document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=347560>Question 347560</A>: <I><SPAN STYLE=\"word-break: break-all;\"> which direction does the parabola for the quadratic equation y = -x2 + 12x + 15 open </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #248456 by </B><A HREF=/tutors/aboutme.mpl?userid=haileytucki><B>haileytucki(390)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=haileytucki><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=haileytucki><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=248456\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y=-x^(2)+12x+15\r<BR>\n" );
document.write( "\n" );
document.write( "To create a trinomial square on the left-hand side of the equation, add a value to both sides of the equation that is equal to the square of half the coefficient of x.  In this problem, add (-6)^(2) to both sides of the equation.<BR>\n" );
document.write( "y=-1(x^(2)-12x+36)-1(-15)-(-1)(0+36)\r<BR>\n" );
document.write( "\n" );
document.write( "Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.<BR>\n" );
document.write( "y=-1(x^(2)-12x+36)-1(-15)-(-1)(36)\r<BR>\n" );
document.write( "\n" );
document.write( "Factor the perfect trinomial square into (x-6)^(2).<BR>\n" );
document.write( "y=-1((x-6)^(2))-1(-15)-(-1)(36)\r<BR>\n" );
document.write( "\n" );
document.write( "Factor the perfect trinomial square into (x-6)^(2).<BR>\n" );
document.write( "y=-1(x-6)^(2)-1(-15)-(-1)(36)\r<BR>\n" );
document.write( "\n" );
document.write( "Multiply -1 by each term inside the parentheses.<BR>\n" );
document.write( "y=-1(x-6)^(2)+15-(-1)(36)\r<BR>\n" );
document.write( "\n" );
document.write( "Multiply -1 by 36 to get -36.<BR>\n" );
document.write( "y=-1(x-6)^(2)+15-(-36)\r<BR>\n" );
document.write( "\n" );
document.write( "Multiply -1 by each term inside the parentheses.<BR>\n" );
document.write( "y=-1(x-6)^(2)+15+36\r<BR>\n" );
document.write( "\n" );
document.write( "Add 36 to 15 to get 51.<BR>\n" );
document.write( "y=-1(x-6)^(2)+51\r<BR>\n" );
document.write( "\n" );
document.write( "This is the form of a paraboloa.  Use this form to determine the values used to find vertex and x-y intercepts.<BR>\n" );
document.write( "y=a(x-h)^(2)+k\r<BR>\n" );
document.write( "\n" );
document.write( "Use the standard form to determine the vertex and x-y intercepts.<BR>\n" );
document.write( "a=-1_k=51_h=6\r<BR>\n" );
document.write( "\n" );
document.write( "The vertex of a parabola is (h,k).<BR>\n" );
document.write( "Vertex: (6,51)\r<BR>\n" );
document.write( "\n" );
document.write( "This formula is used to find the distance from the vertex to the focus.<BR>\n" );
document.write( "(1)/(4p)=a\r<BR>\n" );
document.write( "\n" );
document.write( "Substitute the value of a into the formula.<BR>\n" );
document.write( "(1)/(4p)=-1\r<BR>\n" );
document.write( "\n" );
document.write( "Solve the equation for p.<BR>\n" );
document.write( "p=-(1)/(4)\r<BR>\n" );
document.write( "\n" );
document.write( "Add p to the vertex to find the focus.  If the parabola points up or down add p to the y-coordinate of the vertex, if it points left or right add it to the x-coordinate.<BR>\n" );
document.write( "Focus=(6,51-(1)/(4))\r<BR>\n" );
document.write( "\n" );
document.write( "Find the focus.<BR>\n" );
document.write( "Focus=(6,(203)/(4))\r<BR>\n" );
document.write( "\n" );
document.write( "A parabola can also be defined as locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix).<BR>\n" );
document.write( "y=51-(-(1)/(4))\r<BR>\n" );
document.write( "\n" );
document.write( "Find the directrix.<BR>\n" );
document.write( "Directrix: y=(205)/(4)\r<BR>\n" );
document.write( "\n" );
document.write( "The axis of symmetry is the line that passes through the vertex and focus.  The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.<BR>\n" );
document.write( "Axis of Symmetry: x=6\r<BR>\n" );
document.write( "\n" );
document.write( "These values represent the important values for graphing and analyzing a parabola.<BR>\n" );
document.write( "Vertex: (6,51)_Focus: (6,(203)/(4))_Directrix: y=(205)/(4)_Axis of Symmetry: x=6 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );