Amplitude

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node46.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="node48.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node47.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Amplitude</H2> <P> We first note the following modification to the trigonometric functions we have seen at this point. Consider the function <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="y = 3 sin(x)" SRC="img131.gif">. How is the graph of <IMG WIDTH=60 HEIGHT=24 ALIGN=MIDDLE ALT="y = sin(x)" SRC="img132.gif"> related to the graph of <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="y = 3 sin(x)" SRC="img131.gif">? If we draw the two functions on the same set of axes, we obtain the following: <P> <CENTER> <IMG ALIGN=BOTTOM SRC="plot3.gif" WIDTH=320 HEIGHT=237 ALT="[plot]"> </CENTER> <P> Note that this new function still has the same intercepts, but its range is now [-3,3] rather than [-1,1]. In essence, multiplication of the function <IMG WIDTH=34 HEIGHT=12 ALIGN=BOTTOM ALT="sin x" SRC="img128.gif"> by 3 has ``stretched'' the graph of <IMG WIDTH=34 HEIGHT=12 ALIGN=BOTTOM ALT="sin x" SRC="img128.gif"> vertically by a factor of 3, while retaining the same <I>x</I>-intercepts. <P> This vertical scaling factor is known as the <B>amplitude</B> of the function. Thus, in general, we say the following: <P> <CENTER> <IMG WIDTH=492 HEIGHT=67 ALIGN=BOTTOM ALT="[amplitude defn]" SRC="img133.gif"> </CENTER> <P> We now see how we can stretch a <IMG WIDTH=17 HEIGHT=12 ALIGN=BOTTOM ALT="sin" SRC="img20.gif"> or <IMG WIDTH=19 HEIGHT=7 ALIGN=BOTTOM ALT="cos" SRC="img21.gif"> function vertically. How would we shift it vertically? This is the subject of our next section. <P> <HR><center> <table border=1> <td><A HREF="node46.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="node48.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node47.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>