Horizontal Translations

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node48.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node44.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node50.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node49.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Horizontal Translations</H2> <P> If adding a constant to the outside of a trigonometric function causes a vertical shift in the graph, how can we add a constant to cause a horizontal shift of the graph? The answer is implied in the following example. <P> Consider the graphs of <IMG WIDTH=60 HEIGHT=24 ALIGN=MIDDLE ALT="y = sin(x)" SRC="img132.gif"> and <IMG WIDTH=123 HEIGHT=39 ALIGN=MIDDLE ALT="y = sin(x - pi/4)" SRC="img137.gif"> They appear in the adjacent figure. <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot5.gif" WIDTH=322 HEIGHT=233 ALT="[plot]"> </CENTER> <P> What is apparent here? We see that the graph of <IMG WIDTH=113 HEIGHT=39 ALIGN=MIDDLE ALT="y = sin(x - pi/4)" SRC="img138.gif"> is the same as the graph of <IMG WIDTH=34 HEIGHT=12 ALIGN=BOTTOM ALT="y = sin(x)" SRC="img128.gif"> except that it has been shifted to the <B>right</B> by a factor of <IMG WIDTH=11 HEIGHT=36 ALIGN=MIDDLE ALT="pi/4" SRC="img139.gif"> units. In general, we can say the following: <P> <CENTER> <IMG WIDTH=492 HEIGHT=84 ALIGN=BOTTOM ALT="[shift defn]" SRC="img140.gif"> </CENTER> <P> Hence, for example, the graph of the function <IMG WIDTH=111 HEIGHT=23 ALIGN=MIDDLE ALT="y = cos(x + 3pi)" SRC="img141.gif"> is essentially the graph of <IMG WIDTH=62 HEIGHT=14 ALIGN=MIDDLE ALT="y = cos(x)" SRC="img134.gif"> shifted to the left by a factor of <IMG WIDTH=15 HEIGHT=11 ALIGN=BOTTOM ALT="3 pi" SRC="img142.gif"> since <IMG WIDTH=45 HEIGHT=22 ALIGN=MIDDLE ALT="x + 3pi" SRC="img143.gif"> is the same as <IMG WIDTH=75 HEIGHT=23 ALIGN=MIDDLE ALT="x - (-3pi)" SRC="img144.gif"> So the graph of <IMG WIDTH=111 HEIGHT=23 ALIGN=MIDDLE ALT="y = cos(x + 3pi)" SRC="img141.gif"> looks like the following: <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot6.gif" WIDTH=320 HEIGHT=235 ALT="[plot]"> </CENTER> <P> <HR><center> <table border=1> <td><A HREF="node48.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node44.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node50.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node49.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I><A HREF="http://www.math.psu.edu/sellersj/">James A. Sellers</A> </I> </ADDRESS> </BODY>