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<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
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<title>Linear Functions and Their Graphs</title>
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<div id="book-content">
<div class="section" id="fwk-redden-ch02_s02" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">2.2</span> Linear Functions and Their Graphs</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch02_s02_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch02_s02_o01" numeration="arabic">
<li>Graph a line by plotting points.</li>
<li>Determine the slope of a line.</li>
<li>Identify and graph a linear function using the slope and <em class="emphasis">y</em>-intercept.</li>
<li>Interpret solutions to linear equations and inequalities graphically.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch02_s02_s01" version="5.0" lang="en">
<h2 class="title editable block">A Review of Graphing Lines</h2>
<p class="para block" id="fwk-redden-ch02_s02_s01_p01">Recall that the set of all solutions to a linear equation can be represented on a rectangular coordinate plane using a straight line through at least two points; this line is called its graph. For example, to graph the linear equation <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0457" display="inline"><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>12</mn></mrow></math></span> we would first solve for <em class="emphasis">y</em>.</p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0458" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>12</mn><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mn>8</mn><mi>x</mi><mtext> </mtext><mi>o</mi><mi>n</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><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>12</mn><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</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><mn>4.</mn></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow><mn>4</mn></mfrac><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><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></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>8</mn><mi>x</mi></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><mn>12</mn></mrow><mn>4</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p03">Written in this form, we can see that <em class="emphasis">y</em> depends on <em class="emphasis">x</em>; in other words, <em class="emphasis">x</em> is the <span class="margin_term"><a class="glossterm">independent variable</a><span class="glossdef">The variable that determines the values of other variables. Usually we think of the <em class="emphasis">x</em>-value of an ordered pair (<em class="emphasis">x</em>, <em class="emphasis">y</em>) as the independent variable.</span></span> and <em class="emphasis">y</em> is the <span class="margin_term"><a class="glossterm">dependent variable</a><span class="glossdef">The variable whose value is determined by the value of the independent variable. Usually we think of the <em class="emphasis">y</em>-value of an ordered pair (<em class="emphasis">x</em>, <em class="emphasis">y</em>) as the dependent variable.</span></span>. Choose at least two <em class="emphasis">x</em>-values and find the corresponding <em class="emphasis">y</em>-values. It is a good practice to choose zero, some negative numbers, as well as some positive numbers. Here we will choose five <em class="emphasis">x</em> values, determine the corresponding <em class="emphasis">y</em>-values, and then form a representative set of ordered pair solutions.</p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p04"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">x</em></p></th>
<th align="center"><p class="para"><em class="emphasis">y</em></p></th>
<th align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0459" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><em class="emphasis">Solutions</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">−2</p></td>
<td align="center"><p class="para">7</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0460" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>4</mn><mo>+</mo><mn>3</mn><mo>=</mo><mn>7</mn></mrow></math></span></p></td>
<td align="center"><p class="para">(−2, 7)</p></td>
</tr>
<tr>
<td align="center"><p class="para">−1</p></td>
<td align="center"><p class="para">5</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0461" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>=</mo><mn>5</mn></mrow></math></span></p></td>
<td align="center"><p class="para">(−1, 5)</p></td>
</tr>
<tr>
<td align="center"><p class="para">0</p></td>
<td align="center"><p class="para">3</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0462" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn><mo>+</mo><mn>3</mn><mo>=</mo><mn>3</mn></mrow></math></span></p></td>
<td align="center"><p class="para">(0, 3)</p></td>
</tr>
<tr>
<td align="center"><p class="para">4</p></td>
<td align="center"><p class="para">−5</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0463" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>4</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>8</mn><mo>+</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>5</mn></mrow></math></span></p></td>
<td align="center"><p class="para">(4, −5)</p></td>
</tr>
<tr>
<td align="center"><p class="para">6</p></td>
<td align="center"><p class="para">−9</p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0464" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>6</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>12</mn><mo>+</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>9</mn></mrow></math></span></p></td>
<td align="center"><p class="para">(6, −9)</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p05">Plot the points and draw a line through the points with a straightedge. Be sure to add arrows on either end to indicate that the graph extends indefinitely.</p>
<div class="informalfigure large block">
<img src="section_05/0244f9b50115f258989e40e7e426dc63.png">
</div>
<p class="para block" id="fwk-redden-ch02_s02_s01_p07">The resulting line represents all solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0465" display="inline"><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>12</mn></mrow></math></span>, of which there are infinitely many. The above process describes the technique for graphing known as <span class="margin_term"><a class="glossterm">plotting points</a><span class="glossdef">A way of determining a graph using a finite number of representative ordered pair solutions.</span></span>. This technique will be used to graph more complicated functions as we progress in this course.</p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p08">The steepness of any incline can be measured as the ratio of the vertical change to the horizontal change. For example, a 5% incline can be written as <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0466" display="inline"><mrow><mfrac><mn>5</mn><mrow><mn>100</mn></mrow></mfrac></mrow></math></span>, which means that for every 100 feet forward, the height increases 5 feet.</p>
<div class="informalfigure large block">
<img src="section_05/44f82796f7f071a80cad03b3a908622e.png">
</div>
