Question

What is the equation of the parabola shown below, given a focus at F(1, 2) and a directrix of x = 5? In addition, identify the vertex and the equation of the axis of symmetry for the parabola.

Answer 1

The distance from the focus F(1,2) to P(x,y) is the same as the distance from P(x,y) to the directrix, x=5, so
$$
\sqrt{(x-1)^2+(y-2)^2}=\sqrt{(x-5)^2}=\\
x = -y^2/8+y/2+5/2
$$
You can get all the properties of this horizontal parabola from this equation:

http://www.wolframalpha.com/input/?i=8+x%2B%28y-4%29+y+%3D+20
h https://en.wikipedia.org/wiki/Parabola#Dimensions_of_parabolas_with_axes_of_symmetry_parallel_to_the_y-axis

Just exchange x and y in the wikipedia discussion, since your parabola is parallel to the x-axis not the y-axis

Note that the axis of symmetry is the y-coordinate of the vertex.