document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=176168>Question 176168</A>: <I><SPAN STYLE=\"word-break: break-all;\"> this is one question with different parts <BR>\n" );
document.write( "Find the vertex, line of symmetry, maximum or minimum value of the quadratic function, and graph the function.<BR>\n" );
document.write( "f(x)=-2x^2+2x+3<BR>\n" );
document.write( "x-coordinate of vertex<BR>\n" );
document.write( "y-coordinate of vertex<BR>\n" );
document.write( "equation of line of symmetry<BR>\n" );
document.write( "max/min value of f(x)<BR>\n" );
document.write( "The value of f(1/2)=7/2 is min or max<BR>\n" );
document.write( "Please help I have no idea what I am doing 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 #131271 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=131271\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Quadratic into Vertex Form\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=completing_the_square.solver>Completing the Square to Get a Quadratic into Vertex Form</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2+x%5E2%2B2+x%2B3 ALIGN=MIDDLE  ALT=\"y=-2+x%5E2%2B2+x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=-2+x%5E2%2B2+x ALIGN=MIDDLE  ALT=\"y-3=-2+x%5E2%2B2+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=-2%28x%5E2-1x%29 ALIGN=MIDDLE  ALT=\"y-3=-2%28x%5E2-1x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Take half of the x coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-1%29=-1%2F2 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-1%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  Now square <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;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F2%29%5E2=%28-1%2F2%29%28-1%2F2%29=1%2F4 ALIGN=MIDDLE  ALT=\"%28-1%2F2%29%5E2=%28-1%2F2%29%28-1%2F2%29=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=-2%28x%5E2-1x%2B1%2F4-1%2F4%29 ALIGN=MIDDLE  ALT=\"y-3=-2%28x%5E2-1x%2B1%2F4-1%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not change the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=-2%28%28x-1%2F2%29%5E2-1%2F4%29 ALIGN=MIDDLE  ALT=\"y-3=-2%28%28x-1%2F2%29%5E2-1%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-1x%2B1%2F4 ALIGN=MIDDLE  ALT=\"x%5E2-1x%2B1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-1%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=-2%28x-1%2F2%29%5E2%2B2%281%2F4%29 ALIGN=MIDDLE  ALT=\"y-3=-2%28x-1%2F2%29%5E2%2B2%281%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=-2%28x-1%2F2%29%5E2%2B1%2F2 ALIGN=MIDDLE  ALT=\"y-3=-2%28x-1%2F2%29%5E2%2B1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%28x-1%2F2%29%5E2%2B1%2F2%2B3 ALIGN=MIDDLE  ALT=\"y=-2%28x-1%2F2%29%5E2%2B1%2F2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides to isolate y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%28x-1%2F2%29%5E2%2B7%2F2 ALIGN=MIDDLE  ALT=\"y=-2%28x-1%2F2%29%5E2%2B7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms<BR>\n" );
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document.write( "  Now the quadratic is in vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-2 ALIGN=MIDDLE  ALT=\"a=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=1%2F2 ALIGN=MIDDLE  ALT=\"h=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=7%2F2 ALIGN=MIDDLE  ALT=\"k=7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.<BR>\n" );
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document.write( "  Check:<BR>\n" );
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document.write( "  Notice if we graph the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%5E2%2B2x%2B3 ALIGN=MIDDLE  ALT=\"y=-2x%5E2%2B2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%5E2%2B2x%2B3%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%5E2%2B2x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%5E2%2B2x%2B3 ALIGN=MIDDLE  ALT=\"y=-2x%5E2%2B2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex 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;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2 ALIGN=MIDDLE  ALT=\"7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  Notice if we graph the final equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%28x-1%2F2%29%5E2%2B7%2F2 ALIGN=MIDDLE  ALT=\"y=-2%28x-1%2F2%29%5E2%2B7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2%28x-1%2F2%29%5E2%2B7%2F2%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2%28x-1%2F2%29%5E2%2B7%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%28x-1%2F2%29%5E2%2B7%2F2 ALIGN=MIDDLE  ALT=\"y=-2%28x-1%2F2%29%5E2%2B7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is also (<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;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2 ALIGN=MIDDLE  ALT=\"7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.<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" );
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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%2B3+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-2%2Ax%5E2%2B2%2Ax%2B3+%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" );
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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" );
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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" );
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document.write( "  Maxima / Minima lies at the point where this squared term is equal to zero.<BR>\n" );
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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" );
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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" );
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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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