document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=541311>Question 541311</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have to make a graph of this parabola:\r<BR>\n" );
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document.write( "F(x) = x^2 -2x -15\r<BR>\n" );
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document.write( "I need the details of how each of the principals points are calculated.\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 #354177 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=354177\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The easiest point are the intercepts<BR>\n" );
document.write( "x-intercept: (x,0)<BR>\n" );
document.write( "y-intercept: (0,y)<BR>\n" );
document.write( "To find x-intercept(s) ( may be 1 or 2 or none ),<BR>\n" );
document.write( "set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+0+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( f(x) is the same as y )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+x%5E2+-+2x+-+15+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+x%5E2+-+2x+-+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+x%5E2+-+2x+-+15+ ALIGN=MIDDLE  ALT=\"+0+=+x%5E2+-+2x+-+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+-+2x++=+15+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+2x++=+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Complete the square by dividing the coefficient<BR>\n" );
document.write( "of the x term by 2, square it, and add it to both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+-+2x+%2B+%28-2%2F2%29%5E2+=+15+%2B+%28-2%2F2%29%5E2+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+2x+%2B+%28-2%2F2%29%5E2+=+15+%2B+%28-2%2F2%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+-+2x+%2B+1+=+15+%2B+1+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+2x+%2B+1+=+15+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x+-+1%29%5E2+=+4%5E2+ ALIGN=MIDDLE  ALT=\"+%28x+-+1%29%5E2+=+4%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "take the square root of both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+-+1+=+4+ ALIGN=MIDDLE  ALT=\"+x+-+1+=+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+5+ ALIGN=MIDDLE  ALT=\"+x+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And, taking the negative root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4%5E2+ ALIGN=MIDDLE  ALT=\"+4%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+-+1+=+-4+ ALIGN=MIDDLE  ALT=\"+x+-+1+=+-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+-3+ ALIGN=MIDDLE  ALT=\"+x+=+-3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the x-intercepts ( also called roots ) are at<BR>\n" );
document.write( "(5,0) and (-3,0)<BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "To find the y-intercept ( only 1 ), set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=0+ ALIGN=MIDDLE  ALT=\"+x=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+x%5E2+-+2x+-+15+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+x%5E2+-+2x+-+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+0%5E2+-+2%2A0+-+15+ ALIGN=MIDDLE  ALT=\"+y+=+0%5E2+-+2%2A0+-+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+-15+ ALIGN=MIDDLE  ALT=\"+y+=+-15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y-intercept = (0, -15)<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Now you have 3 important points, also since the<BR>\n" );
document.write( "coefficient of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term is positive, the<BR>\n" );
document.write( "parabola has a minimum, not a maximum.<BR>\n" );
document.write( "The minimum is always exactly between the<BR>\n" );
document.write( "x-intercepts, or at ( (5 - 3)/2, y ) = ( 1, y), and to find y,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+1%5E2+-+2%2A1+-+15+ ALIGN=MIDDLE  ALT=\"+y+=+1%5E2+-+2%2A1+-+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+1+-+2+-+15+ ALIGN=MIDDLE  ALT=\"+y+=+1+-+2+-+15+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+-16+ ALIGN=MIDDLE  ALT=\"+y+=+-16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the vertex which is a minimum is at (1, -16) </SPAN>\n" );
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