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is a nonzero constant in the quadratic equation above. When graphed in the 
-plane, the equation is a parabola with a vertex at point 
. What is the value of 
?
, where the vertex is at point 
. To employ vertex form, recognize that you will need to get the quadratic to be a perfect square, using the 
term to balance. Here if you were to multiply out the quadratic in the given equation, you'd have:
as the middle term is 
, which factors to the perfect square 
. So in order to put this equation in vertex form, you need to convert 
into 
, where 
equals out the step you took to turn 
within the parentheses to 
. To do that step, you added 
(remember that the entire parenthetical is multiplied by 
). So you need to subtract 
on the outside of the parentheses to keep the equation balanced. That means that you now have:
, or when factored directly to vertex form 
. 
, or more directly that 
. So the correct answer is 
.
, will be at 
and 
, as those are the points at which each parenthetical term equals zero, so 
would in turn equal zero. Because parabolas are symmetrical, that tells you that the vertex will lie halfway between those two points, at 
. If you then plug in the 
-coordinate of the vertex, 
, you'll get the corresponding 
-coordinate:
.
and has a vertex at point 
. Which of the following is true of a parabola defined by the equation 
?
, where the point 
is the vertex of the parabola, and the sign of 
determines which way the parabola opens. A negative 
opens downward, and a positive 
opens upward. Here the given parabola has established that the value of 
without the negative sign would point upward, so multiplying 
by 
will flip the parabola to point downward. And in Vertex Form, 
and 
are 
and 
, so the vertex will be at 
.
Teresa
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