Horizontal ``Stretching''

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node49.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="node51.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node50.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Horizontal ``Stretching''</H2> <P> We now come to the final type of graphical adjustment that can be made to the basic sine and cosine functions. It is known as horizontal scaling or ``stretching''. At this point, we have seen addition to the function on the outside, as well as addition within the argument of the function. Both of these operations cause a shift or translation, either vertically or horizontally. We have also seen multiplication on the outside of the function and noted that this brings about vertical scaling. We will now see the related multiplication and note its affect. <P> If we multiply the argument of the function by a constant, we find that horizontal stretching takes place. Note from our previous discussion that this will affect the <B>period</B> of the function in question. Let's turn to an example to see this affect. <P> Consider the graphs of the functions <IMG WIDTH=62 HEIGHT=14 ALIGN=MIDDLE ALT="y = cos(x)" SRC="img134.gif"> and <IMG WIDTH=80 HEIGHT=23 ALIGN=MIDDLE ALT="y = cos(2x)" SRC="img145.gif">. (Remember that <IMG WIDTH=48 HEIGHT=23 ALIGN=ABSMIDDLE ALT="cos(2x)" SRC="img146.gif"> naturally appeared in the double angle formulas above.) How do the graphs of these two differ? Let's look at the graphs and determine the answer to this question. <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot7.gif" WIDTH=322 HEIGHT=236 ALT="[plot]"> </CENTER> <P> We see that one cycle of <IMG WIDTH=80 HEIGHT=23 ALIGN=MIDDLE ALT="y = cos(2x)" SRC="img145.gif"> is achieved after a distance of <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="pi" SRC="img58.gif"> on the <I>x</I>-axis, whereas one cycle of <IMG WIDTH=62 HEIGHT=14 ALIGN=MIDDLE ALT="y = cos(x)" SRC="img134.gif"> requires a distance of <IMG WIDTH=15 HEIGHT=11 ALIGN=BOTTOM ALT="2 pi" SRC="img122.gif">. Hence, we see that the period of <IMG WIDTH=80 HEIGHT=23 ALIGN=MIDDLE ALT="y = cos(2x)" SRC="img145.gif"> is <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="pi" SRC="img58.gif"> rather than <IMG WIDTH=15 HEIGHT=11 ALIGN=BOTTOM ALT="2 pi" SRC="img122.gif">. You may not see the pattern yet, but we will note it here officially. <P> <CENTER> <IMG WIDTH=492 HEIGHT=68 ALIGN=BOTTOM ALT="[period defn]" SRC="img147.gif"> </CENTER> <HR><center> <table border=1> <td><A HREF="node49.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="node51.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node50.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>