<p class="para block" id="fwk-redden-ch02_s02_s01_p10">In mathematics, we call the incline of a line the <span class="margin_term"><a class="glossterm">slope</a><span class="glossdef">The incline of a line measured as the ratio of the vertical change to the horizontal change, often referred to as “rise over run.”</span></span>, denoted by the letter <em class="emphasis">m</em>. The vertical change is called the <span class="margin_term"><a class="glossterm">rise</a><span class="glossdef">The vertical change between any two points on a line.</span></span> and the horizontal change is called the <span class="margin_term"><a class="glossterm">run</a><span class="glossdef">The horizontal change between any two points on a line.</span></span>. Given any two points <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0467" display="inline"><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0468" display="inline"><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span>, we can obtain the rise and run by subtracting the corresponding coordinates.</p>
<div class="informalfigure large block">
<img src="section_05/fbf2e7be33395f3360392a9bb9c72aa8.png">
</div>
<p class="para block" id="fwk-redden-ch02_s02_s01_p12">This leads us to the <span class="margin_term"><a class="glossterm">slope formula</a><span class="glossdef">The slope of the line through the points <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0469" display="inline"><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0470" display="inline"><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span> is given by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0471" display="inline"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>.</mo></math></span></span></span>. Given any two points <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0472" display="inline"><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0473" display="inline"><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math></span>, the slope is given by:</p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p13"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0474" display="block"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>l</mi><mi>o</mi><mi>p</mi><mi>e</mi></mstyle><mtext> </mtext><mi>m</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>i</mi><mi>s</mi><mi>e</mi></mrow><mrow><mi>r</mi><mi>u</mi><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>←</mo><mtext> </mtext><mi>C</mi><mi>h</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>y</mi></mstyle></mtd></mtr><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>←</mo><mtext> </mtext><mi>C</mi><mi>h</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>x</mi></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p14">The Greek letter delta (<span class="inlineequation"><math xml:id="fwk-redden-ch02_m0475" display="inline"><mi mathvariant="normal">Δ</mi></math></span>) is often used to describe the change in a quantity. Therefore, the slope is sometimes described using the notation <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0476" display="inline"><mrow><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac></mrow></math></span>, which represents the change in <em class="emphasis">y</em> divided by the change in <em class="emphasis">x</em>.</p>
<div class="callout block" id="fwk-redden-ch02_s02_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch02_s02_s01_p15">Find the slope of the line passing through (−3, −5) and (2, 1).</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s01_p16">Given (−3, −5) and (2, 1), calculate the difference of the <em class="emphasis">y</em>-values divided by the difference of the <em class="emphasis">x</em>-values. Take care to be consistent when subtracting the coordinates:</p>
<p class="para" id="fwk-redden-ch02_s02_s01_p17"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0477" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0478" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mi>m</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>1</mn><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>1</mn><mo>+</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>+</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>6</mn><mn>5</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p19">It does not matter which point you consider to be the first and second. However, because subtraction is not commutative, you must take care to subtract the coordinates of the first point from the coordinates of the second point in the same order. For example, we obtain the same result if we apply the slope formula with the points switched:</p>
<p class="para" id="fwk-redden-ch02_s02_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0479" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0480" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>m</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><mo>−</mo><mn>1</mn></mrow><mrow><mo>−</mo><mn>3</mn><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>6</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p22">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0481" display="inline"><mrow><mi>m</mi><mo>=</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch02_s02_s01_p23">Verify that the slope is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0482" display="inline"><mrow><mfrac><mn>6</mn><mrow><mn>5</mn></mrow></mfrac></mrow></math></span> by graphing the line described in the previous example.</p>
<div class="informalfigure large block">
<img src="section_05/47a6a3a8626ad58361ab3f4f7f4db808.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p25">Certainly the graph is optional; the beauty of the slope formula is that, given any two points, we can obtain the slope using only algebra.</p>
<div class="callout block" id="fwk-redden-ch02_s02_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch02_s02_s01_p26">Find the <em class="emphasis">y</em>-value for which the slope of the line passing through <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0483" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0484" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>9</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0485" display="inline"><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><mn>3</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s01_p27">Substitute the given information into the slope formula.</p>
<p class="para" id="fwk-redden-ch02_s02_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0486" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mtext> </mtext><mi>S</mi><mi>l</mi><mi>o</mi><mi>p</mi><mi>e</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>m</mi><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>9</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0487" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>m</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mtext> </mtext><mi>y</mi><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mtext> </mtext></mrow><mrow><mo>−</mo><mn>9</mn><mo>−</mo><mn>6</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mi>y</mi><mo>+</mo><mn>3</mn><mtext> </mtext></mrow><mrow><mo>−</mo><mn>15</mn></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p30">After substituting in the given information, the only variable left is <em class="emphasis">y</em>. Solve.</p>
<p class="para" id="fwk-redden-ch02_s02_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0488" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mo>−</mo><mn>15</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>−</mo><mn>15</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mi>y</mi><mo>+</mo><mn>3</mn><mtext> </mtext></mrow><mrow><mn>15</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>y</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>y</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s01_p32">Answer: <em class="emphasis">y</em> = 7</p>
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p33">There are four geometric cases for the value of the slope.</p>
<div class="informalfigure large block">
