document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=421228>Question 421228</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the vertex, the line of symmetry, the max or minimum value<BR>\n" );
document.write( "of the quadratic function, and graph the function.\r<BR>\n" );
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document.write( "f(x) -2x^2 + 2x + 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #294186 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=294186\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> find the vertex, the line of symmetry, the max or minimum value<BR>\n" );
document.write( "of the quadratic function, and graph the function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=-2x%5E2+%2B+2x+%2B+4 ALIGN=MIDDLE  ALT=\"f%28x%29+=-2x%5E2+%2B+2x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the equation for a parabola can also be written in \"vertex form\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=+-2%28x-h%29%5E2+%2B+k ALIGN=MIDDLE  ALT=\"f%28x%29=+-2%28x-h%29%5E2+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >....the vertex of the parabola is the point (h, k)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-coordinate ALIGN=MIDDLE  ALT=\"x-coordinate\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the vertex\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2F2%28-2%29=-2%2F-4=1%2F2 ALIGN=MIDDLE  ALT=\"-2%2F2%28-2%29=-2%2F-4=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-coordinate ALIGN=MIDDLE  ALT=\"x-coordinate\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substituting in the original equation to get the y-coordinate, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=+-2%281%2F2%29%5E2+%2B+2%281%2F2%29+%2B+4 ALIGN=MIDDLE  ALT=\"f%28x%29=+-2%281%2F2%29%5E2+%2B+2%281%2F2%29+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-2%281%2F4%29+%2B+1+%2B+4 ALIGN=MIDDLE  ALT=\"y=+-2%281%2F4%29+%2B+1+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-1%2F2+%2B+1+%2B+4 ALIGN=MIDDLE  ALT=\"y=+-1%2F2+%2B+1+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+1%2F2++%2B+4 ALIGN=MIDDLE  ALT=\"y=+1%2F2++%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+4.5 ALIGN=MIDDLE  ALT=\"y=+4.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, the vertex of the parabola is at (1/2, 4.5). \r<BR>\n" );
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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=hummingbird_min_max_test1.solver>Min/Max of a Quadratic Function</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The min/max of a quadratic equation is always at a point where its first differential is zero. This means that in our case, the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > at which the given equation has a <A  rel=nofollow HREF=Maxima_and_minima.wikipedia>maxima/minima</A> must satisfy the following equation:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A-2%2Ax%2B2=0 ALIGN=MIDDLE  ALT=\"2%2A-2%2Ax%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2%2F%282%2A-2%29+=+0.5 ALIGN=MIDDLE  ALT=\"x+=+-2%2F%282%2A-2%29+=+0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  This point is a minima if value of coefficient of x<sup>2</sup> is positive and vice versa. For our function the point x=0.5 is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maxima ALIGN=MIDDLE  ALT=\"maxima\"   DISABLED_style= \"max-width:90%; height:auto;\" > The graph of the equation is :<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+-2%2Ax%5E2%2B2%2Ax%2B4+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-2%2Ax%5E2%2B2%2Ax%2B4+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Alternate method<BR>\n" );
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document.write( "  In this method, we will use the <A  rel=nofollow HREF=Perfect_square.wikipedia>perfect square</A> method.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Step one:<BR>\n" );
document.write( "  Make the coefficient of <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;\" > positive by multiplying it by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > in case<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  Maxima / Minima is decided from the sign of 'a'. <BR>\n" );
document.write( "  If 'a' is positive then we have Minima and for 'a'negative we have Maxima.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Step two:<BR>\n" );
document.write( "  Now make the perfect square with the same <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient. <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281.4142135623731%2Ax+%2B+%28-2%2F2%29%29%5E2 ALIGN=MIDDLE  ALT=\"%281.4142135623731%2Ax+%2B+%28-2%2F2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Maxima / Minima lies at the point where this squared term is equal to zero.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence,<BR>\n" );
document.write( "  =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-%28-2%2F2%29%2F1.4142135623731%29=+0.707106781186547 ALIGN=MIDDLE  ALT=\"x=%28-%28-2%2F2%29%2F1.4142135623731%29=+0.707106781186547\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  This point is a minima if value of coefficient of x<sup>2</sup> is positive and vice versa. For our function the point x=0.5 is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maxima ALIGN=MIDDLE  ALT=\"maxima\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For more on this topic, refer to <A  rel=nofollow HREF=Quadratic_equation.wikipedia>Min/Max of a Quadratic equation</A>.</TD></TR>\n" );
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