mo>−</mo><mn>12</mn><mo>+</mo><mn>5</mn><mo>=</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>11</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>2</mn><mi> </mi></mrow></mtd><mtd columnalign="left"><mi> </mi><mn>5</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn><mo>=</mo><mn>8</mn><mo>−</mo><mn>8</mn><mo>+</mo><mn>5</mn><mo>=</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>1</mn><mi> </mi></mtd><mtd columnalign="left"><mrow><mi> </mi><mn>11</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>5</mn><mo>=</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>11</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p51">To summarize, we have</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p52"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0934" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>y</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>s</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>None</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi>a</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>11</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mn>11</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p53">Plot the points and sketch the graph.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p54">Answer:</p>
<div class="informalfigure large">
<img src="section_09/82b2e581081310037251baf4405c7726.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch06_s04_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch06_s04_s01_p56">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0935" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p57">Since <em class="emphasis">a</em> = 1, the parabola opens upward. Furthermore, <em class="emphasis">c</em> = −1, so the <em class="emphasis">y</em>-intercept is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0936" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To find the <em class="emphasis">x</em>-intercepts, set <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0937" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p58"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0938" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p59">In this case, solve using the quadratic formula with <em class="emphasis">a</em> = 1, <em class="emphasis">b</em> = −2, and <em class="emphasis">c</em> = −1.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p60"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0939" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle><mo stretchy="false">)</mo><mo>±</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mo>−</mo><mn>2</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow></msqrt></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mo>±</mo><msqrt><mn>8</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>±</mo><msqrt><mn>2</mn></msqrt><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>±</mo><msqrt><mn>2</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p61">Here we obtain two real solutions for <em class="emphasis">x</em>, and thus there are two <em class="emphasis">x</em>-intercepts:</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p62"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0940" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>and</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>x</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>0.41</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>2.41</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>p</mi><mi>p</mi><mi>r</mi><mi>o</mi><mi>x</mi><mi>i</mi><mi>m</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p63">Approximating the <em class="emphasis">x</em>-intercepts using a calculator will help us plot the points. However, we will present the exact <em class="emphasis">x</em>-intercepts on the graph. Next, find the vertex.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p64"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0941" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>2</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p65">Given that the <em class="emphasis">x</em>-value of the vertex is 1, substitute into the original equation to find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p66"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0942" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mo>−</mo><mn>2</mn><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p67">The vertex is (1, −2). We need one more point.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p68"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0943" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left" columnlines="solid none" rowlines="solid none"><mtr columnalign="left"><mtd columnalign="right" style="border-bottom:1pt solid black"><mi>x</mi><mi> </mi></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mi> </mi><mi>y</mi></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>Point</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>2</mn><mi> </mi></mtd><mtd columnalign="left"><mrow><mi> </mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>=</mo><mn>4</mn><mo>−</mo><mn>4</mn><mo>−</mo><mn>1</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p69">To summarize, we have</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p70"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0944" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>y</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>-</mo><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>c</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>s</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi>a</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p71">Plot the points and sketch the graph.</p>
<p class="para" id="fwk-redden-ch06_s04_s01_p72">Answer:</p>
<div class="informalfigure large">
<img src="section_09/82f18defc1c3e80a64004f4599fc4eeb.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch06_s04_s01_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s04_s01_p74"><strong class="emphasis bold">Try this!</strong> Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0945" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s01_p75">Answer:</p>
<div class="informalfigure large">
<img src="section_09/02f6d92b66d61a15d13dd85c7e419961.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/tQiDXNhn7ik" condition="http://img.youtube.com/vi/tQiDXNhn7ik/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/tQiDXNhn7ik" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch06_s04_s02" version="5.0" lang="en">
<h2 class="title editable block">Finding the Maximum or Minimum</h2>
<p class="para editable block" id="fwk-redden-ch06_s04_s02_p01">It is often useful to find the maximum and/or minimum values of functions that model real-life applications. To find these important values given a quadratic function, we use the vertex. If the leading coefficient <em class="emphasis">a</em> is positive, then the parabola opens upward and there will be a minimum <em class="emphasis">y</em>-value. If the leading coefficient <em class="emphasis">a</em> is negative, then the parabola opens downward and there will be a maximum <em class="emphasis">y</em>-value.</p>
