document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316858>Question 316858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the quadratic function in the form y = a(x - h)2 + k. Find the veertex, axis of symmetry, domain and range<BR>\n" );
document.write( "y = -2x2 - 8x - 5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226818 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=226818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -2x^2\ -\ 8x\ -\ 5\">\r<BR>\n" );
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document.write( "Factor out -2 from the variable terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -2\left(x^2\ +\ 4x\right) -\ 5\">\r<BR>\n" );
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document.write( "Divide the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term by 2 and square the result.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{4}{2}\right)^2\ =\ 4\">\r<BR>\n" );
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document.write( "Add 4 inside the parentheses and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -(-2)(4)\ =\ 8\"> outside the parentheses\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -2\left(x^2\ +\ 4x\ +\ 4\right) -\ 5\ +\ 8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -2\left(x^2\ +\ 4x\ +\ 4\right)\ +\ 3\">\r<BR>\n" );
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document.write( "Factor the perfect square in the parentheses:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -2\left(x\ +\ 2\right)^2\ +\ 3\">\r<BR>\n" );
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document.write( "Vertex:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-2,3\right)\">\r<BR>\n" );
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document.write( "Axis of symmetry:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ -2\">\r<BR>\n" );
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document.write( "Domain: (All real numbers, just like any other polynomial function) <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \{x\,|\,x\,\in\,\mathbb{R}\}\">\r<BR>\n" );
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document.write( "Range:  The lead coefficient is negative, therefore the parabola opens downward and the vertex represents a maximum value of the function.  That maximum value is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of the vertex, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\">. The range is unconstrained less than the maximum, so Range:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \{y\, |\,y\,\in\,\mathbb{R},\ y\ \leq\ 3\}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A600%2C+600%2C+-6%2C+6%2C+-6%2C+6%2C%0D%0Agrid%281%29%2C%0D%0Acircle%28-3.2247%2C0%2C.2%29%2C%0D%0Acircle%28-0.7752%2C0%2C.2%29%2C%0D%0Acircle%280%2C-5%2C.2%29%2C%0D%0Agraph%28%0D%0A600%2C+600%2C+-6%2C+6%2C+-6%2C+6%2C%0D%0A-2x%5E2-8x-5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You didn't ask for the intercepts, but here they are anyway:\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept is at the value of the function when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 0\">, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(0)\ =\ -2(0)^2\ -\ 8(0)\ -\ 5\ =\ -5\">\r<BR>\n" );
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document.write( "and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,-5)\">\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts are the roots of:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -2x^2\ -\ 8x\ -\ 5\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{4\ \pm\ \sqrt{6}}{-2}\">\r<BR>\n" );
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document.write( "Verification of that last step is left as an exercise for the student.\r<BR>\n" );
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document.write( "Hence the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{4\ +\ \sqrt{6}}{-2},0\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{4\ -\ \sqrt{6}}{-2},0\right)\">\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" );
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