Question

Using a directrix of y = −3 and a focus of (2, 1), what quadratic function is created?

Answer 1

@Answers101

Answer 2

@ganeshie8

Answer 3

you there?

Answer 4

By a quadratic function I'll guess they speak of a parabola.
A parabola is defined as the set of points equidistant from a fixed point called "foci" and a line called "directrix".
Now, this will mean, that if we have a foci \(F( \alpha, \beta)\) and a directrix with an equation of d) y=a where "a" can be any real number.
We will then define a generic point T(x,y) that satisfied the conditions:
\[dist(F,T)=dist(T,d)\]
Now there are equations that allows us to calculate the distance from a point to a line, but we will define a point belonging to the directrix d) y=a as the point \(U(k,a)\), not that U will beling in the directrix because it has a y-component of "a" and "k" is just an arbitrary x-value.
Now, let's rewrite those distances:
\[dist(F,T)=dist(T,U)\]
Rearranging to the distance formulas:
\[\sqrt{(x- \alpha)^2+(y- \beta)^2}=\sqrt{(k-x)^2+(a-y)^2}\]
And from here on it's just a matter of simplifying and we will end up with a equation with the form:
\[y=ax^2+bx+c\]
And that is a full quadratic and the equation of the parabola.

So, recapping the steps you have to take in order to find the equation of a parabola given the foci and the directrix is:
- Establish the definition of parabola
- Define a generic point (x,y)
- Spot a point belonging in the directrix
- Utilize the distance formula
- simplify the expression.

Answer 5

thx nice work.

Answer 6

there is an easier way

Answer 7

Indeed there is a practical but still sound shortcut towards finding the equation of this parabola.

It involves "p," which is the distance from the vertex to the directrix of a parabola, OR the distance between the vertex and the focus. Your job is to determine the value of p. I'd suggest you graph all of the given info, incl. the focus, the directrix and the location of the vertex.

If (h,k) represents the vertex, then the equation will be

y-k=4p(x-h)^2

If the parabola opens up, this is the correct equation;
if the parabola opens down, then place a (-) sign in front of 4p(x-h)^2.