Graphing Absolute Value Equations

<p><span style="font-size:22px"><a href="https://www.prepswift.com/quizzes/quiz/prepswift-graphing-absolute-value-equations" target="_blank">Graphing Absolute Value Equations Exercise</a></span></p><p><span style="color:#8e44ad;">Graphing Linear Absolute Value Equations</span> can be a bit tricky, but it IS doable if you follow these steps.</p> <p style="margin-left: 40px;"><strong>Step 1</strong>: Split the linear absolute value equation into its two separate equations (example below).</p> <p>$$y = |x+3|$$</p> <p style="text-align: center;"><span style="color:#8e44ad;">Equation 1</span>: $y=x+3$</p> <p style="text-align: center;"><span style="color:#8e44ad;">Equation 2</span>: $y=-x-3$</p> <p style="margin-left: 40px;"><strong>Step 2</strong>: Find the value of $x$ that makes the $y$ value AS SMALL AS POSSIBLE. Another way to to think about it is find the $x$ value that makes the &quot;stuff&quot; inside the absolute value lines equal to $0$. In this case, that is $x=-3$. Think of this $x$ value as the &quot;boundary&quot; between the two equations above.&nbsp;</p> <p style="margin-left: 40px;"><strong>Step 3</strong>: On the $xy$-coordinate plane, plot BOTH equations from&nbsp;<strong>Step 1</strong>, using the $x=-3$ as the boundary point, or the &quot;v&quot;-point of the graph. You&#39;ll see below.</p> <p style="text-align: center;"><span style="color:#8e44ad;">Equation 1</span>: $y=x+3$</p> <p style="text-align: center;"><span style="color:#27ae60;">for $x$ values that are $\geq -3$</span></p> <p style="text-align: center;"><span style="color:#8e44ad;">Equation 2</span>: $y=-x-3$</p> <p style="text-align: center;"><span style="color:#e74c3c;">for $x$ values that are $&lt; -3$</span></p> <center><svg height="207.6571502685547" style=" width:661.8285522460938px; height:207.6571502685547px; background: #FFF; fill: none; " width="661.8285522460938" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape axis2d" style="stroke-width: 1; stroke: rgb(0, 0, 0);"><path class="real" d=" M201.6,134.12 L461.04,134.12 M332.04,7 L332.04,186.26"></path><path d=" M454.04,129.12 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630 344 689 233 689C165 689 120 670 72 622L43 498L74 498L92 554C103 588 166 622 218 622C283 622 336 569 336 506C336 431 277 368 206 368C198 368 187 369 174 370L159 371L147 318L154 312C192 329 211 334 238 334C321 334 369 281 369 190C369 88 308 21 215 21C169 21 128 36 98 64C74 86 61 109 42 163L15 153C36 92 44 56 50 6C103 -12 147 -20 184 -20C307 -20 462 87 462 224Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g></svg> </svg></center> <hr /> <p><span style="color:#8e44ad;">Graphing Quadratic Absolute Value Equations</span> can be a tricky mofo. Read below to find out how.</p> <p><strong><span style="color:#27ae60;">The Good News</span></strong></p> <p style="margin-left: 40px;">In some cases, absolute value signs on a qaudratic change nothing. The two quadratic equations below are <strong><u>EQUAL</u></strong> to each other. Can you see why?</p> <p>$$y = x^2 +3$$</p> <p>$$y=|x^2 +3|$$</p> <p style="margin-left: 40px;">The two equations are equal to each other because it&#39;s impossible for the quadratic $y = x^2 + 3$ to be negative, so the absolute value signs are redundant. In cases such as these, graph the quadratic as if the absolute value signs were not there.</p> <p><strong><span style="color:#e74c3c;">The Bad News</span></strong></p> <p style="margin-left: 40px;">Of course, many quadratic equations CAN spit out negative values. For example...</p> <p>$$y = x^2 - 4$$</p> <p style="margin-left: 40px;">So how would we graph something like this? Follow the steps below to find out.</p> <p>$$y = |x^2-4|$$</p> <p style="margin-left: 40px;"><strong>Step 1</strong>: Split the quadratic absolute value equation into its two separate equations.</p> <p style="text-align: center;"><span style="color:#8e44ad;">Equation 1</span>: $y=x^2-4$</p> <p style="text-align: center;"><span style="color:#8e44ad;">Equation 2</span>: $y=-x^2+4$</p> <p style="margin-left: 40px;"><strong>Step 2</strong>: Find the $x$ values that make the $y$ value AS SMALL as possible in the original equation. In other words, find the $x$ values that make the &quot;stuff&quot; inside the absolute value lines equal to $0$.&nbsp;</p> <p style="text-align: center;"><span style="color:#8e44ad;">Value 1</span>: $x=2$</p> <p style="text-align: center;"><span style="color:#8e44ad;">Value 2</span>: $x=-2$&nbsp;</p> <p style="margin-left: 40px;"><strong>Step 3:&nbsp;</strong>Identify the two values from Step 2 as the &quot;boundary points,&quot; the points where the graph suddenly shifts directions.&nbsp;</p> <p style="margin-left: 40px;"><strong>Step 4a:&nbsp;</strong>For $x$ values $\leq -2$ and $\geq 2$, graph <span style="color:#27ae60;">equation 1</span>, making sure not to go beyond the boundary points.</p> <p style="margin-left: 40px;"><strong>Step 4b</strong>: For $x$ values between $-2$ and $2$, graph <span style="color:#e74c3c;">equation 2</span>. 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