Understanding vertex of a parabola

Every parabola has a single turning point called the vertex.
If the parabola opens upward the vertex is the minimum; if it opens downward the vertex is the maximum.

The vertex sits on the axis of symmetry, the vertical line that splits the parabola into mirror halves.
Because the two $x$-intercepts are mirror images, the axis is exactly halfway between them.
For intercepts at $x = 1$ and $x = 5$, the vertex is at $x = \frac{1 + 5}{2} = 3$.

Once you have the $x$-coordinate, the $y$-coordinate comes from substituting it into the equation.
Plugging that $x$ back in gives the height of the turning point, which is the function's maximum or minimum value.
So a maximum value of $12$ at $x = 1$ means the vertex is $(1, 12)$.

When the equation is in vertex form $y = a(x - h)^2 + k$, you can read the vertex directly as $(h, k)$.
The axis of symmetry is also useful for finding a matching point: any point on the parabola has a mirror twin the same distance on the other side of the axis.
A point at $x = 2$ on the axis $x = 4$ has its twin at $x = 6$.

A parabola has $x$-intercepts at $x = -2$ and $x = 4$. What is the $x$-coordinate of its vertex?
The vertex lies midway between the intercepts.
So $x = \frac{-2 + 4}{2} = 1$.

A downward parabola reaches a maximum value of $8$ at $x = 3$. What is its vertex?
The vertex is the turning point, here the maximum, at $x = 3$.
Since the maximum value is $8$, the vertex is $(3, 8)$.

The vertex of $f(x) = 2x^2 - 8x + 5$ is at $(2, -3)$, and point A on the parabola is at $x = 1$. What is the $x$-coordinate of its mirror point B?
A and B are symmetric about the axis $x = 2$, and A is $1$ unit to the left.
So B is $1$ unit to the right, at $x = 3$.