document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=975336>Question 975336</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are interested in the maximum height the skier will reach after becoming airborne at (7,4). A series of photos of a jumper performing his jump shows the skier at the points (7,4) (9,7) (14,5.75) and (15,5). Find the equation of this parabolic path in height vs time.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #597252 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=597252\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Two points determine a straight line.<BR>\n" );
document.write( "Three points determine a parabolic path.<BR>\n" );
document.write( "We have four points.<BR>\n" );
document.write( "We could use some statistics software to determine the parabola that most closely fits the data.<BR>\n" );
document.write( "We can also use three of those points to determine a quadratic function and see if the fourth point would fit.<BR>\n" );
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document.write( "USING THE FIRST THREE POINTS:<BR>\n" );
document.write( "The parabolic path will be determine by a  quadratic function of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For (7,4) , substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=7%2Cy=4%29 ALIGN=MIDDLE  ALT=\"system%28x=7%2Cy=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , gives us the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4=49a%2B7b%2Bc ALIGN=MIDDLE  ALT=\"4=49a%2B7b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For (9,7) , substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=9%2Cy=7%29 ALIGN=MIDDLE  ALT=\"system%28x=9%2Cy=7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , gives us the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7=81a%2B9b%2Bc ALIGN=MIDDLE  ALT=\"7=81a%2B9b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For (14,5.75) , substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=14%2Cy=5.75%29 ALIGN=MIDDLE  ALT=\"system%28x=14%2Cy=5.75%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , gives us the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.75=196a%2B14b%2Bc ALIGN=MIDDLE  ALT=\"5.75=196a%2B14b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284=49a%2B7b%2Bc%2C7=81a%2B9b%2Bc%2C5.75=196a%2B14b%2Bc%29 ALIGN=MIDDLE  ALT=\"system%284=49a%2B7b%2Bc%2C7=81a%2B9b%2Bc%2C5.75=196a%2B14b%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=-0.25%2Cb=5.5%2Cc=-22.25%29 ALIGN=MIDDLE  ALT=\"system%28a=-0.25%2Cb=5.5%2Cc=-22.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.25x%5E2%2B5.5x-22.25 ALIGN=MIDDLE  ALT=\"y=-0.25x%5E2%2B5.5x-22.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the equation for the parabolic path .<BR>\n" );
document.write( "For that equation, the maximum would happen at<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a=-5.5%2F%282%2A%28-0.25%29%29=-5.5%2F%28-0.5%29=11 ALIGN=MIDDLE  ALT=\"x=-b%2F2a=-5.5%2F%282%2A%28-0.25%29%29=-5.5%2F%28-0.5%29=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.25%2A11%5E2%2B5.5%2A11-22.25=-30.25%2B60.5-22.25=8 ALIGN=MIDDLE  ALT=\"y=-0.25%2A11%5E2%2B5.5%2A11-22.25=-30.25%2B60.5-22.25=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the maximum height the skier would reach is estimated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Also, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15 ALIGN=MIDDLE  ALT=\"x=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.25x%5E2%2B5.5x-22.25 ALIGN=MIDDLE  ALT=\"y=-0.25x%5E2%2B5.5x-22.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > would give us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.25%2A15%5E2%2B5.5%2A15-22.25+=-0.25%2A225%2B82.5-22.25=-56.25%2B82.5-22.25=4 ALIGN=MIDDLE  ALT=\"y=-0.25%2A15%5E2%2B5.5%2A15-22.25+=-0.25%2A225%2B82.5-22.25=-56.25%2B82.5-22.25=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , meaning that the skier would go through point (15.4) rather than 15,5).<BR>\n" );
document.write( "Is there a typo, or did they mean for the four points not to fit perfectly into a quadratic function?<BR>\n" );
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document.write( "USING STATISTICS TO FIND THE BEST FIT:<BR>\n" );
document.write( "Plugging points (7,4) (9,7) (14,5.75) and (15,5) into a spreadsheet a linear regression gives us a best-fit parabolic path as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.2096x%5E3%2B4.6872x-18.426 ALIGN=MIDDLE  ALT=\"y=-0.2096x%5E3%2B4.6872x-18.426\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which predicts a maximum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=7.78%28rounded%29 ALIGN=MIDDLE  ALT=\"y=7.78%28rounded%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11.18 ALIGN=MIDDLE  ALT=\"x=11.18\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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