document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=618720>Question 618720</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for the quadratic equation -16t^2+125t what is the maximum and minimum? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #389063 by </B><A HREF=/tutors/aboutme.mpl?userid=math-vortex><B>math-vortex(648)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math-vortex><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=math-vortex><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=389063\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi, there--<BR>\n" );
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document.write( "A quadratic equation only has one, either a maximum or a minimum. It won't have both. <BR>\n" );
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document.write( "The shape of the graph of a quadratic is a parabola. If the parabola opens upward, the the equation has a minimum at the vertex. If the parabola opens downward, the equation has a maximum at the vertex.<BR>\n" );
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document.write( "There are many ways to solve this problem. I'm not sure what math level you are studying, so I'll show an Algebra I method.<BR>\n" );
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document.write( "Since the minimum or maximum is located at the vertex, we'll end the vertex of this equation. The vertex will be an ordered pair (t,h). We have<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B125t ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B125t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that I added an h to be the 2nd variable of the equation. You could use any variable. When a quadratic equation is in standard form,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=at%5E2%2Bbt%2Bc ALIGN=MIDDLE  ALT=\"h=at%5E2%2Bbt%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "the t-value of the vertex is the value -b/2a, where a and b are the coefficients of the t^2 term and the t term in your equation. In your case a=-16 and b=125. Therefore,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a=%28-125%29%2F%282%2A-16%29=125%2F32 ALIGN=MIDDLE  ALT=\"-b%2F2a=%28-125%29%2F%282%2A-16%29=125%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the t-value of the vertex is 125/32. To find the h-value, we use substitution:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16%28125%2F32%29%5E2%2B125%28125%2F32%29 ALIGN=MIDDLE  ALT=\"h=-16%28125%2F32%29%5E2%2B125%28125%2F32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-250000%2F1024%2B15625%2F32 ALIGN=MIDDLE  ALT=\"h=-250000%2F1024%2B15625%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-250000%2F1024%2B500000%2F1024 ALIGN=MIDDLE  ALT=\"h=-250000%2F1024%2B500000%2F1024\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=250000%2F1024=15625%2F64 ALIGN=MIDDLE  ALT=\"h=250000%2F1024=15625%2F64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we know that the vertex of the parabola is at the point (125/32, 15625/64). <BR>\n" );
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document.write( "When a quadratic equation is in standard form, we can use the leading coefficient---the a-value---to determine if the parabola opens upward or downward. If a>0, the parabola opens up; if a<0, it opens down. In your case the parabola opens downward. This means that the vertex is a maximum for your parabola.<BR>\n" );
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document.write( "Hope this helps. Feel free to email if you have questions about this.<BR>\n" );
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document.write( "Ms.Figgy<BR>\n" );
document.write( "math.in.the.vortex@gmail.com\r<BR>\n" );
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