document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=320079>Question 320079</A>: <I><SPAN STYLE=\"word-break: break-all;\"> could you please show me how i would go about sketching the graph of the function f given by f(x) = -1/2ln(x+1)<BR>\n" );
document.write( "what steps need to taken? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #229217 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=229217\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> function is y = f(x) = -(1/2)*ln(x+1)\r<BR>\n" );
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document.write( "Graph of this function looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C-%281%2F2%29%2Aln%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C-%281%2F2%29%2Aln%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You find that by just plotting the function using graphing software or a graphing calculator.\r<BR>\n" );
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document.write( "If you don't have either, then you would need to draw the graph manually.\r<BR>\n" );
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document.write( "You do this the same way you would draw any other graph.\r<BR>\n" );
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document.write( "You plot values of y for values of x and then draw a curve in between.\r<BR>\n" );
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document.write( "You need to be able to find ln(x+1) so you would need a scientific calculator for that, or you would need to be able to consult natural log tables.\r<BR>\n" );
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document.write( "LN (x+1) means take the natural log of (x+1).\r<BR>\n" );
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document.write( "Natural log of a number is the log of the number to the base of e.\r<BR>\n" );
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document.write( "e is the scientific constant of 2.718281828.....\r<BR>\n" );
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document.write( "At any rate, I'll assume you have a scientific calculator, if not a graphing calculator.\r<BR>\n" );
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document.write( "You will be plotting y = -(1/2) * ln(x+1) for values of x.\r<BR>\n" );
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document.write( "Start with x = -5 to 5 in increments of 1.\r<BR>\n" );
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document.write( "Since x+1 is 1 more than x, when you plot from x = -5 to x = 5, you will be plotting for the value of -(1/2)*ln(-4) to y = -1/2*ln(6).\r<BR>\n" );
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document.write( "Use your calculator to find ln(-4) to ln(6) and then take 1/2 of that to get your values\r<BR>\n" );
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document.write( "To be able to take a log of number, the number have to be greater than 0, otherwise the log will be undefined.\r<BR>\n" );
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document.write( "This works for logs to any base, including the base of e.\r<BR>\n" );
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document.write( "So your value of y will be undefined up to x = -1.\r<BR>\n" );
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document.write( "This means your graph will start plotting for numbers greater than -1.\r<BR>\n" );
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document.write( "Numbers will be rounded to 2 decimal places.\r<BR>\n" );
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document.write( "When x = -1 or lower, y is undefined.<BR>\n" );
document.write( "When x = 0, y = -(1/2)*ln(1) = 0<BR>\n" );
document.write( "When x = 1, y = -(1/2)*ln(2) = -.35 <BR>\n" );
document.write( "When x = 2, y = (-1/2)*ln(3) = -.55<BR>\n" );
document.write( "When x = 3, y = (-1/2)*ln(4) = -.69<BR>\n" );
document.write( "When x = 4, y = (-1/2)*ln(5) = -.80<BR>\n" );
document.write( "When x = 5, y = (-1/2)*ln(6) = -.90\r<BR>\n" );
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document.write( "The graph looks like it stops at y = 2 when x = -1, but that isn't the case.\r<BR>\n" );
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document.write( "The closer that x gets to -1, the higher the value of y.\r<BR>\n" );
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document.write( "This suggests that the limit will approach infinity as the value of x gets closer and closer to x = -1.\r<BR>\n" );
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document.write( "You won't be able to calculate this, but I can show you that the value of y will definitely be greater than 2.\r<BR>\n" );
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document.write( "Understand that the closer that x gets to -1, the closer (x+1) gets to 0.\r<BR>\n" );
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document.write( "You are taking the natural log of (x+1), not x.\r<BR>\n" );
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document.write( "Any value of x greater than -1 will get you a value of (x+1) greater than 0.\r<BR>\n" );
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document.write( "The closest I can get to 0 appears to be .000000001 * 10^(-50).\r<BR>\n" );
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document.write( "That's a very small number.\r<BR>\n" );
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document.write( "The natural log of that becomes ln (.000000001 * 10^(-50)) = 68.\r<BR>\n" );
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document.write( "While not infinity, this is considerable larger than the value of 2 that was shown on the graph.\r<BR>\n" );
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document.write( "As x approaches -1, the value of ln (x+1) gets higher and higher.\r<BR>\n" );
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document.write( "The closer x gets to -1, the closer ln(x+1) gets to ln(0).\r<BR>\n" );
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document.write( "ln(0) is not valid, but anything close to that on the positive side is.\r<BR>\n" );
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document.write( "The problem is that you can't calculate close enough to see that the value of y would get very very large, but the theoretical limit appears to be infinity since there is nothing to prevent y from getting larger and larger the closer that (x+1) bets to 0.\r<BR>\n" );
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document.write( "As a practical matter, you would probably plot no closer than .01 to x = -1.\r<BR>\n" );
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document.write( "That makes x = -.99 which makes (x+1) = .01.\r<BR>\n" );
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document.write( "When x = -.99, y equals -(1/2)*ln(.01) = 2.302585093.\r<BR>\n" );
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document.write( "This appears to be close to the resolution of the graph shown since the value the graph showed was y = 2.\r<BR>\n" );
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document.write( "The graphing software simply doesn't have the a finer resolution to show you the values any closer than that.\r<BR>\n" );
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document.write( "If you look at the graph I showed you and the values that we plotted, you'll see that they are consistent with each other.\r<BR>\n" );
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document.write( "Without knowing what the graph looked like in advance, your curve plotted manually would have been close to that.<BR>\n" );
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