![\"\"](\"/images/stars/stars-13.gif\")
You can put this solution on YOUR website!
You can work this in a couple of ways.
\n" ); document.write( ".
\n" ); document.write( "First way. Solve for x. If you get two solutions, then the graph crosses the x-axis at
\n" ); document.write( "two points ... that is at two different values for x. The peak of the graph (or in some problems
\n" ); document.write( "the lowest point of the graph) occurs at the midway point between the two values of x.
\n" ); document.write( "Once you have the value of x at that midway point, you can substitute that value into the
\n" ); document.write( "original equation and find y. Let's apply this method to solving the problem:
\n" ); document.write( ".
\n" ); document.write( "Given
\n" ); document.write( ".
\n" ); document.write( "Factor the right side of this equation:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "Then if you set y equal to zero, you can identify the values where the graph crosses the
\n" ); document.write( "x-axis. (Think about graphs. If a point has a y value of zero, its x value must be on the
\n" ); document.write( "x-axis). Setting y equal to zero results in:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "The right side will be equal to zero if either of the two factors are zero ... that is
\n" ); document.write( "if either x = 0 or -x +6 = 0. And by adding x to both sides of the second factor
\n" ); document.write( "we get that this factor will be zero if 6 = x. So now we know the graph crosses the x-axis
\n" ); document.write( "at the points where x = 0 and x = +6. Midway between these points is x = 3. (You can
\n" ); document.write( "find the midway point by averaging the two values of x ... that is by adding the two values
\n" ); document.write( "together and dividing by 2. In this case, add 0 + 6 to get +6 and divide by 2 to get +3.)
\n" ); document.write( "Now that you have the value of x at the midway point between the two x-axis crossing
\n" ); document.write( "points, just substitute that value into the original problem you were given and compute the
\n" ); document.write( "resulting value of y. So we take:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "and substitute +3 for x to get:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "The maximum value of the graph is at the point (3, 9).
\n" ); document.write( ".
\n" ); document.write( "The second way is to use the quadratic formula to find the value of x that causes the
\n" ); document.write( "value of y to be at the maximum (or minimum) point. The quadratic formula says that for
\n" ); document.write( "a quadratic of the form:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "The two values of x that satisfy this equation (they are the x-axis crossing points because
\n" ); document.write( "y is equal to zero) are given by the relationship:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "The portion of this answer that we are looking for is the
\n" ); document.write( "midpoint value of x. The plus and minus signed answers are equally spaced about that point,
\n" ); document.write( "the plus answer to the right of it, and the minus one equally to the left of it.
\n" ); document.write( ".
\n" ); document.write( "If you compare the given problem with y presumed to be zero
\n" ); document.write( "form
\n" ); document.write( "zero because it does not exist in our problem. Now recall that we are looking for the
\n" ); document.write( "midway point which we said was given by the equation
\n" ); document.write( "the values we have found for \"a\" and \"b\" we get:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "Note that by using this way, we also find that the midway point is x = +3, just as we found
\n" ); document.write( "by averaging the two values of x where the graph crosses the x-axis. Therefore, by substituting
\n" ); document.write( "+3 into the original equation we again get that the maximum value of the graph is when
\n" ); document.write( "y = 9 so the exact point of the maximum is at (+3, +9).
\n" ); document.write( ".
\n" ); document.write( "How do you know that the graph has a maximum point. You can tell because the squared term
\n" ); document.write( "in the original equation is preceded by a minus sign. If it instead was positive,
\n" ); document.write( "the graph would have a minimum point instead of a maximum.
\n" ); document.write( ".
\n" ); document.write( "Hope this helps you understand how this problem can be worked. \n" ); document.write( "