document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208130>Question 1208130</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let P = (x,y) be a point on the graph of y = 1/x.\r<BR>\n" );
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document.write( "A. Express the distance d from P to the origin as a function of x. \r<BR>\n" );
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document.write( "B. Use a graphing utility to graph d = d(x). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846322 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846322\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part A\r<BR>\n" );
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document.write( "(x1,y1) = origin = (0,0)<BR>\n" );
document.write( "(x2,y2) = point P = (x,y) = (x,1/x) since y = 1/x\r<BR>\n" );
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document.write( "Distance formula from (x1,y1) to (x2,y2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x2-x1%29%5E2+%2B+%28y2-y1%29%5E2+%29+ ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x2-x1%29%5E2+%2B+%28y2-y1%29%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x-0%29%5E2+%2B+%28y-0%29%5E2+%29+ ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x-0%29%5E2+%2B+%28y-0%29%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+x%5E2+%2B+y%5E2+%29+ ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+x%5E2+%2B+y%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28x%29+=+sqrt%28+x%5E2+%2B+%281%2Fx%29%5E2+%29+ ALIGN=MIDDLE  ALT=\"d%28x%29+=+sqrt%28+x%5E2+%2B+%281%2Fx%29%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Optionally you can simplify this, but I'll let the student take over from here.<BR>\n" );
document.write( "Note that the distance formula is based on the Pythagorean Theorem.\r<BR>\n" );
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document.write( "Part B\r<BR>\n" );
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document.write( "This is what the graph of d(x) looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2Csqrt%28+x%5E2+%2B+%281%2Fx%29%5E2+%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2Csqrt%28+x%5E2+%2B+%281%2Fx%29%5E2+%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are many graphing tools to pick from. <BR>\n" );
document.write( "My favorite is GeoGebra, but Desmos is really good as well. <BR>\n" );
document.write( "If you are familiar with something like a TI83, then it might be best to stick to that.\r<BR>\n" );
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document.write( "Things to pay attention to:<ul><li>x = 0 isn't allowed since it causes a division by zero error. This explains the vertical asymptote at x = 0.</li><li>The curve is symmetric about the y axis. The function d(x) is even. We can prove this algebraically by showing that d(-x) = d(x) for all x in the domain. I'll leave the proof for the reader.</li><li>The smallest distance possible is d = sqrt(2) = 1.41421356 approximately, which occurs when either x = 1 or x = -1. Use calculus to find this, or use the graphing calculator's minimum locator feature.</li></ul><BR>\n" );
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