<div class="callout block" id="fwk-redden-ch06_s04_s02_n01">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch06_s04_s02_p02">Determine the maximum or minimum: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0946" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>−</mo><mn>35</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p03">Since <em class="emphasis">a</em> = −4, we know that the parabola opens downward and there will be a maximum <em class="emphasis">y</em>-value. To find it, first find the <em class="emphasis">x</em>-value of the vertex.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0947" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mtext>-</mtext><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>v</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mstyle color="#007f3f"><mrow><mn>24</mn></mrow></mstyle></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>u</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mo>-</mo><mn mathvariant="italic">4</mn><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>b</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mn mathvariant="italic">24.</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mn>24</mn></mrow><mrow><mo>−</mo><mn>8</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s02_p05">The <em class="emphasis">x</em>-value of the vertex is 3. Substitute this value into the original equation to find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0948" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>−</mo><mn>35</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>u</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mn mathvariant="italic">3.</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn><msup><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>24</mn><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>−</mo><mn>35</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>36</mn><mo>+</mo><mn>72</mn><mo>−</mo><mn>35</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s02_p07">The vertex is (3, 1). Therefore, the maximum <em class="emphasis">y</em>-value is 1, which occurs where <em class="emphasis">x</em> = 3, as illustrated below:</p>
<div class="informalfigure large">
<img src="section_09/a174540aade4ed6f840a6a189f3104f4.png">
</div>
<p class="para" id="fwk-redden-ch06_s04_s02_p09"><strong class="emphasis bold">Note</strong>: The graph is not required to answer this question.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p10">Answer: The maximum is 1.</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s04_s02_n02">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch06_s04_s02_p11">Determine the maximum or minimum: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0949" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn><mi>x</mi><mo>+</mo><mn>62</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p12">Since <em class="emphasis">a</em> = 4, the parabola opens upward and there is a minimum <em class="emphasis">y</em>-value. Begin by finding the <em class="emphasis">x</em>-value of the vertex.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p13"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0950" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>32</mn></mrow></mstyle></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>4</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>u</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mn mathvariant="italic">4</mn><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>b</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mo>-</mo><mn mathvariant="italic">32.</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mo>−</mo><mn>32</mn></mrow><mn>8</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s02_p14">Substitute <em class="emphasis">x</em> = 4 into the original equation to find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0951" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn><mi>x</mi><mo>+</mo><mn>62</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><msup><mrow><mo>(</mo><mstyle color="#007f3f"><mn>4</mn></mstyle><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>32</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>4</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>62</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>64</mn><mo>−</mo><mn>128</mn><mo>+</mo><mn>62</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s02_p16">The vertex is (4, −2). Therefore, the minimum <em class="emphasis">y</em>-value of −2 occurs where <em class="emphasis">x</em> = 4, as illustrated below:</p>
<div class="informalfigure large">
<img src="section_09/9a415ffd6ad693eb1d7116d868e7943c.png">
</div>
<p class="para" id="fwk-redden-ch06_s04_s02_p18">Answer: The minimum is −2.</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s04_s02_n03">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch06_s04_s02_p19">The height in feet of a projectile is given by the function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0952" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>72</mn><mi>t</mi></mrow></math></span>, where <em class="emphasis">t</em> represents the time in seconds after launch. What is the maximum height reached by the projectile?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p20">Here <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0953" display="inline"><mrow><mi>a</mi><mo>=</mo><mo>−</mo><mn>16</mn></mrow></math></span>, and the parabola opens downward. Therefore, the <em class="emphasis">y</em>-value of the vertex determines the maximum height. Begin by finding the time at which the vertex occurs.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p21"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0954" display="block"><mrow><mi>t</mi><mo>=</mo><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mn>72</mn></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>16</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>72</mn></mrow><mrow><mn>32</mn></mrow></mfrac><mo>=</mo><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s02_p22">The maximum height will occur in <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0955" display="inline"><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></math></span> seconds (or <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0956" display="inline"><mrow><mn>2</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> seconds). Substitute this time into the function to determine the maximum height attained.</p>
<p class="para" id="fwk-redden-ch06_s04_s02_p23"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0957" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>h</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>16</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>72</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></mstyle></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>16</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mn>81</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mn>72</mn><mrow><mo>(</mo><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>81</mn><mo>+</mo><mn>162</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>81</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s02_p24">Answer: The maximum height of the projectile is 81 feet.</p>
</div>
</div>
<div class="section" id="fwk-redden-ch06_s04_s03" version="5.0" lang="en">
<h2 class="title editable block">Finding the Vertex by Completing the Square</h2>
