Question

For medal and fan please help (:
Write an equation of a parabola that opens upward, has a vertex at the origin, and a focus at (0, 1).

Answer 1

the possible answers are \[y=-\frac{ 1 }{ 4 }x^2, y=\frac{ 1 }{ 4 }x^2, y=\frac{ 1 }{ 12}x^2, y=-\frac{ 1 }{ 12}x^2\]

Answer 2

Vertex form for a parabola is
\[y = a(x-h)^2 + k\]where the vertex is at \((h,k)\)
That means our equation is of the form
\[y = a(x-0)^2 +0\]or\[y = ax^2\]So far, that doesn't help us at all :-)

Parabolas that open upward have a value of \(a>0\). We still have 2 choices left.

We can find \(a\) by \[a = \frac{1}{4p}\]where \(p\) is the distance from the focus to the vertex. That distance for this parabola is the distance from (0,1) to (0,0), or 1.\[a = \frac{1}{4*1} = \frac{1}{4}\]

Therefore, our answer is \[y = \frac{1}{4}x^2\]