document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=57794>Question 57794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the domain and range of the function -x^2 + 6x - 8. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #39507 by </B><A HREF=/tutors/aboutme.mpl?userid=funmath><B>funmath(2933)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=funmath><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=39507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Determine the domain and range of the function y=-x^2 + 6x - 8.<BR>\n" );
document.write( "Domain has to do with what your x values are allowed to be.  The domain of all polynomial functions is: all real numbers.  There are no restrictions on what x can be.<BR>\n" );
document.write( "Set builder notation for the domain would be: {x|x is all real numbers}<BR>\n" );
document.write( "Interval notation would be: x=(-infinity,infinity)  I can't do the infinity symbols yet.<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Range has to do with what the y values are given that the function's domain is what it is.<BR>\n" );
document.write( "For quadratic functions, the range is found through the vertex and whether the graph is opening up or down.<BR>\n" );
document.write( "The coefficient of x^2 is negative, therefore our parabola opens downward and goes to negative infinity.  It's vertex is at its maximum height.  We can find the vertex with the formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-b%2F2a%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-b%2F2a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, our a=-1, b=6 and c=-8<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%286%29%2F%282%28-1%29%29 ALIGN=MIDDLE  ALT=\"x=-%286%29%2F%282%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6%2F-2 ALIGN=MIDDLE  ALT=\"x=-6%2F-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The maximum y value will happen when x=3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1%283%29%5E2%2B6%283%29-8 ALIGN=MIDDLE  ALT=\"y=-1%283%29%5E2%2B6%283%29-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1%289%29%2B6%283%29-8 ALIGN=MIDDLE  ALT=\"y=-1%289%29%2B6%283%29-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-9%2B18-8 ALIGN=MIDDLE  ALT=\"y=-9%2B18-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore the range is: - infinity to 1<BR>\n" );
document.write( "(-infinity,1] <---- interval notation<BR>\n" );
document.write( ":<BR>\n" );
document.write( "If you don't have to do this algebraically and you can use a graphing calculator, then graph the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C-x%5E2%2B6x-8%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C-x%5E2%2B6x-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The x's extend forever left and right, the y' go from 1 down to -infinity.<BR>\n" );
document.write( "Happy Calculating!!!  </SPAN>\n" );
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