<img src="section_05/1937e900a5722c4c34b039ba0cc01e05.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p35">Reading the graph from left to right, lines with an upward incline have positive slopes and lines with a downward incline have negative slopes. The other two cases involve horizontal and vertical lines. Recall that if <em class="emphasis">k</em> is a real number we have</p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p36"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0489" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mi>y</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>k</mi></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><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>L</mi><mi>i</mi><mi>n</mi><mi>e</mi></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>k</mi></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><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>L</mi><mi>i</mi><mi>n</mi><mi>e</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p37">For example, if we graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0490" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn></mrow></math></span> we obtain a horizontal line, and if we graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0491" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>4</mn></mrow></math></span> we obtain a vertical line.</p>
<div class="informalfigure large block">
<img src="section_05/c9a3bfd24ed1288a534c5597f0e47413.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p39">From the graphs we can determine two points and calculate the slope using the slope formula.</p>
<p class="para block" id="fwk-redden-ch02_s02_s01_p40"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Horizontal Line</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Vertical Line</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0492" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>2</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0493" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>m</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>2</mn><mo>−</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>2</mn><mo>−</mo><mn>2</mn></mrow><mrow><mn>3</mn><mo>+</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>0</mn><mn>6</mn></mfrac><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math></span></p>
</td>
<td align="center">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0494" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0495" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>m</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>1</mn><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>4</mn><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow><mrow><mo>−</mo><mn>4</mn><mo>+</mo><mn>4</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>2</mn><mn>0</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>U</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>f</mi><mi>i</mi><mi>n</mi><mi>e</mi><mi>d</mi></mstyle></mtd></mtr></mtable></math></span></p>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s01_p41">Notice that the points on the horizontal line share the same <em class="emphasis">y</em>-values. Therefore, the rise is zero and hence the slope is zero. The points on the vertical line share the same <em class="emphasis">x</em>-values. Consequently, the run is zero, leading to an undefined slope. In general,</p>
<div class="informalfigure large block">
<img src="section_05/a7c8b429b81c4191b69e332116d192b5.png">
</div>
</div>
<div class="section" id="fwk-redden-ch02_s02_s02" version="5.0" lang="en">
<h2 class="title editable block">Linear Functions</h2>
<p class="para block" id="fwk-redden-ch02_s02_s02_p01">Given any linear equation in <span class="margin_term"><a class="glossterm">standard form</a><span class="glossdef">Any nonvertical line can be written in the standard form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0496" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>=</mo><mi>c</mi></mrow><mo>.</mo></math></span></span></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0497" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>=</mo><mi>c</mi></mrow></math></span>, we can solve for <em class="emphasis">y</em> to obtain <span class="margin_term"><a class="glossterm">slope-intercept form</a><span class="glossdef">Any nonvertical line can be written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0498" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></math></span>, where <em class="emphasis">m</em> is the slope and (0, <em class="emphasis">b</em>) is the <em class="emphasis">y</em>-intercept.</span></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0499" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>.</mo></math></span> For example,</p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0500" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>←</mo><mtext> </mtext><mi>S</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>a</mi><mi>r</mi><mi>d</mi><mtext> </mtext><mi>F</mi><mi>o</mi><mi>r</mi><mi>m</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>3</mn><mi>x</mi></mrow><mrow><mo>−</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mn>8</mn><mrow><mo>−</mo><mn>4</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mn>2</mn><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>←</mo><mtext> </mtext><mi>S</mi><mi>l</mi><mi>o</mi><mi>p</mi><mi>e</mi><mtext>-</mtext><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><mtext> </mtext><mi>F</mi><mi>o</mi><mi>r</mi><mi>m</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p03">Where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0501" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0502" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span> and thus <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0503" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> is an ordered pair solution. This is the point where the graph intersects the <em class="emphasis">y</em>-axis and is called the <span class="margin_term"><a class="glossterm"><em class="emphasis">y</em>-intercept</a><span class="glossdef">The point (or points) where a graph intersects the <em class="emphasis">y</em>-axis, expressed as an ordered pair (0, <em class="emphasis">y</em>).</span></span>. We can use this point and the slope as a means to quickly graph a line. For example, to graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0504" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span>, start at the <em class="emphasis">y</em>-intercept <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0505" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> and mark off the slope to find a second point. Then use these points to graph the line as follows:</p>
<div class="informalfigure large block">
<img src="section_05/6b9765d69c08031feace9f698f6a8887.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s02_p05">The vertical line test indicates that this graph represents a function. Furthermore, the domain and range consists of all real numbers.</p>
<div class="informalfigure large block">
<img src="section_05/8c6a1121fa317dd71ca28ba409ab2c41.png">
</div>
<p class="para block" id="fwk-redden-ch02_s02_s02_p07">In general, a <span class="margin_term"><a class="glossterm">linear function</a><span class="glossdef">Any function that can be written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0506" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></math></span></span></span> is a function that can be written in the form
