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Question 81843: find the vertex and intercepts for each parabola and graph if you could please help
h(x)=-3x^2+6x
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
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It may help you to visualize the graph if you replace h(x) with y to make the equation:
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Since this equation falls into the form  you can tell that its graph is
a parabola.
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In this problem the fact that the  term is preceded by a minus sign tells you something
else. It tells you that in the graph, as you move to the right the graph rises to a peak
and then as you continue to the right, it falls back down.
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Ask yourself, "What is the value of y where the graph intersects the x-axis?" The answer
is that any point on the x-axis has zero as its y value. Therefore, you can find where
the graph intercepts the x-axis by setting y equal to zero and solving for x. In this problem
by setting y equal to zero the equation becomes:
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which we more conventionally flip around to:
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There are a couple of ways that we could solve this equation ... the quadratic formula
being one method. But in this case an either method is factoring. Notice that both of the
terms on the left side contain an x. So you can factor x out to get:
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Notice that the left side will equal the right side if either of the factors is a zero.
So in this case, one at a time you set the two factors equal to zero and solve for the
value of x:
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That's an easy one. Next set the second factor equal to zero:
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Subtract 6 from both sides to get rid of the +6 on the left side and the equation
becomes:
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Solve for x by dividing both sides by -3 which is the multiplier of x. With this division
you get:
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Combine these x values with their corresponding y value of zero, and the two points where
the graph crosses the x-axis are (0,0) and (+2, 0). Between those points the graph rises
to its peak. In fact, the peak will occur when x is at the halfway point between 0 and +2
on the x-axis. This midpoint is at x = 1. All you have to do to find the y value at
the peak is return to the equation for y and let x = 1.
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Substitute 1 for x and you get:
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So the vertex point is (1, 3).
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Notice that the problem asked for the intercepts without specifying whether they are the
x or y intercepts. All you have to do to find the intercept on the y-axis is to set the
value of x in the equation equal to zero because any point on the y-axis has a corresponding
x value of 0. In this problem, start with:
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and setting x equal to zero results in:
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So the y-intercept in this case is (0, 0), but we knew that already. The method here is
the same that you would use in another problem ... set x equal to zero to find the value
of y where the graph crosses the y-axis.
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Hope this helps you to understand this problem a little better and to grasp some of the basic
concepts of graphing quadratic equations.
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