<p class="para block" id="fwk-redden-ch06_s04_s03_p01">In this section, we demonstrate an alternate approach for finding the vertex. Any quadratic function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0958" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> can be rewritten in <span class="margin_term"><a class="glossterm">vertex form</a><span class="glossdef">A quadratic function written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0959" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow><mo>.</mo></math></span></span></span>,</p>
<p class="para block" id="fwk-redden-ch06_s04_s03_p02"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0960" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s04_s03_p03">In this form, the vertex is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0961" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To see that this is the case, consider graphing <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0962" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span> using the transformations.</p>
<p class="para block" id="fwk-redden-ch06_s04_s03_p04"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0963" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>B</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>s</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext> </mtext><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>H</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>z</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mtext> </mtext><mn mathvariant="italic">2</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>u</mi><mi>p</mi><mtext> </mtext><mn mathvariant="italic">3</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s04_s03_p05">Use these translations to sketch the graph,</p>
<div class="informalfigure large block">
<img src="section_09/f324ac41d615d7a8129aa3a9fe230f00.png">
</div>
<p class="para editable block" id="fwk-redden-ch06_s04_s03_p07">Here we can see that the vertex is (2, 3).</p>
<p class="para block" id="fwk-redden-ch06_s04_s03_p08"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0964" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mi>a</mi></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mi>h</mi></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow></mrow></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mn>2</mn></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mn>3</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s04_s03_p09">When the equation is in this form, we can read the vertex directly from it.</p>
<div class="callout block" id="fwk-redden-ch06_s04_s03_n01">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch06_s04_s03_p10">Determine the vertex: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0965" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p11">Rewrite the equation as follows before determining <em class="emphasis">h</em> and <em class="emphasis">k</em>.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p12"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0966" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mi>a</mi></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mi>h</mi></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>2</mn></mtd><mtd><mo>[</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><msup><mo>]</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s03_p13">Here <em class="emphasis">h</em> = −3 and <em class="emphasis">k</em> = −2.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p14">Answer: The vertex is (−3, −2).</p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s04_s03_p15">Often the equation is not given in vertex form. To obtain this form, complete the square.</p>
<div class="callout block" id="fwk-redden-ch06_s04_s03_n02">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch06_s04_s03_p16">Rewrite in vertex form and determine the vertex: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0967" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p17">Begin by making room for the constant term that completes the square.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p18"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0968" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mtext>___</mtext><mo>+</mo><mn>9</mn><mo>−</mo><mtext>___</mtext></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s03_p19">The idea is to add and subtract the value that completes the square, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0969" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>, and then factor. In this case, add and subtract <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0970" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>4</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0971" display="block"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>A</mi><mi>d</mi><mi>d</mi><mi> </mi><mi>a</mi><mi>n</mi><mi>d</mi><mi> </mi><mi>s</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi> </mi><mi>4</mi><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo><mn>4</mn></mstyle><mo>+</mo><mn>9</mn><mstyle color="#007fbf"><mtext> </mtext><mo>−</mo><mn>4</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>+</mo><mn>5.</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span>
<p class="para" id="fwk-redden-ch06_s04_s03_p21">Adding and subtracting the same value within an expression does not change it. Doing so is equivalent to adding 0. Once the equation is in this form, we can easily determine the vertex.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p22"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0972" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mi>a</mi></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mi>h</mi></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow></mrow></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mn>5</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s03_p23">Here <em class="emphasis">h</em> = −2 and <em class="emphasis">k</em> = 5.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p24">Answer: The vertex is (−2, 5).</p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s04_s03_p25">If there is a leading coefficient other than 1, then we must first factor out the leading coefficient from the first two terms of the trinomial.</p>
<div class="callout block" id="fwk-redden-ch06_s04_s03_n03">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch06_s04_s03_p26">Rewrite in vertex form and determine the vertex: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0973" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p27">Since <em class="emphasis">a</em> = 2, factor this out of the first two terms in order to complete the square. Leave room inside the parentheses to add and subtract the value that completes the square.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p28"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0974" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mi> </mi><mi> </mi></mrow><mo>)</mo></mrow><mo>+</mo><mn>8</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s04_s03_p29">Now use −2 to determine the value that completes the square. In this case, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0975" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> Add and subtract 1 and factor as follows:</p>
