Graphing a Complicated Sine Function

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node51.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node51.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node53.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node52.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H3>Graphing a Complicated Sine Function</H3> <P> Consider the function <IMG WIDTH=154 HEIGHT=23 ALIGN=MIDDLE ALT="y = 3 + 2 sin(4x - pi)" SRC="img150.gif"> Let's analyze this function one step at a time, beginning with the most basic function <IMG WIDTH=34 HEIGHT=12 ALIGN=BOTTOM ALT="sin(x)" SRC="img128.gif"> and working our way out to the function in question. <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot8.gif" WIDTH=320 HEIGHT=235 ALT="[plot]"> </CENTER> <P> First, note the presence of <I>D</I>=4. This tells us that our period is <IMG WIDTH=32 HEIGHT=24 ALIGN=MIDDLE ALT="2 pi/4" SRC="img151.gif"> or <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="pi/2" SRC="img57.gif">. Hence, we have taken the function <IMG WIDTH=34 HEIGHT=12 ALIGN=BOTTOM ALT="sin(x)" SRC="img128.gif"> and horizontally ``shrunk'' it so that its period is <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="pi/2" SRC="img57.gif">. We now have a graph of the form <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot9.gif" WIDTH=321 HEIGHT=235 ALT="[plot]"> </CENTER> <P> Next, we see that <IMG WIDTH=42 HEIGHT=11 ALIGN=BOTTOM ALT="C=pi" SRC="img152.gif">. This will horizontally shift our graph <IMG WIDTH=8 HEIGHT=7 ALIGN=BOTTOM ALT="pi" SRC="img58.gif"> units to the right. Thus, we now have <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot10.gif" WIDTH=322 HEIGHT=233 ALT="[plot]"> </CENTER> <P> Thirdly, we note that <I>A</I>=2, which will vertically scale our graph by a factor of 2. Thus, at this point, our range should now be [-2,2]: <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot11.gif" WIDTH=323 HEIGHT=235 ALT="[plot]"> </CENTER> <P> Finally, we must shift our graph vertically 3 units due to the presence of <I>B</I>=3. Hence, our range will now move to [1,5]: <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot12.gif" WIDTH=320 HEIGHT=236 ALT="[plot]"> </CENTER> <P> This is the graph of the function <IMG WIDTH=149 HEIGHT=23 ALIGN=MIDDLE ALT="y = 3 + 2 sin(4x - pi)" SRC="img153.gif">. Note that the graph of this function would be very difficult to draw if we did not take such a step-by-step approach to this problem. <P> <HR><center> <table border=1> <td><A HREF="node51.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node51.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node53.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node52.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>