Question 90254This question is from textbook Algebra and Trigonometry
: Find the maximum value of y=-x squared+6x
This question is from textbook Algebra and Trigonometry
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You can work this in a couple of ways.
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First way. Solve for x. If you get two solutions, then the graph crosses the x-axis at
two points ... that is at two different values for x. The peak of the graph (or in some problems
the lowest point of the graph) occurs at the midway point between the two values of x.
Once you have the value of x at that midway point, you can substitute that value into the
original equation and find y. Let's apply this method to solving the problem:
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Given 
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Factor the right side of this equation:
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Then if you set y equal to zero, you can identify the values where the graph crosses the
x-axis. (Think about graphs. If a point has a y value of zero, its x value must be on the
x-axis). Setting y equal to zero results in:
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The right side will be equal to zero if either of the two factors are zero ... that is
if either x = 0 or -x +6 = 0. And by adding x to both sides of the second factor
we get that this factor will be zero if 6 = x. So now we know the graph crosses the x-axis
at the points where x = 0 and x = +6. Midway between these points is x = 3. (You can
find the midway point by averaging the two values of x ... that is by adding the two values
together and dividing by 2. In this case, add 0 + 6 to get +6 and divide by 2 to get +3.)
Now that you have the value of x at the midway point between the two x-axis crossing
points, just substitute that value into the original problem you were given and compute the
resulting value of y. So we take:
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and substitute +3 for x to get:
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The maximum value of the graph is at the point (3, 9).
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The second way is to use the quadratic formula to find the value of x that causes the
value of y to be at the maximum (or minimum) point. The quadratic formula says that for
a quadratic of the form:
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The two values of x that satisfy this equation (they are the x-axis crossing points because
y is equal to zero) are given by the relationship:
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The portion of this answer that we are looking for is the  because it is the
midpoint value of x. The plus and minus signed answers are equally spaced about that point,
the plus answer to the right of it, and the minus one equally to the left of it.
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If you compare the given problem with y presumed to be zero  to the quadratic
form  you can see that "a" must be -1, "b" must be 6, and "c" must be
zero because it does not exist in our problem. Now recall that we are looking for the
midway point which we said was given by the equation  . Substituting
the values we have found for "a" and "b" we get:
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Note that by using this way, we also find that the midway point is x = +3, just as we found
by averaging the two values of x where the graph crosses the x-axis. Therefore, by substituting
+3 into the original equation we again get that the maximum value of the graph is when
y = 9 so the exact point of the maximum is at (+3, +9).
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How do you know that the graph has a maximum point. You can tell because the squared term
in the original equation is preceded by a minus sign. If it instead was positive,
the graph would have a minimum point instead of a maximum.
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Hope this helps you understand how this problem can be worked.
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