<span class="informalequation"><math xml:id="fwk-redden-ch02_m0507" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>L</mi><mi>i</mi><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</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></mstyle></mrow></mtd></mtr></mtable></mrow></math></span>
where the slope <em class="emphasis">m</em> and <em class="emphasis">b</em> represent any real numbers. Because <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0508" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, we can use <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0509" display="inline"><mi>y</mi></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0510" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> interchangeably, and ordered pair solutions on the graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0511" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> can be written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0512" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0513" display="block"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>⇔</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p11">We know that any <em class="emphasis">y</em>-intercept will have an <em class="emphasis">x</em>-value equal to zero. Therefore, the <em class="emphasis">y</em>-intercept can be expressed as the ordered pair <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0514" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> For linear functions,</p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0515" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>m</mi><mrow><mo>(</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mi>b</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>b</mi></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p13">Hence, the <em class="emphasis">y</em>-intercept of any linear function is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0516" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To find the <span class="margin_term"><a class="glossterm"><em class="emphasis">x</em>-intercept</a><span class="glossdef">The point (or points) where a graph intersects the <em class="emphasis">x</em>-axis, expressed as an ordered pair (<em class="emphasis">x</em>, 0).</span></span>, the point where the function intersects the <em class="emphasis">x</em>-axis, we find <em class="emphasis">x</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0517" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0518" 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>
<div class="callout block" id="fwk-redden-ch02_s02_s02_n01">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch02_s02_s02_p14">Graph the linear function <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0519" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span> and label the <em class="emphasis">x</em>-intercept.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p15">From the function, we see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0520" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>6</mn></mrow></math></span> (or <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0521" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>6</mn></mrow></math></span>) and thus the <em class="emphasis">y</em>-intercept is (0, 6). Also, we can see that the slope <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0522" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>5</mn></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mi>r</mi><mi>i</mi><mi>s</mi><mi>e</mi></mrow><mrow><mi>r</mi><mi>u</mi><mi>n</mi></mrow></mfrac></mrow><mo>.</mo></math></span> Starting from the <em class="emphasis">y</em>-intercept, mark a second point down 5 units and right 3 units. Draw the line passing through these two points with a straightedge.</p>
<div class="informalfigure large">
<img src="section_05/d4830fed42f5b2eb4ba09b1c9a93dfd6.png">
</div>
<p class="para" id="fwk-redden-ch02_s02_s02_p17">To determine the <em class="emphasis">x</em>-intercept, find the <em class="emphasis">x</em>-value where the function is equal to zero. In other words, determine <em class="emphasis">x</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0523" 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-ch02_s02_s02_p18"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0524" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>3</mn><mn>5</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>3</mn><mn>5</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>18</mn></mrow><mn>5</mn></mfrac><mo>=</mo><mn>3</mn><mfrac><mn>3</mn><mn>5</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s02_p19">Therefore, the <em class="emphasis">x</em>-intercept is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0525" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>18</mn></mrow><mn>5</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> The general rule is to label all important points that cannot be clearly read from the graph.</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p20">Answer:</p>
<div class="informalfigure large">
<img src="section_05/d31fac765e2408cdd6a02ff500472947.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch02_s02_s02_n02">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch02_s02_s02_p22">Determine a linear function that defines the given graph and find the <em class="emphasis">x</em>-intercept.</p>
<div class="informalfigure large">
<img src="section_05/7c2d4bce240f9afd2d0d5c28ab2cee2c.png">
</div>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p24">We begin by reading the slope from the graph. In this case, two points are given and we can see that,</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p25"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0526" display="block"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>i</mi><mi>s</mi><mi>e</mi></mrow><mrow><mi>r</mi><mi>u</mi><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mn>3</mn></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s02_p26">In addition, the <em class="emphasis">y</em>-intercept is (0, 3) and thus <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0527" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span> We can substitute into the equation for any linear function.</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0528" display="block"><mtable columnalign="center" columnspacing="0.1em"><mtr><mtd><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mi>m</mi><mi>x</mi></mtd><mtd><mo>+</mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>+</mo></mtd><mtd><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s02_p28">To find the <em class="emphasis">x</em>-intercept, we set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0529" display="inline"><mrow><mi>g</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-ch02_s02_s02_p29"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0530" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>9</mn><mn>2</mn></mfrac><mo>=</mo><mn>4</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s02_p30">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0531" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span>; <em class="emphasis">x</em>-intercept: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0532" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>9</mn><mn>2</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s02_p31">Next, consider horizontal and vertical lines. Use the vertical line test to see that any horizontal line represents a function, and that a vertical line does not.</p>
<div class="informalfigure large block">