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0976" display="block"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mo>_</mo><mo>_</mo><mo>−</mo><mo>_</mo><mo>_</mo></mrow><mo>)</mo></mrow><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>A</mi><mi>d</mi><mi>d</mi><mi> </mi><mi>a</mi><mi>n</mi><mi>d</mi><mi> </mi><mi>s</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi> </mi><mi>1</mi><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mstyle color="#007fbf"><mi> </mi><mo>+</mo><mn>1</mn><mo>−</mo><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mn>8</mn><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mrow><mo>[</mo> <mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow> <mo>]</mo></mrow><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mrow><mo>[</mo> <mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow> <mo>]</mo></mrow><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi> </mi><mi>t</mi><mi>h</mi><mi>e</mi><mi> </mi><mi>2</mi><mtext>.</mtext></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span>
<p class="para" id="fwk-redden-ch06_s04_s03_p31">In this form, we can easily determine the vertex.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p32"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0977" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mi>a</mi></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mi>h</mi></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>2</mn></mtd><mtd><mo stretchy="false">(</mo></mtd><mtd><mi>x</mi></mtd><mtd><mo>−</mo></mtd><mtd><mn>1</mn></mtd><mtd><mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mn>6</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s03_p33">Here <em class="emphasis">h</em> = 1 and <em class="emphasis">k</em> = 6.</p>
<p class="para" id="fwk-redden-ch06_s04_s03_p34">Answer: The vertex is (1, 6).</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s04_s03_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s04_s03_p35"><strong class="emphasis bold">Try this!</strong> Rewrite in vertex form and determine the vertex: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0978" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s04_s03_p36">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0979" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>21</mn></mrow></math></span>; vertex: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0980" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>21</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="mediaobject">
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</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch06_s04_s03_n04">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch06_s04_s03_l01" mark="bullet">
<li>The graph of any quadratic function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0981" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>, where <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> are real numbers and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0982" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, is called a parabola.</li>
<li>When graphing a parabola always find the vertex and the <em class="emphasis">y</em>-intercept. If the <em class="emphasis">x</em>-intercepts exist, find those as well. Also, be sure to find ordered pair solutions on either side of the line of symmetry, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0983" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></math></span>
</li>
<li>Use the leading coefficient, <em class="emphasis">a</em>, to determine if a parabola opens upward or downward. If <em class="emphasis">a</em> is positive, then it opens upward. If <em class="emphasis">a</em> is negative, then it opens downward.</li>
<li>The vertex of any parabola has an <em class="emphasis">x</em>-value equal to <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0984" display="inline"><mrow><mo>−</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></math></span> After finding the <em class="emphasis">x</em>-value of the vertex, substitute it into the original equation to find the corresponding <em class="emphasis">y</em>-value. This <em class="emphasis">y</em>-value is a maximum if the parabola opens downward, and it is a minimum if the parabola opens upward.</li>
<li>The domain of a parabola opening upward or downward consists of all real numbers. The range is bounded by the <em class="emphasis">y</em>-value of the vertex.</li>
<li>An alternate approach to finding the vertex is to rewrite the quadratic function in the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0985" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow><mo>.</mo></math></span> When in this form, the vertex is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0986" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and can be read directly from the equation. To obtain this form, take <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0987" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> and complete the square.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s04_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd01">
<h3 class="title">Part A: The Graph of Quadratic Functions</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch06_s04_qs01_p01"><strong class="emphasis bold">Does the parabola open upward or downward? Explain.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0988" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0989" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>32</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0990" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0991" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0992" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>64</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0993" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd01_qd02" start="7">
<p class="para" id="fwk-redden-ch06_s04_qs01_p14"><strong class="emphasis bold">Determine the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p15"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0994" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p17"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0995" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>13</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p19"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0996" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p21"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0998" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p23"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1000" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p25"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1002" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p27"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1005" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>27</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p29"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1008" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>50</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1011" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1012" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd01_qd03" start="17">
<p class="para" id="fwk-redden-ch06_s04_qs01_p35"><strong class="emphasis bold">Find the vertex and the line of symmetry.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1015" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>−</mo><mn>34</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1017" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1019" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1022" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1025" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1027" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd01_qd04" start="23">
