Identify the vertex, focus, axis of symmetry, and directrix for the parabola (x - 1)^2 = 8(y - 2).

Step-by-step solution

1. 1

   Decide which of the two standard forms applies. The squared variable is $x$, so the parabola opens vertically. That fixes the template as

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

   If $y$ were the squared variable the parabola would open sideways and every formula below would swap roles.

2. 2

   Read off $h$, $k$ and $p$ by matching term for term. Comparing $(x-1)^2 = 8(y-2)$ with the template gives $h = 1$, $k = 2$ and $4p = 8$, so

   $p = 2$

   Since $p > 0$ the parabola opens upward.

3. 3

   The vertex is $(h,k)$. Straight from the match, the vertex is $(1, 2)$.

4. 4

   The focus sits $p$ units from the vertex along the axis. For a vertical parabola the focus is $(h,\; k+p)$:

   $(1,\; 2+2) = (1, 4)$

5. 5

   The directrix sits $p$ units on the other side, and the axis passes through both. The axis of symmetry is the vertical line through the vertex, $x = h = 1$. The directrix is the horizontal line $y = k - p = 2 - 2 = 0$, i.e. the $x$-axis.

6. 6

   Verify with the focus–directrix definition. Take $(5, 4)$: $(5-1)^2 = 16 = 8(4-2)$, so the point really is on the parabola. Its distance to the focus $(1,4)$ is $4$, and its distance to the directrix $y = 0$ is also $4$. Equal distances confirm both the focus and the directrix.