document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=849709>Question 849709</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I do not understand how to even begin on anything like this. I asked my professor and three other students about it but they didn't help me to understand it that much at all. <BR>\n" );
document.write( "Graph the function by substituting and plotting points.<BR>\n" );
document.write( "y=(1/4)e^x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #511704 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=511704\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Make an x_y table.\r<BR>\n" );
document.write( "\n" );
document.write( "x  |  y\r<BR>\n" );
document.write( "\n" );
document.write( "-2   (1/4)e^-2<BR>\n" );
document.write( "-1   (1/4)e^-1<BR>\n" );
document.write( "0    (1/4)<BR>\n" );
document.write( "1    (1/4)e<BR>\n" );
document.write( "2    (1/4)e^2\r<BR>\n" );
document.write( "\n" );
document.write( "If you are not allowed to use a calculator, let's approximate what some of these values are.\r<BR>\n" );
document.write( "\n" );
document.write( "The easy one is we know (0,(1/4)). That's pretty simple to plot, we can even call it (0,.25).\r<BR>\n" );
document.write( "\n" );
document.write( "What about (1/4)*e?  We know that e is 2.71828, so if we were to take a 4th of it we'd get roughly .67. \r<BR>\n" );
document.write( "\n" );
document.write( "What about (1/4)*e^2 We can imagine that the square of e is going to be slightly under 9 since 2.71<3. So, I'll guesstimate and say it's 8.8. It really doesn't matter. What's important is you are close.\r<BR>\n" );
document.write( "\n" );
document.write( "You can see that as we increase x, y is increasing at a much larger rate. It's going to look similar to our exponential function.\r<BR>\n" );
document.write( "\n" );
document.write( "What about the other way?\r<BR>\n" );
document.write( "\n" );
document.write( "(1/4)*e^-1.  Well e^-1 = 1/e  so if we treat e as 3, just to get close, we see that we'd be pretty close to 1/12 = .0833, when we actually plug in (1/4)*e^-1 we get .0919. So off a bit, but we can at least start to see the shape.\r<BR>\n" );
document.write( "\n" );
document.write( "(1/4)*e^-2 = 0.034. Notice we're getting closer and closer to 0 as we approach smaller and smaller numbers. Will we ever reach 0 though? No.\r<BR>\n" );
document.write( "\n" );
document.write( "When we start to put this together our graph looks like this:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src =\"http://www4b.wolframalpha.com/Calculate/MSP/MSP58431bibde6204926c1i0000171i04cbhg263b6b?MSPStoreType=image/gif&s=8&w=300.&h=406.&cdf=Resizeable\" />\r<BR>\n" );
document.write( "\n" );
document.write( "Adding a little fill to this gives us a graph like this.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src =\"http://www4b.wolframalpha.com/Calculate/MSP/MSP300521haf97d3fcaha49000048c61ie30ide1ffe?MSPStoreType=image/gif&s=30&w=300.&h=196.&cdf=RangeControl\"/>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );