document.write( "Question 189964: I am in need of assistance in creating an equation that describes the chart as follows \r
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\n" ); document.write( "I realize the progression of .5,1.5,2.5,4,4.5. I can't figure out the equatio\r
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Algebra.Com's Answer #142582 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Edwin's solution:
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document.write( "First list the numbers like this skipping \r\n" );
document.write( "a line between them:\r\n" );
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document.write( "X    Y\r\n" );
document.write( "0   0.0\r\n" );
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document.write( "1   0.5\r\n" );
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document.write( "2   2.0\r\n" );
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document.write( "3   4.5\r\n" );
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document.write( "4   8.0\r\n" );
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document.write( "5  12.5\r\n" );
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document.write( "Next add a column labeled \"1st differences\" and\r\n" );
document.write( "on each line that was skipped write the difference\r\n" );
document.write( "between the two y-values the skipped line is\r\n" );
document.write( "in between.  \r\n" );
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document.write( "That is, we subtract the 1st value of\r\n" );
document.write( "x which is 0.0 from the 2nd value of x which is 0.5\r\n" );
document.write( "and write that in the line between the first two \r\n" );
document.write( "values.\r\n" );
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document.write( "we subtract the 2nd value of\r\n" );
document.write( "x which is 0.5 from the 3rd value of x which is 2.0,\r\n" );
document.write( "and that gives us 1.5 which we write in the line \r\n" );
document.write( "between the 2nd and 3rd values.  And so forth:\r\n" );
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document.write( "          1st \r\n" );
document.write( "X    Y   diff.   \r\n" );
document.write( "0   0.0\r\n" );
document.write( "          0.5\r\n" );
document.write( "1   0.5\r\n" );
document.write( "          1.5\r\n" );
document.write( "2   2.0\r\n" );
document.write( "          2.5\r\n" );
document.write( "3   4.5\r\n" );
document.write( "          3.5\r\n" );
document.write( "4   8.0\r\n" );
document.write( "          4.5\r\n" );
document.write( "5  12.5\r\n" );
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document.write( "Next we add another column and label it \"2nd\r\n" );
document.write( "differences\", subtracting each value in the\r\n" );
document.write( "1st difference column from the one just before\r\n" );
document.write( "it and writing it in that column between the \r\n" );
document.write( "two values subtracted, like this:\r\n" );
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document.write( "            1st     2nd\r\n" );
document.write( "X    Y     diffs.  diffs.  \r\n" );
document.write( "0   0.0\r\n" );
document.write( "            0.5\r\n" );
document.write( "1   0.5             1.0\r\n" );
document.write( "            1.5\r\n" );
document.write( "2   2.0             1.0 \r\n" );
document.write( "            2.5\r\n" );
document.write( "3   4.5             1.0 \r\n" );
document.write( "            3.5\r\n" );
document.write( "4   8.0             1.0\r\n" );
document.write( "            4.5\r\n" );
document.write( "5  12.5\r\n" );
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document.write( "Now we do not need to make a column for the\r\n" );
document.write( "3rd differences, because all the 2nd differences\r\n" );
document.write( "are the same.\r\n" );
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document.write( "Since the second differences came out all the\r\n" );
document.write( "same number, we know to assume the equation is a\r\n" );
document.write( "second degree polynomial,\r\n" );
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document.write( "    y = Ax˛ + Bx + C\r\n" );
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document.write( "That has 3 values A,B, and C which we must find,\r\n" );
document.write( "so we substitute 3 of the six given points.\r\n" );
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document.write( "First we substitute x = 0, y = 0, and get\r\n" );
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document.write( "    0 = A(0)˛ + B(0) + C\r\n" );
document.write( "    0 = 0 + 0 + C\r\n" );
document.write( "    0 = C\r\n" );
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document.write( "Now we substitute x = 1, y = 0.5, and get\r\n" );
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document.write( "  0.5 = A(1)˛ + B(1) + C\r\n" );
document.write( "  0.5 = A + B + C\r\n" );
document.write( "  \r\n" );
document.write( "Now we substitute x = 2, y = 2.0, and get\r\n" );
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document.write( "  2.0 = A(2)˛ + B(2) + C\r\n" );
document.write( "  2.0 = 4A + 2B + C\r\n" );
document.write( "      \r\n" );
document.write( "So we have three equations in three unknowns:\r\n" );
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document.write( "\"system%280=C%2C0.5=A%2BB%2BC%2C2.0=4A%2B2B%2BC%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"system%28C=0%2CA%2BB%2BC=0.5%2C4A%2B2B%2BC=2.0%29\"\r\n" );
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document.write( "Do you know how to solve that for A, B and C?\r\n" );
document.write( "If not post again asking how.  The answers here\r\n" );
document.write( "are \r\n" );
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document.write( "\"A=0.5\", \"B=0\", \"C=0\"\r\n" );
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document.write( "So we substitute those values in\r\n" );
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document.write( "\"y+=+Ax%5E2+%2B+Bx+%2B+C\"\r\n" );
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document.write( "and get\r\n" );
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document.write( "\"y+=+0.5x%5E2+%2B+0x+%2B+0\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"y=+0.5x%5E2\"\r\n" );
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document.write( "That's the equation you were looking for. \r\n" );
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document.write( "Now check by substituting x=0,1,2,3,4,5\r\n" );
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document.write( "and you will get the corresponding y-values. \r\n" );
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document.write( "Edwin
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