https://wiki.math.ucr.edu/index.php?action=history&feed=atom&title=Graph_Transformations 2026-04-22T17:29:40Z Revision history for this page on the wiki MediaWiki 1.35.0 https://wiki.math.ucr.edu/index.php?title=Graph_Transformations&diff=1099&oldid=prev 2015-10-13T05:10:15Z

<p>Created page with &quot;&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt; ==Introduction== In this section we will learn how to graph modified versions of the functions from the library of functions. The transfor...&quot;</p> <p><b>New page</b></p><div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> ==Introduction==<br /> In this section we will learn how to graph modified versions of the functions from the library of functions. The transformations we will be focusing on are left/right shifts, up/down shifts, and vertical stretching/compression.<br /> Even though these properties hold for functions in general, we will focus on functions from the library of functions.<br /> ==Vertical shift==<br /> <br /> A vertical shift comes from taking a graph and moving every point up, or down, by some uniform amount. Algebraically, we take a point (x, f(x)) and move it to (x, f(x) + c), where c is the shift up or down.<br /> This means the function f(x) + c is a shift up by c units if c is positive, and down by c units if c is negative.<br /> <br /> Example: Graph &lt;math&gt; f(x) = x^2 + 4&lt;/math&gt;<br /> <br /> Here we can recognize that the function from the library of functions is &lt;math&gt; x^2&lt;/math&gt;. This tells us that the graph of f(x) will have the same shape as &lt;math&gt;x^2&lt;/math&gt;. Since we are in the vertical shift section, we can<br /> guess that the function has been shifted vertically by some amount. Since we are adding 4 to the output of every point from &lt;math&gt; x^2&lt;/math&gt; the graph of &lt;math&gt;x^2 + 4&lt;/math&gt; is the graph of &lt;math&gt;x^2&lt;/math&gt; shifted up by 4 units.<br /> <br /> ==Horizontal Shift==<br /> Horizontal shifts are a little harder to describe algebraically. Graphically, it is exactly what you think it is. We take the graph of f(x) and move all of the points horizontally by a uniform amount.<br /> For identifying horizontal shifts start by attempting which function from the library of functions is the base function. Then observe where a point on the base function moves. I usually keep track of the y-intercept. So the approach is focused on determining<br /> what value of x is required to plug into the shifted function to get the same value that evaluating the base function at 0 is.<br /> <br /> Example:<br /> <br /> Graph &lt;math&gt; f(x) = (x - 5)^3&lt;/math&gt;<br /> <br /> The first thing to notice is the base function from the library of functions is &lt;math&gt; x^3.&lt;/math&gt; The y-intercept is (0, 0). Now we want to find a value of x such that &lt;math&gt;(x - 5)^3 = 0&lt;/math&gt;. Solving this we find that we need x = 5.<br /> This means the point (0, 0) moved to (5, 0) and every point moved to the right 5 units.<br /> <br /> <br /> [[Math_5|'''Return to Topics Page]]</div>

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