The vertex form of a quadratic function is written as $f(x) = a(x - h)^2 + k$. This structural setup lets us immediately read off the most important landmark of a parabola: its vertex, which is located at the coordinates $(h, k)$. Because the parabola is perfectly symmetrical, it has a vertical line running straight through its center called the axis of symmetry, defined by the equation $x = h$.

The behavior of the parabola is also controlled by the coefficient $a$. If $a$ is positive, the parabola opens upward like a smile; if $a$ is negative, it opens downward. The magnitude of $a$ stretches or compresses the curve. When $|a| > 1$, the function values grow faster, making the parabola narrower than the parent function $f(x) = x^2$. Conversely, if $0 < |a| < 1$, the curve is wider and flatter.