document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=224612>Question 224612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the parabola y = -(x – 6)^2 – 1, determine each of the following.<BR>\n" );
document.write( "Identify whether the parabola y = -(x – 6)^2 – 1 opens up or down. <BR>\n" );
document.write( "Identify the vertex of the parabola y = -(x – 6)^2 – 1.  <BR>\n" );
document.write( "Identify the x-intercept(s) of the parabola y = -(x – 6)^2 – 1. <BR>\n" );
document.write( "Identify the y-intercept(s) of the parabola y = -(x – 6)^2 – 1. <BR>\n" );
document.write( "Please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #167771 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=167771\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the parabola y = -(x – 6)^2 – 1, determine each of the following.<BR>\n" );
document.write( "Identify whether the parabola y = -(x – 6)^2 – 1 opens up or down.<BR>\n" );
document.write( "It opens down, the x^2 term is negative.<BR>\n" );
document.write( "------------------<BR>\n" );
document.write( "Identify the vertex of the parabola y = -(x – 6)^2 – 1<BR>\n" );
document.write( "y = -x^2 + 12x - 37<BR>\n" );
document.write( "The vertex is at x = -b/2a = -12/-2 = 6<BR>\n" );
document.write( "At x = 6, y = -1 --> vertex at (6,-1)<BR>\n" );
document.write( "-------------------- <BR>\n" );
document.write( "Identify the x-intercept(s)<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1x%5E2%2B12x%2B-37+=+0 ALIGN=MIDDLE  ALT=\"-1x%5E2%2B12x%2B-37+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2812%29%5E2-4%2A-1%2A-37=-4 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2812%29%5E2-4%2A-1%2A-37=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The discriminant -4 is less than zero. That means that there are no solutions among real numbers.<BR><BR>\n" );
document.write( "  If you are a student of advanced school algebra and are aware about <u>imaginary numbers<u>, read on.<BR><BR>\n" );
document.write( "  <BR>\n" );
document.write( "      In the field of imaginary numbers, the square root of -4 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+4%29+=+2 ALIGN=MIDDLE  ALT=\"sqrt%28+4%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "      The solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28-12%2B-i%2Asqrt%28+-4+%29%29%2F2%5C-1+=++%28-12%2B-i%2A2%29%2F2%5C-1+ ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28-12%2B-i%2Asqrt%28+-4+%29%29%2F2%5C-1+=++%28-12%2B-i%2A2%29%2F2%5C-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or<BR>\n" );
document.write( "  Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B12%2Ax%2B-37+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B12%2Ax%2B-37+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "---------------<BR>\n" );
document.write( "Identify the y-intercept(s)<BR>\n" );
document.write( "-37<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );