document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=571736>Question 571736</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A square centered at the origin has its vertices on the x and y axes.  The graph of the function<BR>\n" );
document.write( "f(x) = ax2 - 4, a > 0<BR>\n" );
document.write( "passes through three of the square's vertices.  Find a.<BR>\n" );
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document.write( "I tried to draw a graph of it, but didn't really help, since I don't know the exact measures of the square... But being that its vertices are on the x and y axes I figured that it is like, diagonal, but other than that, I am completely lost!<BR>\n" );
document.write( "Please help, I would definitely appreciate it :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #368172 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=368172\">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( "Let the four vertices of the square (starting at the negative <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> axis and going clockwise) be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A: \left(-\alpha,\,0\right)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B: \left(0,\,\beta\right)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C: \left(\alpha,\,0\right)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D: \left(0,\,-\beta\right)\">.\r<BR>\n" );
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document.write( "Since the second degree term variable in the function is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> we know that the axis of symmetry of the parabola graph is a vertical line.  Furthermore, since we are given that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ >\ 0\">, we know that the parabola opens upward, and the vertex is a minimum local extrema.  That fixes the three points of the square that are in common with the graph of the function as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D\"> as the vertex of the parabola.\r<BR>\n" );
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document.write( "The standard form of a parabola is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ \neq\ 0\">\r<BR>\n" );
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document.write( "So if the parabola passes through the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_i,\,y_i\right)\"> then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_i\ =\ a(x_i)^2\ +\ b(x_i)\ +\ c\">\r<BR>\n" );
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document.write( "In the case of your function at point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D: \left(0,\,-\beta\right)\">,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(0)^2\ +\ (0)(0)\ -\ 4\ =\ -\beta\">\r<BR>\n" );
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document.write( "from which we deduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \beta\ =\ 4\">\r<BR>\n" );
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document.write( "But since the given figure was a square, we know that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \alpha\ =\ \beta\ =\ 4\">\r<BR>\n" );
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document.write( "Subsituting the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C: \left(\alpha,\,0\right)\"> which we now know to be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C: \left(4,\,0\right)\"> into your given function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(4)^2\ +\ (0)(4)\ -\ 4\ =\ 0\">\r<BR>\n" );
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document.write( "from which we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ \frac{1}{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<BR>\n" );
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