<img src="section_05/5ad671aeecab21526320fced06da0ecf.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s02_p33">Given any horizontal line, the vertical line test shows that every <em class="emphasis">x</em>-value in the domain corresponds to exactly one <em class="emphasis">y</em>-value in the range; it is a function. A vertical line, on the other hand, fails the vertical line test; it is not a function. A vertical line represents a set of ordered pairs where all of the elements in the domain are the same. This violates the requirement that functions must associate exactly one element in the range to each element in the domain. We summarize as follows:</p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p34"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para"> </p></th>
<th align="center"><p class="para"><em class="emphasis">Horizontal Line</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Vertical Line</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="right"><p class="para"><em class="emphasis">Equation:</em></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0533" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0534" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><em class="emphasis">x-intercept:</em></p></td>
<td align="center"><p class="para">None</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0535" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><em class="emphasis">y-intercept:</em></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0536" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para">None</p></td>
</tr>
<tr>
<td align="right"><p class="para"><em class="emphasis">Domain:</em></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0537" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0538" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>}</mo></mrow></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><em class="emphasis">Range:</em></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0539" display="inline"><mrow><mrow><mo>{</mo><mn>2</mn><mo>}</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0540" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><em class="emphasis">Function:</em></p></td>
<td align="center"><p class="para">Yes</p></td>
<td align="center"><p class="para">No</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s02_s02_p35">A horizontal line is often called a <strong class="emphasis bold">constant function</strong>. Given any real number <em class="emphasis">c</em>,</p>
<p class="para block" id="fwk-redden-ch02_s02_s02_p36"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0541" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>t</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></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch02_s02_s02_n03">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch02_s02_s02_p37">Graph the constant function <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0542" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span> and state the domain and range.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p38">Here we are given a constant function that is equivalent to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0543" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><mo>.</mo></math></span> This defines a horizontal line through <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0544" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large">
<img src="section_05/a90d830abb8ae95ee20344c4f4eec389.png">
</div>
<p class="para" id="fwk-redden-ch02_s02_s02_p40">Answer: Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0545" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0546" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch02_s02_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s02_s02_p41"><strong class="emphasis bold">Try this!</strong> Graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0547" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span> and label the <em class="emphasis">x</em>-intercept.</p>
<p class="para" id="fwk-redden-ch02_s02_s02_p42">Answer:</p>
<div class="informalfigure large">
<img src="section_05/802ac96a297617787cf6a402dc3d2545.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/iyePPmPaMlA" condition="http://img.youtube.com/vi/iyePPmPaMlA/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/iyePPmPaMlA" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch02_s02_s03" version="5.0" lang="en">
<h2 class="title editable block">Linear Equations and Inequalities: A Graphical Interpretation</h2>
<p class="para block" id="fwk-redden-ch02_s02_s03_p01">We can use the ideas in this section to develop a geometric understanding of what it means to solve equations of the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0548" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, where <em class="emphasis">f</em> and <em class="emphasis">g</em> are linear functions. Using algebra, we can solve the linear equation <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0549" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn></mrow></math></span> as follows:</p>
<p class="para block" id="fwk-redden-ch02_s02_s03_p02"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0550" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s03_p03">The solution to this equation is <em class="emphasis">x</em> = 4. Geometrically, this is the <em class="emphasis">x</em>-value of the intersection of the two graphs <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0551" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0552" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span> The idea is to graph the linear functions on either side of the equation and determine where the graphs coincide.</p>
<div class="callout block" id="fwk-redden-ch02_s02_s03_n01">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch02_s02_s03_p04">Graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0553" display="inline"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0554" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> on the same set of axes and determine where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0555" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s03_p05">Here <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0556" display="inline"><mi>f</mi></math></span> is a linear function with slope <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0557" display="inline"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <em class="emphasis">y</em>-intercept (0,1). The function <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0558" display="inline"><mrow><mi>g</mi></mrow></math></span> is a constant function and represents a horizontal line. Graph both of these functions on the same set of axes.</p>
<div class="informalfigure large">
<img src="section_05/9c0d4ecab240afdf9d8c4a8f0d387f29.png">
</div>
<p class="para" id="fwk-redden-ch02_s02_s03_p07">From the graph we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0559" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0560" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span> In other words, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0561" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0562" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s03_p08">Answer: <em class="emphasis">x</em> = 4</p>
</div>
<p class="para block" id="fwk-redden-ch02_s02_s03_p09">We can extend the geometric interpretation a bit further to solve inequalities. For example, we can solve the linear inequality <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0563" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>3</mn></mrow></math></span>, using algebra, as follows:</p>
