Question 123835
Given:
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{{{y = -x^2 - 6x +2}}}
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Before we solve this problem, let's take a look at its graph:
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{{{graph(600,600,-10,10,-20,20,-x^2 - 6x + 2)}}}
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From this look at the graph you can tell that the axis of symmetry will be a vertical line
that crosses the x-axis somewhere near the point x = -3.
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There are two ways we could solve this problem. One way is to find the two values of x where
the graph crosses the x-axis. From the graph it looks as if those points are approximately
at x = +0.5 and at x = -6.5. We can find the two points exactly by going to the equation and
setting y equal to zero and then applying the quadratic formula to find the exact values of x.
But suppose that they are +0.5 and -6.5. The axis of symmetry will go through the midway point 
between those two values.  So we could find the value of x for the line of symmetry by 
averaging +0.5 and -6.5.  The average will be (+0.5 - 6.5)/2 = -6/2 = -3. And this tells you
that the equation for the axis of symmetry would be x = -3. But there are two things wrong
with this. First the answer is only approximate and to get the exact answer you have to work
your way through the entire quadratic formula to get two values of x and then average the two
answers that you do get.
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A shorter way to work this problem is to use recognize that the quadratic formula applies
to equations of the standard form:
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{{{ax^2 + bx + c = 0}}}
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If we go to the given equation for this problem and set y = 0 we get:
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{{{-x^2 -6x + 2 = 0}}}
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By comparing the standard form with our equation we can see that:
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a = -1
b = -6
c = +2
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The first part of the answer to the quadratic formula is {{{-b/(2*a)}}} and this gives you the
equation:
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{{{x = -b/(2*a)}}}
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which will be the equation for the line of symmetry. For our equation we found that a = -1
and b = -6. Substituting these values results in:
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{{{x = -b/(2*a) = -(-6)/(2*(-1)) = 6/-2 = -3}}}
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Our graph was pretty good. It told us that the axis of symmetry was a vertical line going
crossing the x-axis at about x = -3 and that turns out to be correct. (Notice that the
axis of symmetry goes through the peak of the graph.)
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So the answer to your problem is that the equation for the axis of symmetry is:
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{{{x = -3}}}
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and on the graph it looks like the green line on this graph:
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{{{graph(600,600,-10,10,-20,20,-x^2 - 6x + 2,6000(x+3))}}}
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Hope this helps you to understand the problem a little better.
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