<p class="para" id="fwk-redden-ch06_s04_qs01_p48"><strong class="emphasis bold">Graph. Find the vertex and the <em class="emphasis">y</em>-intercept. In addition, find the <em class="emphasis">x</em>-intercepts if they exist.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1029" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1030" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1031" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1032" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1033" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1034" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1035" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1036" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1037" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1038" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1039" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1040" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1041" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1042" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1043" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1044" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1045" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1046" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p85"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1047" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1048" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1049" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1050" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1051" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1052" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1053" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1054" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1055" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1056" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd02">
<h3 class="title">Part B: Finding the Maximum or Minimum</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd02_qd01" start="51">
<p class="para" id="fwk-redden-ch06_s04_qs01_p105"><strong class="emphasis bold">Determine the maximum or minimum <em class="emphasis">y</em>-value.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1057" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1058" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1059" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1060" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1061" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1062" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>1</mn><mo>−</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1063" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>20</mn><mi>x</mi><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1064" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1065" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1067" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1069" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1071" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd02_qd02" start="63">
<p class="para" id="fwk-redden-ch06_s04_qs01_p130"><strong class="emphasis bold">Given the following quadratic functions, determine the domain and range.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1073" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>50</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1076" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1079" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1082" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1085" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1088" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p143">The height in feet reached by a baseball tossed upward at a speed of 48 feet per second from the ground is given by the function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1091" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mi>t</mi></mrow></math></span>, where <em class="emphasis">t</em> represents the time in seconds after the ball is thrown. What is the baseball’s maximum height and how long does it take to attain that height?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p145">The height in feet of a projectile launched straight up from a mound is given by the function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1092" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>96</mn><mi>t</mi><mo>+</mo><mn>4</mn></mrow></math></span>, where <em class="emphasis">t</em> represents seconds after launch. What is the maximum height?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p147">The profit in dollars generated by producing and selling <em class="emphasis">x</em> custom lamps is given by the function <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1093" display="inline"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>800</mn><mi>x</mi><mo>−</mo><mn>12,000</mn></mrow><mo>.</mo></math></span> What is the maximum profit?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p149">The profit in dollars generated from producing and selling a particular item is modeled by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1094" display="inline"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>100</mn><mi>x</mi><mo>−</mo><mn>0.0025</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span>, where <em class="emphasis">x</em> represents the number of units produced and sold. What number of units must be produced and sold to maximize revenue?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p151">The average number of hits to a radio station Web site is modeled by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1095" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>450</mn><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>3,600</mn><mi>t</mi><mo>+</mo><mn>8,000</mn></mrow></math></span>, where <em class="emphasis">t</em> represents the number of hours since 8:00 a.m. At what hour of the day is the number of hits to the Web site at a minimum?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p153">The value in dollars of a new car is modeled by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1096" display="inline"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>125</mn><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>3,000</mn><mi>t</mi><mo>+</mo><mn>22,000</mn></mrow></math></span>, where <em class="emphasis">t</em> represents the number of years since it was purchased. Determine the minimum value of the car.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p155">The daily production cost in dollars of a textile manufacturing company producing custom uniforms is modeled by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1097" display="inline"><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.02</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>x</mi><mo>+</mo><mn>10,000</mn></mrow></math></span>, where <em class="emphasis">x</em> represents the number of uniforms produced.</p>
<p class="para" id="fwk-redden-ch06_s04_qs01_p156">
</p>
