document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460108>Question 460108</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I am new to this site, can someone help me. I have not had math in years and I am terrible at it.<BR>\n" );
document.write( "My question is: Find the quadratic equations that fits the following set of data points. ((-1,5),(-4,59),(-3,37))\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #315574 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=315574\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since we're forcing a parabola to go through these three points, this means that when we plug in each 'x' into the general formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we'll get 'y'. So essentially, we're plugging in corresponding values of 'x' and 'y' and forming equations, which we'll use to solve for 'a', 'b', and 'c'.\r<BR>\n" );
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document.write( "So for instance, if we plug in x=-1, then we'll get the output of y=-5. This means that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=a%28-1%29%5E2%2Bb%28-1%29%2Bc ALIGN=MIDDLE  ALT=\"5=a%28-1%29%5E2%2Bb%28-1%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-b%2Bc=5 ALIGN=MIDDLE  ALT=\"a-b%2Bc=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Do the same with the second point to get the second equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16a-4b%2Bc=59 ALIGN=MIDDLE  ALT=\"16a-4b%2Bc=59\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Finally, do the same with the third point to get the third equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a-3b%2Bc=37 ALIGN=MIDDLE  ALT=\"9a-3b%2Bc=37\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "After doing all that, we have the system\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a-b%2Bc=5%2C16a-4b%2Bc=59%2C9a-3b%2Bc=37%29 ALIGN=MIDDLE  ALT=\"system%28a-b%2Bc=5%2C16a-4b%2Bc=59%2C9a-3b%2Bc=37%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which translates into this augmented matrix\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C-1%2C1%2C5%2C16%2C-4%2C1%2C59%2C9%2C-3%2C1%2C37%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C1%2C-1%2C1%2C5%2C16%2C-4%2C1%2C59%2C9%2C-3%2C1%2C37%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Row reduce this matrix (see <a  rel=nofollow HREF=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/rref6-11-20112.png\">this image</a> for full steps on how to do that) to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C0%2C0%2C2%2C0%2C1%2C0%2C-8%2C0%2C0%2C1%2C-5%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C1%2C0%2C0%2C2%2C0%2C1%2C0%2C-8%2C0%2C0%2C1%2C-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: alternatively you can use a calculator to do the row reduction.\r<BR>\n" );
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document.write( "The last column represents the values of 'a', 'b', and 'c' respectively.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-8 ALIGN=MIDDLE  ALT=\"b=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-5 ALIGN=MIDDLE  ALT=\"c=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the quadratic equation that goes through the three given points is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%5E2-8x-5 ALIGN=MIDDLE  ALT=\"y=2x%5E2-8x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "As a check, if we plug in x = -1, then....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%28-1%29%5E2-8%28-1%29-5 ALIGN=MIDDLE  ALT=\"y=2%28-1%29%5E2-8%28-1%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%281%29-8%28-1%29-5 ALIGN=MIDDLE  ALT=\"y=2%281%29-8%28-1%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2B8-5 ALIGN=MIDDLE  ALT=\"y=2%2B8-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So plugging in x = -1 gives us y = 5, meaning that the point (-1,5) lies on the parabola. This partially verifies our answer.\r<BR>\n" );
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document.write( "Do the same with the other two points to verify this equation actually works.\r<BR>\n" );
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document.write( "Let me know if this helps. </SPAN>\n" );
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