ANSWER
![f(x)= \frac{1}{4} {(x - 1)}^{2}](https://tex.z-dn.net/?f=f%28x%29%3D%20%5Cfrac%7B1%7D%7B4%7D%20%7B%28x%20-%201%29%7D%5E%7B2%7D%20)
EXPLANATION
Since the directrix is
![y = - 1](https://tex.z-dn.net/?f=y%20%3D%20-%201)
the axis of symmetry of the parabola is parallel to the y-axis.
Again, the focus being,
![(1,1)](https://tex.z-dn.net/?f=%281%2C1%29)
also means that the parabola will open upwards.
The equation of parabola with such properties is given by,
![{(x - h)}^{2} = 4p(y - k)](https://tex.z-dn.net/?f=%20%7B%28x%20-%20h%29%7D%5E%7B2%7D%20%3D%204p%28y%20-%20k%29)
where
![(h,k)](https://tex.z-dn.net/?f=%28h%2Ck%29)
is the vertex of the parabola.
The directrix and the axis of symmetry of the parabola will intersect at
![(1, - 1)](https://tex.z-dn.net/?f=%281%2C%20-%201%29)
The vertex is the midpoint of the focus and the point of intersection of the axis of the parabola and the directrix.
This implies that,
![h = \frac{1 + 1}{2} = 1](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B1%20%2B%201%7D%7B2%7D%20%3D%201)
and
![k = \frac{ - 1 + 1}{2} = 0](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B%20-%201%20%2B%201%7D%7B2%7D%20%3D%200)
The equation of the parabola now becomes,
![(x - 1) ^{2} = 4p(y - 0)](https://tex.z-dn.net/?f=%28x%20-%201%29%20%5E%7B2%7D%20%3D%204p%28y%20-%200%29)
![|p| = 1](https://tex.z-dn.net/?f=%20%7Cp%7C%20%3D%201)
Thus, the distance between the vertex and the directrix.
This means that,
![p = - 1 \: or \: 1](https://tex.z-dn.net/?f=p%20%3D%20-%201%20%5C%3A%20or%20%5C%3A%201)
Since the parabola opens up, we choose
![p = 1](https://tex.z-dn.net/?f=p%20%3D%201)
Our equation now becomes,
![{(x - 1)}^{2} = 4(1)(y - 0)](https://tex.z-dn.net/?f=%20%7B%28x%20-%201%29%7D%5E%7B2%7D%20%3D%204%281%29%28y%20-%200%29)
This simplifies to
![{(x - 1)}^{2} = 4y](https://tex.z-dn.net/?f=%20%7B%28x%20-%201%29%7D%5E%7B2%7D%20%3D%204y)
or
![y = \frac{1}{4} {(x - 1)}^{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%7B%28x%20-%201%29%7D%5E%7B2%7D%20)
This is the same as,
![f(x)= \frac{1}{4} {(x - 1)}^{2}](https://tex.z-dn.net/?f=f%28x%29%3D%20%5Cfrac%7B1%7D%7B4%7D%20%7B%28x%20-%201%29%7D%5E%7B2%7D%20)
The correct answer is D .