<ol class="orderedlist" id="fwk-redden-ch06_s04_qs01_o01" numeration="loweralpha"> <li>How many uniforms should be produced to minimize the daily production costs?</li>
<li>What is the minimum daily production cost?</li>
</ol>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p157">The area in square feet of a certain rectangular pen is given by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1098" display="inline"><mrow><mi>A</mi><mo>=</mo><mn>14</mn><mi>w</mi><mo>−</mo><msup><mi>w</mi><mn>2</mn></msup></mrow></math></span>, where <em class="emphasis">w</em> represents the width in feet. Determine the width that produces the maximum area.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd03">
<h3 class="title">Part C: Finding the Vertex by Completing the Square</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd03_qd01" start="77">
<p class="para" id="fwk-redden-ch06_s04_qs01_p159"><strong class="emphasis bold">Determine the vertex.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1099" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1100" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1101" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1102" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1103" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1104" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd03_qd02" start="83">
<p class="para" id="fwk-redden-ch06_s04_qs01_p172"><strong class="emphasis bold">Rewrite in vertex form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1105" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></math></span> and determine the vertex.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p173"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1106" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>+</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p175"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1108" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>40</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p177"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1110" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p179"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1112" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p181"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1114" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p183"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1116" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p185"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1118" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p187"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1120" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd03_qd03" start="91">
<p class="para" id="fwk-redden-ch06_s04_qs01_p189"><strong class="emphasis bold">Graph. Find the vertex and the <em class="emphasis">y</em>-intercept. In addition, find the <em class="emphasis">x</em>-intercepts if they exist.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1122" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1123" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1124" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1125" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1126" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1127" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p202"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1128" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p204"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1129" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p206"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1130" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p208"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1131" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p210"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1132" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p212"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1133" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p214"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1134" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p216"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1135" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p218"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1136" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>22</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p220"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1137" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>13</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s04_qs01_qd04_qd01" start="107">
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p222">Write down your plan for graphing a parabola on an exam. What will you be looking for and how will you present your answer? Share your plan on the discussion board.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p223">Why is any parabola that opens upward or downward a function? Explain to a classmate how to determine the domain and range.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch06_s04_qs01_p224">Research and discuss ways of finding a quadratic function that has a graph passing through any three given points. Share a list of steps as well as an example of how to do this.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s04_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p03_ans">Upward</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p07_ans">Downward</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p11_ans">Downward</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p16_ans"><em class="emphasis">x</em>-intercepts: (−6, 0), (2, 0);<em class="emphasis">y</em>-intercept: (0, −12)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p20_ans"><em class="emphasis">x</em>-intercepts: (−3, 0), <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0997" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercept: (0, −3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p24_ans"><em class="emphasis">x</em>-intercepts: (−1, 0), <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1001" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>,</mo><mtext> </mtext><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercept: (0, 2)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p28_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1006" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1007" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercept: (0, −27)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p32_ans"><em class="emphasis">x</em>-intercepts: none; <em class="emphasis">y</em>-intercept: (0, 1)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p37_ans">Vertex: (5, −9); line of symmetry: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1016" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p41_ans">Vertex: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1020" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>; line of symmetry: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1021" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s04_qs01_p45_ans">Vertex: (0, −1); line of symmetry: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1026" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s04_qs01_qa23_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_09/9b9dcd736c50a8b3677f067365e567e9.png">
</div>
</div>