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### Finding the Equation of a Parabola

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Date: 04/28/2001 at 17:57:42
From: Rakesh
Subject: Finding equation of a parabola

This is the question:

The zeros of a quadratic relation are 0 and 6. The relation has a
minimum value of -9. Find the equation of the parabola.

I have already graphed the parabola, but I do not know how to find the
equation of a parabola. All of the questions I've done already provide
you with the equation. How can you do it?
```

```
Date: 04/28/2001 at 21:53:05
From: Doctor Scott
Subject: Re: Finding equation of a parabola

Hi Rakesh.

This is a very interesting problem. Remember that the zeros of the
equal to zero. But, by the Factor Theorem, if 6 is a zero of a
polynomial function, then (x-6) is a factor of the polynomial. So,
since your zeros are 0 and 6, that means that (x-0) and (x-6) are two

So, if P(x) is the polynomial function, in factored form it would look
like this: a(x-0)(x-6) = P(x). This is just a(x^2-6x) = P(x). We need
the a because the parabola could be "stretched" without changing its
zeros.

So, we now have P(x) = ax^2 - 6ax + 0 as our polynomial. The
information given says that the vertex occurs when y = -9 (since
that's the minimum value). So, since the minimum is at the vertex, we
know that the x-coordinate of the vertex of a quadratic function of
the form ax^2 + bx + c is given by -b/(2a). So, for our quadratic, the
x-coordinate of the vertex is at (6a/2a) or x = 3. So, the point
(3,-9) must lie on the function:

-9 = a(3^2) - 6a(3) + 0
-9 = 9a - 18a
-9 = -9a
so
a = 1

So, the quadratic must be y = x^2 - 6x + 0

- Doctor Scott, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

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