document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=244678>Question 244678</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Were learning how to find/create functions from patterns. I don't get how you<BR>\n" );
document.write( "analyze your data and figure out the function for the pattern. Please explain what the x pattern and y pattern effects the function. I just practically need the steps on how to find the function. Please reply as soon as possible! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178916 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=178916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You realize, of course, that you have asked a very general question and that means that you are going to get a very general answer.\r<BR>\n" );
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document.write( "First of all, in general there is no set of \"steps to find the function.\"  In general, it takes some creativity to recognize how the pattern evolves and how to describe it mathematically.\r<BR>\n" );
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document.write( "A function is nothing more than a little machine.  On one end is a slot and on the other is a hopper.  You put numbers (the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> values) in the slot, and the function machine does its magic, and other numbers (the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> values) come out in the hopper.  The only restriction that makes it a function machine rather than just an ordinary relationship machine is that you can only get one <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> value out of the hopper for any given <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> value submitted in the slot.  It is ok if two different <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> values give you the same <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> value out -- just not the other way around.\r<BR>\n" );
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document.write( "The process of pattern recognition and defining functions by looking at the patterns requires that you put a set of values into the machine (the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> column or row of your table of values) and see what comes out the other end of the machine (the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> column or row of your table of values).\r<BR>\n" );
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document.write( "Then you have to try to determine what process happens each time inside your machine to create the result you see each and every time.  If you have 10 pairs of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> values in your table, and the rule you develop works for nine of them but not the tenth one, then your rule is invalid.  Back to the drawing board.\r<BR>\n" );
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document.write( "Sometimes it helps to plot the points on a graph.  Each pair of values makes an ordered pair (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">,<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">).  Once you have plotted the points, draw a smooth curve through them.\r<BR>\n" );
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document.write( "If you have a straight line, then your function is going to have the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ mx\ + b\">.  If you have something that looks like a parabola, then you might want to try a quadratic (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\">) relationship.  There are way to many to describe here, but I think you get the idea.\r<BR>\n" );
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document.write( "Example\r<BR>\n" );
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document.write( "First the set of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> values:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \{0, 1, 2, 3, 4\}\">\r<BR>\n" );
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document.write( "Then the set of corresponding <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> values:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \{4, 5, 8, 13, 20\}\">\r<BR>\n" );
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document.write( "Hmmm, every one of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> values is 4 greater than a perfect square, and coincidentally, it is 4 greater than the square of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> value, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ f(x)\ =\ x^2\ + 4\">\r<BR>\n" );
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document.write( "Let's check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0^2\ +\ 4\ =\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1^2\ +\ 4\ =\ 5\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2^2\ +\ 4\ =\ 4\ +\ 4\ = 8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3^2\ +\ 4\ =\ 9\ +\ 4\ = 13\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4^2\ +\ 4\ =\ 16\ +\ 4\ = 20\">\r<BR>\n" );
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document.write( "To all of the above add a little imagination, creativity, intuition, and mostly hard work at trial and error.  Can't tell you much more than that.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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