document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=123835>Question 123835</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Find the axis of symmetry of y = -x squared - 6x + 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #90901 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=90901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-x%5E2+-+6x+%2B2 ALIGN=MIDDLE  ALT=\"y+=+-x%5E2+-+6x+%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Before we solve this problem, let's take a look at its graph:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C-x%5E2+-+6x+%2B+2%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C-x%5E2+-+6x+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "From this look at the graph you can tell that the axis of symmetry will be a vertical line<BR>\n" );
document.write( "that crosses the x-axis somewhere near the point x = -3.<BR>\n" );
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document.write( "There are two ways we could solve this problem. One way is to find the two values of x where<BR>\n" );
document.write( "the graph crosses the x-axis. From the graph it looks as if those points are approximately<BR>\n" );
document.write( "at x = +0.5 and at x = -6.5. We can find the two points exactly by going to the equation and<BR>\n" );
document.write( "setting y equal to zero and then applying the quadratic formula to find the exact values of x.<BR>\n" );
document.write( "But suppose that they are +0.5 and -6.5. The axis of symmetry will go through the midway point <BR>\n" );
document.write( "between those two values.  So we could find the value of x for the line of symmetry by <BR>\n" );
document.write( "averaging +0.5 and -6.5.  The average will be (+0.5 - 6.5)/2 = -6/2 = -3. And this tells you<BR>\n" );
document.write( "that the equation for the axis of symmetry would be x = -3. But there are two things wrong<BR>\n" );
document.write( "with this. First the answer is only approximate and to get the exact answer you have to work<BR>\n" );
document.write( "your way through the entire quadratic formula to get two values of x and then average the two<BR>\n" );
document.write( "answers that you do get.<BR>\n" );
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document.write( "A shorter way to work this problem is to use recognize that the quadratic formula applies<BR>\n" );
document.write( "to equations of the standard form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If we go to the given equation for this problem and set y = 0 we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2+-6x+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"-x%5E2+-6x+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By comparing the standard form with our equation we can see that:<BR>\n" );
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document.write( "a = -1<BR>\n" );
document.write( "b = -6<BR>\n" );
document.write( "c = +2<BR>\n" );
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document.write( "The first part of the answer to the quadratic formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this gives you the<BR>\n" );
document.write( "equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which will be the equation for the line of symmetry. For our equation we found that a = -1<BR>\n" );
document.write( "and b = -6. Substituting these values results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F%282%2Aa%29+=+-%28-6%29%2F%282%2A%28-1%29%29+=+6%2F-2+=+-3 ALIGN=MIDDLE  ALT=\"x+=+-b%2F%282%2Aa%29+=+-%28-6%29%2F%282%2A%28-1%29%29+=+6%2F-2+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Our graph was pretty good. It told us that the axis of symmetry was a vertical line going<BR>\n" );
document.write( "crossing the x-axis at about x = -3 and that turns out to be correct. (Notice that the<BR>\n" );
document.write( "axis of symmetry goes through the peak of the graph.)<BR>\n" );
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document.write( "So the answer to your problem is that the equation for the axis of symmetry is:<BR>\n" );
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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" );
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document.write( "and on the graph it looks like the green line on this graph:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C-x%5E2+-+6x+%2B+2%2C6000%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C-x%5E2+-+6x+%2B+2%2C6000%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better.<BR>\n" );
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