<p class="para block" id="fwk-redden-ch02_s02_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch02_m0564" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch02_s02_s03_p11">The solution set consists of all real numbers greater than or equal to 4. Geometrically, these are the <em class="emphasis">x</em>-values for which the graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0565" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span> lies above the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0566" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch02_s02_s03_n02">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch02_s02_s03_p12">Graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0567" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0568" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> on the same set of axes and determine where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0569" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s02_s03_p13">On the graph we can see this shaded.</p>
<div class="informalfigure large">
<img src="section_05/3c382de66a8835a4c11154dee0841ead.png">
</div>
<p class="para" id="fwk-redden-ch02_s02_s03_p15">From the graph we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0570" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0571" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>3</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0572" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s02_s03_p16">Answer: The <em class="emphasis">x</em>-values that solve the inequality, in interval notation, are <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0573" display="inline"><mrow><mrow><mo>[</mo><mrow><mn>4</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
<div class="key_takeaways block" id="fwk-redden-ch02_s02_s03_n03">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch02_s02_s03_l01" mark="bullet">
<li>We can graph lines by plotting points. Choose a few values for <em class="emphasis">x</em>, find the corresponding <em class="emphasis">y</em>-values, and then plot the resulting ordered pair solutions. Draw a line through the points with a straightedge to complete the graph.</li>
<li>Given any two points on a line, we can calculate the slope algebraically using the slope formula, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0574" display="inline"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>i</mi><mi>s</mi><mi>e</mi></mrow><mrow><mi>r</mi><mi>u</mi><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac></mrow><mo>.</mo></math></span>
</li>
<li>Use slope-intercept form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0575" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></math></span> to quickly sketch the graph of a line. From the <em class="emphasis">y</em>-intercept <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0576" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span>, mark off the slope to determine a second point. Since two points determine a line, draw a line through these two points with a straightedge to complete the graph.</li>
<li>Linear functions have the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0577" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></math></span>, where the slope <em class="emphasis">m</em> and <em class="emphasis">b</em> are real numbers. To find the <em class="emphasis">x</em>-intercept, if one exists, set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0578" 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>.</li>
<li>Since <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0579" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> we can use <em class="emphasis">y</em> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0580" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> interchangeably. Any point on the graph of a function can be expressed using function notation <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0581" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span>
</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s02_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd01">
<h3 class="title">Part A: Graphing Lines by Plotting Points</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch02_s02_qs01_p01"><strong class="emphasis bold">Find five ordered pair solutions and graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0582" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0583" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0584" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0585" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0586" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0587" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0588" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0589" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0590" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0591" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0592" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0593" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0594" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0595" display="inline"><mrow><mn>8</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0596" display="inline"><mrow><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0597" display="inline"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0598" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>y</mi><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0599" display="inline"><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0600" display="inline"><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0601" display="inline"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd01_qd02" start="21">
<p class="para" id="fwk-redden-ch02_s02_qs01_p42"><strong class="emphasis bold">Find the slope of the line passing through the given points.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0602" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0603" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0604" 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-ch02_m0605" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0607" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0608" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0610" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0611" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0612" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>9</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0613" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>9</mn><mo>,</mo><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa26">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0614" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0615" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd01_qd03" start="27">
<p class="para" id="fwk-redden-ch02_s02_qs01_p55"><strong class="emphasis bold">Find the <em class="emphasis">y</em>-value for which the slope of the line passing through given points has the given slope.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0617" display="inline"><mrow><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0618" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0619" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0621" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0622" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0623" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>9</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0625" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mn>4</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0626" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0627" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0629" display="inline"><mrow><mi>m</mi><mo>=</mo><mn>3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0630" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0631" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0633" display="inline"><mrow><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0634" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0635" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0637" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0638" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0639" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd01_qd04" start="33">
