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

Step-by-step solution

1. 1

   Match the equation to a standard form. The squared variable is $x$, so this is a vertical parabola and the template is $(x-h)^2=4p(y-k)$. Which variable is squared decides the whole orientation, so identify it before anything else.

2. 2

   Read off h, k and 4p. Comparing $(x-1)^2=8(y-2)$ with the template gives $h=1$, $k=2$ and $4p=8$, hence $p=2$. Because $p>0$, the parabola opens upward.

3. 3

   State the vertex. The vertex is $(h,k)=(1,2)$; it is the point where the squared term is zero.

4. 4

   Locate the focus. For a vertical parabola the focus sits $p$ units from the vertex along the axis, in the direction of opening: $(h,k+p)=(1,2+2)=(1,4)$.

5. 5

   Write the axis of symmetry and directrix. The axis is the vertical line through the vertex, $x=h=1$. The directrix is the horizontal line $p$ units on the opposite side: $y=k-p=2-2=0$, which is the $x$-axis here.

6. 6

   Check with the focus-directrix definition. Take the point $(5,4)$, which satisfies $(5-1)^2=16=8(4-2)$. Its distance to the focus $(1,4)$ is $4$, and its distance to the line $y=0$ is also $4$. Equal distances confirm the focus and directrix.