Question 383953: Hello! Time for homework!!
I am trying to solve:
Identify the equation of a quadratic function with the vertex (0,0.5), that passes through the point (-2, -15.5) and opens downward.
Thanks so much!!
Denise
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
In order to uniquely determine a quadratic function, you need three points. But fortunately one of your given points was the vertex. That means we can determine the 3rd point by symmetry. That is to say that since the vertex is at (0,0.5), the axis of symmetry must be  . From that fact we can determine that if there is a point on the parabola at (-2,-15.5), then by symmetry (2, -15.5) must also be on the parabola.
We are looking for a quadratic function of the form:
\ =\ ax^2\ +\ bx\ +\ c)
The fact that (0,0.5) is on the graph means that
\ =\ 0.5)
which is to say:
^2\ +\ b(0)\ +\ c\ =\ 0.5)
from which it is clear that 
Likewise, using the point (-2,-15.5) we can say:
^2\ +\ b(-2)\ +\ c\ =\ -15.5)
and using (2,-15.5) we can say:
^2\ +\ b(2)\ +\ c\ =\ -15.5)
Substituting and simplifying leaves us with the following pair of two-variable linear equations:
Solving the system (left as an exercise for the student) results in:
from which we can substitute into:
to write:
\ =\ -4x^2\ +\ 0.5)
Check:
John
My calculator said it, I believe it, that settles it
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