The standard form of the equation for a parabola is y = a x^2. You should be very familiar with the graph of y = a x^2 and the characteristics of this graph.
The focus of the resulting parabola has focus (0, a/4) and directrix (0, -a/4).
In general the vertex and directric lie on the axis of symmetry of the parabola, on opposite sides of the vertex, with the parabola opening in the direction of the ray originating at the vertex and passing through the focus (roughly speaking, the parabola opens on the same 'side' of the vertex as the focus).
If x and y are replaced by x - h and y - k the result is a shift of h units in the x direction and k units in the y direction, so that the equation (y - k) = a ( x - h)^2 has its vertex at the point (h, k). The focus and directrix will be displaced from the vertex in the y direction by a/4 units and -a/4 units, respectively, so that the focus is at (h, k+a/4) and the directrix is the line y = k - a/4.
If the equation is x = a y^2, then the roles of x and y are reversed, with the parabola opening to the right or left rather than up or down. The focus is again displaced from the vertex a / 4 units, the directrix -a/4 units.
If the equation is (x - h) = a ( y - k)^2 then the vertex is again (h, k), with the focus displaced a/4 units in the x direction from the vertex and the directrix -a/4 units. The focus is therefore (h + a/4, k) and the directrix is the line x = h - a/4.