document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161075>Question 1161075</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the quadratic polynomial whose graph passes through the points (1,14) (2,28) (3,48)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #784524 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=784524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The standard method for solving a problem like this is as shown by the other two tutors: use the x and y coordinates of the given points in the standard equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get three equations in the coefficients a, b, and c; then solve the system of equations.<br><BR>\n" );
document.write( "Here is a completely different method that I find a bit less work.  This alternative method can be used because the y values are given for three consecutive x values.<br><BR>\n" );
document.write( "So think of the problem as a sequence of numbers produced by a quadratic polynomial.<br><BR>\n" );
document.write( "In a sequence produced by the quadratic polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the second differences are the constant 2a.  (If you know some basic calculus, you can see that is true, since the second derivative of the polynomial is 2a.)<br><BR>\n" );
document.write( "In this sequence, the second difference is 6:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   14  18  48\r\n" );
document.write( "     14  20\r\n" );
document.write( "        6</PRE><BR>\n" );
document.write( "Since we now that second difference is 2a, we know that the leading coefficient a is 3, and the quadratic polynomial is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"3x%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The \"bx+c\" is a linear polynomial; it can be determined easily by comparing the actual y values to the value of 3x^2:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   x   3x^2   y   y-3x^2 (= bx+c)\r\n" );
document.write( "  -------------------------------\r\n" );
document.write( "   1     3   14     11\r\n" );
document.write( "   2    12   28     16\r\n" );
document.write( "   3    27   48     21</PRE><BR>\n" );
document.write( "It is easy to see that the values in the last column are bx+c = 5x+6.<br><BR>\n" );
document.write( "So, already knowing that the leading term of the quadratic is 3x^2, we now know that the complete quadratic is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B5x%2B6 ALIGN=MIDDLE  ALT=\"3x%5E2%2B5x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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