document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=160957>Question 160957</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ihave three problems that my father brought to my attention. H etried to do them and could not figure them out. I for the life of me can't either. Hetold me someone said they some sort of algebra called grinding.Can someone help? Question is: Write the minimum or maximum point for each function,and give the coordinates for that point. 1. y=x^2-4x+1 hint start at 0 and increase. 2. y=x^2-8x-5 hint start at 2 and increase. 3. y=-2^2+2x+6 hint start at -1 and increase. Can anyone help us with these. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #118619 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=118619\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I think the procedure is just start at a point get a value for the function, move over get another point, etc. until you see the pattern that the function has either maxed out (reached a maximum) or bottomed out (reached a minimum). Plug and chug we used to call it.<BR>\n" );
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document.write( "a)<BR>\n" );
document.write( "y=x^2-4x+1<BR>\n" );
document.write( "y(0)=0-0+1=1<BR>\n" );
document.write( "y(1)=1-4+1=-2<BR>\n" );
document.write( "y(2)=4-8+1=-3<BR>\n" );
document.write( "y(3)=9-12+1=-2<BR>\n" );
document.write( "Looks like (2,-3) is the minimum.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2-4x%2B1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2-4x%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "y=x^2-8x-5<BR>\n" );
document.write( "y(2)=4-16-5=-17<BR>\n" );
document.write( "y(3)=9-24-5=-20<BR>\n" );
document.write( "y(4)=16-32-5=-21<BR>\n" );
document.write( "y(5)=25-40-5=-20<BR>\n" );
document.write( "Looks like (4,-21) is the minimum.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-2%2C+7%2C+-25%2C+5%2C+x%5E2-8x-5%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-2%2C+7%2C+-25%2C+5%2C+x%5E2-8x-5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "y=-2x^2+2x+6<BR>\n" );
document.write( "y(-1)=-2-2+6=2<BR>\n" );
document.write( "y(0)=0+0+6=6<BR>\n" );
document.write( "y(1)=-2+2+6=6<BR>\n" );
document.write( "y(2)=-8+4+6=2<BR>\n" );
document.write( "The maximum is between 0 and 1, at x=1/2.<BR>\n" );
document.write( "y(1/2)=-1/2+1+6=6.5<BR>\n" );
document.write( "(0.5,6.5)<BR>\n" );
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document.write( "Just a hint, for quadratic equations in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the max or min always occurs at<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so if we look at your examples again,\r<BR>\n" );
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document.write( "y=x^2-4x+1, a=1, b=-4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%28-4%29%2F%282%281%29%29=2 ALIGN=MIDDLE  ALT=\"x=-%28-4%29%2F%282%281%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "y=x^2-8x-5, a=1, b=-8, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%28-8%29%2F%282%281%29%29=4 ALIGN=MIDDLE  ALT=\"x=-%28-8%29%2F%282%281%29%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "y=-2x^2+2x+6, a=-2, b=2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%282%29%2F%282%28-2%29%29=1%2F2 ALIGN=MIDDLE  ALT=\"x=-%282%29%2F%282%28-2%29%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Same answers just much faster than guessing. </SPAN>\n" );
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