document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1028268>Question 1028268</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result.\r<BR>\n" );
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document.write( "Vertex: <BR>\n" );
document.write( "(&#8722;6, &#8722;4)\r<BR>\n" );
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document.write( "Point: <BR>\n" );
document.write( "(&#8722;4, 1)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #643452 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=643452\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The answer depends on what you mean by standard form.  Some people call <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ +\ bx\ +\ c\"> standard form and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ a(x\ -\ h)^2\ +\ k\"> vertex form.  And some people refer to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ a(x\ -\ h)^2\ +\ k\"> as standard form.\r<BR>\n" );
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document.write( "Either way there are two ways to go about this.\r<BR>\n" );
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document.write( "Since you are given the vertex and one other point, you can find a third point on the graph by considering symmetry.  Since the point (-4,1) is two units to the right of the axis of symmetry, then the function value two units to the left of the axis must be the same, hence the point (-8,1) must also be on the graph.\r<BR>\n" );
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document.write( "Assuming <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ +\ bx\ +\ c\"> as standard form, if (-4,1) is on the graph, then it must be true that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  16a\ -\ 4b\ +\ 1\ =\ 1\">\r<BR>\n" );
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document.write( "Likewise, considering the other two points,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  36a\ -\ 6b\ +\ 1\ =\ -4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  64a\ -\ 8b\ +\ 1\ =\ 1\">\r<BR>\n" );
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document.write( "The solution to the 3X3 system of equations yields the standard form coefficients.\r<BR>\n" );
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document.write( "I'll leave it to you to verify that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ \frac{5}{4}\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 15\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\ =\ 41\">.\r<BR>\n" );
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document.write( "On the other hand, if by standard form you mean what most folks call the vertex form, you can proceed as follows:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ a(x\ -\ h)^2\ +\ k\">\r<BR>\n" );
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document.write( "Substituting the vertex coordinates:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ a(x\ +\ 6)^2\ -\ 4\">\r<BR>\n" );
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document.write( "Then, since you know that the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-4,1)\"> is on the graph,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(-4\ +\ 6)^2\ -\ 4\ =\ 1\">\r<BR>\n" );
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document.write( "And all you need to do is solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">.  Any doubt in your mind that the result will be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ \frac{5}{4}\">?\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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