<p><span style="font-size:22px"><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-graphing-by-completing-the-square">Graphing by Completing the Square Exercise</a></span></p><p>If a quadratic is not factorable, or even if it is (lol), you can use <span style="color:#27ae60;">Graphing by Completing the Square</span>. Let's do it with the example below:</p>
<p>$$y = x^2 -8x + 4$$</p>
<p>Notice how the equation above is <span style="color:#e74c3c;">NOT</span> factorable.</p>
<p><strong>Step 1</strong>: Complete the square. If you forgot how to do that, revisit that mountain entry in Algebra.</p>
<p>$$y = x^2 -8x + 4$$</p>
<p>$$y + 16 = x^2 -8x + 16 + 4$$</p>
<p>$$y + 16 = (x^2 -8x + 16) + 4$$</p>
<p>$$y + 16 = (x-4)^2 + 4$$</p>
<p>$$y = (x-4)^2 - 12$$</p>
<p><strong>Step 2</strong>: Use the equation in which we completed the square to find the minimum value. That's easy. We know the $(x-4)^2$ is always going to be positive except in the case in which it's $0$. Let's make it $0$ by saying $x=4$. And when $x=4$, $y = -12$. </p>
<p style="text-align: center;"><strong>vertex (minimum value)</strong>: $(4, -12)$</p>
<p><strong>Step 3</strong>: Find two to four other points (to make graphing easier). I recommend moving one or two points to the left and right of the vertex. Let's see what happens when $x=3$ and when $x=5$. Hell, let's also see what happens when $x=2$ and $x=6$.</p>
<p style="text-align: center;"><strong>other point$_1$</strong>: $(3, -11)$</p>
<p style="text-align: center;"><strong>other point$_2$</strong>: $(5, -11)$</p>
<p style="text-align: center;"><strong>other point$_3$</strong>: $(2, -7)$</p>
<p style="text-align: center;"><strong>other point$_4$</strong>: $(6, -7)$</p>
<p><strong>Step 4</strong>: Graph it.</p>