<p class="para" id="fwk-redden-ch02_s02_qs01_p68"><strong class="emphasis bold">Given the graph, determine the slope.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa33">
<div class="question">
<div class="informalfigure large">
<img src="section_05/956d7b6b4031a799fc9b6d12b722e140.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa34">
<div class="question">
<div class="informalfigure large">
<img src="section_05/434dbaf198fb18199a9cfa2c0e8d0766.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa35">
<div class="question">
<div class="informalfigure large">
<img src="section_05/7b5a473ccbc9dd7057897771a874bf53.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa36">
<div class="question">
<div class="informalfigure large">
<img src="section_05/db48810d3eacf46e19d2ba15ac9155cc.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa37">
<div class="question">
<div class="informalfigure large">
<img src="section_05/41661f8a2aab3b531eada7499f707111.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa38">
<div class="question">
<div class="informalfigure large">
<img src="section_05/c0c8eedb901784cb201bb1cae119cc52.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa39">
<div class="question">
<div class="informalfigure large">
<img src="section_05/833b9adf04e15e6f72a3801336f5cf4c.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa40">
<div class="question">
<div class="informalfigure large">
<img src="section_05/224ad953d35f57d53c40bb241498714a.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd02">
<h3 class="title">Part B: Linear Functions</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd02_qd01" start="41">
<p class="para" id="fwk-redden-ch02_s02_qs01_p85"><strong class="emphasis bold">Find the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts and use them to graph the following functions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0648" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0649" display="inline"><mrow><mn>8</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0650" display="inline"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p92"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0651" display="inline"><mrow><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0652" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0653" display="inline"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0654" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0655" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0656" display="inline"><mrow><mn>9</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>30</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0657" display="inline"><mrow><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>28</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0658" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0659" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0660" display="inline"><mrow><mfrac><mn>7</mn><mn>9</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mrow><mn>14</mn></mrow><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0661" display="inline"><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0662" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>9</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0663" display="inline"><mrow><mfrac><mn>2</mn><mrow><mn>15</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0664" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0665" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>3</mn><mn>8</mn></mfrac><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-ch02_s02_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0666" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0667" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd02_qd02" start="61">
<p class="para" id="fwk-redden-ch02_s02_qs01_p126"><strong class="emphasis bold">Graph the linear function and label the <em class="emphasis">x</em>-intercept.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p127"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0668" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p129"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0669" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0670" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0671" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0672" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0673" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0674" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0675" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0676" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>6</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0677" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0678" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0679" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd02_qd03" start="73">
<p class="para" id="fwk-redden-ch02_s02_qs01_p151"><strong class="emphasis bold">Determine the linear function that defines the given graph and find the <em class="emphasis">x</em>-intercept.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa73">
<div class="question">
<div class="informalfigure large">
<img src="section_05/f1637826b168fe76cf0fc6cf7860d9a8.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa74">
<div class="question">
<div class="informalfigure large">
<img src="section_05/1e327adbdf8b40a7199a50c77886b3ee.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa75">
<div class="question">
<div class="informalfigure large">
<img src="section_05/cc4269012d902ffad867ca47e7df3c91.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa76">
<div class="question">
<div class="informalfigure large">
<img src="section_05/fb91d9cf79934c55e322730372676a70.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa77">
<div class="question">
<div class="informalfigure large">
<img src="section_05/eb753f457bbee549fb53bf9881b36b7d.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa78">
<div class="question">
<div class="informalfigure large">
<img src="section_05/8c34ae0b680160da94a78fd86b58f73b.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa79">
<div class="question">
<div class="informalfigure large">
<img src="section_05/b054eb5453faaf4b49091d5e8f2e8cd6.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa80">
<div class="question">
<div class="informalfigure large">
<img src="section_05/a8870e33cc8e3acf2fffdd515b1d986f.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd03">
<h3 class="title">Part C: A Graphical Interpretation of Linear Equations and Inequalities</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd03_qd01" start="81">
