document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=28313>Question 28313</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The general equation os a quadratic polynomial is p(x)=ax(squared)+bx+c with a,b,c all real numbers and a doesn't = 0. In this problem we show that three points in the plane uniquely determine a quad polynomial, )just as two points in aplane uniquely determine a line).\r<BR>\n" );
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document.write( "Step 1. Suppose that we were given the three point (-1,1),(1,5), and (2,10). We want to find the quad poly P(x) which passes through these points. So, evaluate p at each x value. For example, since p(-1)must be 1 (why?), we get\r<BR>\n" );
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document.write( "p(-1)=a(-1) squared+b(-1)+c=1 (arrow) a-b+c=1.\r<BR>\n" );
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document.write( "Evaluating p(1) and p(2), we get two more equations. What are they?\r<BR>\n" );
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document.write( "p(1)=<BR>\n" );
document.write( "p(2)=\r<BR>\n" );
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document.write( "Step 2: We now have _________________ linear equations in the _______________ unknowns___,__________, and _______. Thus, we can try to solve for _,_____, and ____. Once we have values for ___,______, and __, our polynomials will be determined.\r<BR>\n" );
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document.write( "Step 3: set up the augmented coefficient matrix. We can start with\r<BR>\n" );
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document.write( "M= [1  -1  1  1]<BR>\n" );
document.write( "   [?   ?  ?  ?]<BR>\n" );
document.write( "   [?   ?  ?  10].\r<BR>\n" );
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document.write( "There are 6 values missing. Where do they come from? What are they? What variable does the first column represent? The second? The third?\r<BR>\n" );
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document.write( "Step 4:<BR>\n" );
document.write( "Using elementary row operations, we transform M into the matrix M' that looks like\r<BR>\n" );
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document.write( "M'= [1  0  0  1]<BR>\n" );
document.write( "    [0  1  0  2]<BR>\n" );
document.write( "    [0  0  1  2].\r<BR>\n" );
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document.write( "I need to be able to explicity show the computations -- i.e the row of operations - leading from M tp M'.\r<BR>\n" );
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document.write( "From this it follows that a=____, b____, and c=___. Thus our polynomial p(x) is p(x)=?\r<BR>\n" );
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document.write( "Step 5:<BR>\n" );
document.write( "Check the answer by evaluating p(x) with the a, b and c values determined in step 4. We should have p(-1)=1, p(1)=5, and p(2)=10. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15388 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><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=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=15388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you have 3 points that are on a quadratic curve, (-1,1),(1,5), and (2,10).\r<BR>\n" );
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document.write( "If you put x=-1 into the quadratic, the answer for y is 1...this is the coordinate (-1,1). Similarly when x=1, then y=5 and also when x=2, then y=10.\r<BR>\n" );
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document.write( "We do not know the quadratic, so we are going build it up from first principles...\r<BR>\n" );
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document.write( "the quadratic is of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y=ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so using (-1,1) gives a-b+c=1<BR>\n" );
document.write( "and using (1,5) gives a+b+c=5<BR>\n" );
document.write( "and using (2,10) gives 4a+2b+c=10\r<BR>\n" );
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document.write( "these are your three \"p\"'s.\r<BR>\n" );
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document.write( "--> got 3 polynomials in 3 unknowns --> a,b,c.\r<BR>\n" );
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document.write( "matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C%0D%0A1%2C-1%2C1%2C1%2C%0D%0A1%2C1%2C1%2C5%2C%0D%0A4%2C2%2C1%2C10%29+%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C%0D%0A1%2C-1%2C1%2C1%2C%0D%0A1%2C1%2C1%2C5%2C%0D%0A4%2C2%2C1%2C10%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the columns are the coefficients of a,b and c respectively and the y-value.<BR>\n" );
document.write( "the rows are the 3 equations.\r<BR>\n" );
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document.write( "OK, now you use Gaussian Elimination (GE) to get the matrix M'...look into your books for this. I hate GE - i spent years doing it at uni.\r<BR>\n" );
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document.write( "I have done GE for you... you struggle with it. It is the only way!<BR>\n" );
document.write( "c=2<BR>\n" );
document.write( "b=2<BR>\n" );
document.write( "a=1\r<BR>\n" );
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document.write( "and i have checked the quadratic with the 3 sets of coordinates...they match.\r<BR>\n" );
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document.write( "jon.<BR>\n" );
document.write( " </SPAN>\n" );
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