document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=36840>Question 36840</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for the funtion f(x)=-x^2+6x-8\r<BR>\n" );
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document.write( "(a) Find the line of symmetry<BR>\n" );
document.write( "(b) state whether there is a minimum or a maximum and find the value<BR>\n" );
document.write( "(c) Find the range </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #22644 by </B><A HREF=/tutors/aboutme.mpl?userid=martinr><B>martinr(6)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=martinr><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=22644\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, you did not indicate whether you could use a calculator or not, so I will explain it by hand.<BR>\n" );
document.write( "First, you know that this graph is a parabola (U-Shaped) due to the fact the the largest exponent it 2.  Therefore, there are at most two solutions (points where the graph crosses the x-axis).  Also, since the leading coefficient (the number in front of x^2), is negative, you know know that the graph opens down.<BR>\n" );
document.write( "Standard form for a quadratic:  y = ax^2 + bx + c<BR>\n" );
document.write( "Identify a = , b = , and c =  .   For right now, the c value will not be used in the formulas, but identify it now, because you will use it soon with the quadratic formula.<BR>\n" );
document.write( "So a = -1, b = 6 and c = -8.<BR>\n" );
document.write( "Find the x-coordinate of the vertex (the turning point of the graph) by using the formula x = -b/2a.   This is also the formula for the axis of symmetry (where the graph turns and is a mirror image of itself).<BR>\n" );
document.write( "In this case, the x-coordinate of the vertex is x = -6/2(-1) = 3.<BR>\n" );
document.write( "To find the y-coordinate, substitute x = 3 into the original equation and find the y-coordinate for the vertex which is -(3)^2 + 6(3) - 8 = 1.  The vertes is (3,1).<BR>\n" );
document.write( "Now I would choose 2 points greater than the x-coordinate of the vertex and two values less than it as well.  <BR>\n" );
document.write( "Make a table of values<BR>\n" );
document.write( "x      y = -x^2 + 6x - 8    y<BR>\n" );
document.write( "1       -(1)^2 + 6(1) - 8  = -3 <BR>\n" );
document.write( "2       -(2)^2 + 6(2) - 8   =0<BR>\n" );
document.write( "3       -(3)^2 + 6(3) - 8   =1 <BR>\n" );
document.write( "4       -(4)^2 + 6(4) - 8   =0<BR>\n" );
document.write( "5       -(5)^2 + 6(5) - 8   =-3<BR>\n" );
document.write( "Points (1, -3), (2, 0), (3, 1), (4, 0), and (5, -3).<BR>\n" );
document.write( "Graph the five points.  Remember that this is a U-shaped graph, so there needs to be a curve to it.<BR>\n" );
document.write( "The graph is opening down, and the axisof symmetry is a dashed line (vertical) that goes through the vertex.<BR>\n" );
document.write( "The maximum value is the vertex (3, 1).  It is the highest point that is on the graph.<BR>\n" );
document.write( "The range is the y values.  Since the graph goes on forever, and it goes negative, therefore the range is highest <= y <= -Infinity.  The sign <= means greater than or equal to.<BR>\n" );
document.write( "Range:  3<= y <= -Infinity (negative infinity sideways 8).<BR>\n" );
document.write( "Hope that this is helpful\r<BR>\n" );
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