<p class="para" id="fwk-redden-ch02_s02_qs01_p168"><strong class="emphasis bold">Graph the functions</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0694" display="inline"><mi>f</mi></math></span> <strong class="emphasis bold">and</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0695" display="inline"><mi>g</mi></math></span> <strong class="emphasis bold">on the same set of axes and determine where</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0696" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span> <strong class="emphasis bold">Verify your answer algebraically.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p169"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0697" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0698" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p171"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0700" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0701" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p173"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0703" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0704" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p175"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0706" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0707" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p177"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0709" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0710" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p179"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0712" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0713" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p181"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0715" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0716" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p183"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0718" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0719" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p185"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0721" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0722" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p187"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0724" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0725" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd03_qd02" start="91">
<p class="para" id="fwk-redden-ch02_s02_qs01_p189"><strong class="emphasis bold">Graph the functions</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0727" display="inline"><mi>f</mi></math></span> <strong class="emphasis bold">and</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0728" display="inline"><mi>g</mi></math></span> <strong class="emphasis bold">on the same set of axes and determine where</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0729" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span> <strong class="emphasis bold">Verify your answer algebraically.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0730" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0731" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0733" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0734" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0736" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0737" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0739" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0740" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0742" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0743" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0745" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0746" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p202"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0748" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0749" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p204"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0751" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0752" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd03_qd03" start="99">
<p class="para" id="fwk-redden-ch02_s02_qs01_p206"><strong class="emphasis bold">Graph the functions</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0754" display="inline"><mi>f</mi></math></span> <strong class="emphasis bold">and</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0755" display="inline"><mi>g</mi></math></span> <strong class="emphasis bold">on the same set of axes and determine where</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0756" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span> <strong class="emphasis bold">Verify your answer algebraically.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p207"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0757" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0758" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p209"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0760" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0761" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p211"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0763" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mi>x</mi></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0764" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p213"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0766" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0767" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p215"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0769" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0770" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p217"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0772" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0773" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p219"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0775" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0776" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p221"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m0778" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m0779" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s02_qs01_qd04_qd01" start="107">
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p223">Do all linear functions have <em class="emphasis">y</em>-intercepts? Do all linear functions have <em class="emphasis">x</em>-intercepts? Explain.</p>
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<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p224">Can a function have more than one <em class="emphasis">y</em>-intercept? Explain.</p>
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<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch02_s02_qs01_p225">How does the vertical line test show that a vertical line is not a function?</p>
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<div class="qandaset block" id="fwk-redden-ch02_s02_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
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<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa01_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/f3d050fa1174f80fdcd15b1b0a9785d1.png">
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<li class="qandaentry" id="fwk-redden-ch02_s02_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">
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<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa03_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/17848b86dfcb89c8761e39a24ef7d418.png">
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<li class="qandaentry" id="fwk-redden-ch02_s02_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">
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</li>
<li class="qandaentry" id="fwk-redden-ch02_s02_qs01_qa05_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/9b2e8d28c18ad6e18d942ad21bf8a224.png">
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<li class="qandaentry" id="fwk-redden-ch02_s02_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-ch02_s02_qs01_qa07_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/6bd691b4c4b33e5dd039acb92dcdf493.png">
</div>
</div>