document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=909355>Question 909355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi. First of all I'm not so sure if this problem goes here but it is for functions and their graphs. I have been looking for help on this problem for hours but it just doesn't make sense anymore and this is my last resort. \"Let P= (x,y) be a point on the graph y= 1/x.<BR>\n" );
document.write( "a. express the distance d from p to the origin as a function of x<BR>\n" );
document.write( "b. use a graphing utility to graph d=d(x)<BR>\n" );
document.write( "c/ for what values of x is d smallest?\"<BR>\n" );
document.write( "Do I have to use any formulas or anything, I am so sorry for asking.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #551844 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=551844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your equation is y = 1/x\r<BR>\n" );
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document.write( "this is a graph of your equation.\r<BR>\n" );
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document.write( "p is a point on the graph of y = 1/x.\r<BR>\n" );
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document.write( "d = the distance from p to the origin.\r<BR>\n" );
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document.write( "on the graph, p is represented by y which is the result of the equation of y = 1/x for any value of x.\r<BR>\n" );
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document.write( "since the equation is y = 1/x, then you want to find the formula for the distance from any point on that graph  to the origin.\r<BR>\n" );
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document.write( "the distance from any point on the graph to the origin is given by the generic equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28x%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28x%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "when y = 1/x, this equation becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28x%5E2+%2B+%281%2Fx%29%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28x%5E2+%2B+%281%2Fx%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\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%5E4%2B1%29%2Fx%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28%28x%5E4%2B1%29%2Fx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28x%5E4%2B1%29%2Fx ALIGN=MIDDLE  ALT=\"d+=+sqrt%28x%5E4%2B1%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "you want to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28x%5E4%2B1%29%2Fx%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28x%5E4%2B1%29%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "to graph it, replace d with y to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28x%5E4%2B1%29%2Fx%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28x%5E4%2B1%29%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the graph of that equation is shown below.\r<BR>\n" );
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document.write( "the particular software i am using marks the max/min points on each curve of the graph.\r<BR>\n" );
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document.write( "it is showing me that the minimum point if at x = 1 and the maximum point is at x = -1.\r<BR>\n" );
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document.write( "it is also showing me that the distance to the origin will be 1.41414 at that point, which is the samee as sqrt(2).\r<BR>\n" );
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document.write( "the minimum distances from any point on the graph of y = 1/x to the origin will occur when x = -1 and when x = 1.\r<BR>\n" );
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document.write( "when x = -1, y = -1, and when x = 1, y = 1.\r<BR>\n" );
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document.write( "the distance to the origin in both cases will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28x%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28x%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%281%5E2+%2B+1%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%281%5E2+%2B+1%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%282%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28%28-1%29%5E2+%2B+%28-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28%28-1%29%5E2+%2B+%28-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which still becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%282%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "the solution to your problem is that d is the smallest when x = -1 and when x = 1.\r<BR>\n" );
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document.write( "the particular calculator that i used is at:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.desmos.com/calculator\" target=\"_blank\">http://www.desmos.com/calculator</a>\r<BR>\n" );
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