Vertex

Intuition and core definition

The vertex of a parabola is its turning point and lies on the axis of symmetry. For $y=a(x-h)^2+k$, the vertex is $(h,k)$. If $a>0$, it is a minimum; if $a<0$, it is a maximum. The vertex is a point, whereas the axis is the line $x=h$.

Notation, language, and conditions

For standard form $y=ax^2+bx+c$, $h=-b/(2a)$ and $k=f(h)$. Completing the square converts standard to vertex form. In an application, the vertex’s coordinates carry input and output units and must be checked against the model’s domain.

Why this idea matters

The vertex is a parabola's turning point and packages its extremum value with the axis of symmetry.

A dependable method

1. Identify the quadratic’s form and coefficient $a$.
2. Read $(h,k)$ from vertex form, or compute $h=-b/(2a)$ in standard form.
3. Evaluate the function at $h$ to find $k$.
4. Classify the point as maximum or minimum using the sign of $a$.
5. Check symmetry with inputs $h-d$ and $h+d$ and interpret the coordinates.

Worked example

Common mistakes