Vertex, axis and concavity

From the equation $y=ax^2+bx+c$ one immediately obtains the elements needed to draw the curve: the vertex, the axis of symmetry and the direction of the concavity.

Property — Vertex, axis and concavity

• Vertex: $V=\left(-\dfrac{b}{2a},\ -\dfrac{\Delta}{4a}\right)$
• Axis of symmetry: $x=-\dfrac{b}{2a}$
• $a>0$: concavity upwards; $a<0$: concavity downwards.

For example the parabola $y=x^2-2x-3=(x-1)^2-4$ has $a=1>0$ (concavity upwards), vertex $V(1,-4)$ and axis of symmetry $x=1$. The graph is symmetric with respect to this axis.

Graph of the parabola $y=x^2-2x-3=(x-1)^2-4$, with vertex $V(1,-4)$ and axis of symmetry $x=1$.

Try it yourself: drag the sliders $a$, $b$, $c$ and watch how the vertex $V$ (red) and the axis of symmetry (dashed) move, and how the sign of $a$ flips the concavity.

Drag the sliders $a$, $b$, $c$: the vertex, axis of symmetry and concavity update in real time.

Links

Topics: Parabola
Concepts: Axis of symmetry · Concavity · Discriminant · Vertex
Functions: Parabola
Methods: Parabola vertex
Skills: Drawing a graph · Using formulae