how to find the vertex of a parabola

Vertex of a Parabola

Main Concept

The vertex of a parabola is the point at the intersection of the parabola and its line of symmetry.

For a parabola whose equation is given in standard form $y = a x^{2} + b x + c$ , the vertex will be the minimum (lowest point) of the graph if $a > 0$ and the maximum (highest point) of the graph if $a < 0$ .

Vertex form

The vertex form of a parabola is $y = a {(x - h)}^{2} + k$ where $(h, k)$ is the vertex. The variable $a$ has the same value and function as the variable in the standard form. If $a > 0$ (positive) the parabola opens up, and if $a < 0$ (negative) the parabola opens down.

The vertex form of a parabola is usually not provided. To convert from the standard form $y = {ax}^{2} + bx + c$ to vertex form, you must complete the square.

Example of completing the square:

1. Factor the leading coefficient out of the first two terms.	$4 x^{2} + 8 x + 4 =$	$4 (x^{2} + 2 x) + 4$
2. Complete the square by addition and subtracting the magic number - the square of half the coefficient of $x$ .	$=$	$4 (x^{2} + 2 x + 1 - 1) + 4$
3. Factor out the constant ${(\frac{b}{2 a})}^{2}$ .	$=$	$4 (x^{2} + 2 x + 1) + 4 - 4$
4. Factor the perfect square.	$=$	$4 {(x^{} + 1)}^{2}$

Use the sliders to change the vertex $(h, k)$ and observe how your changes affect the graph of the parabola.

Observe that when the graph opens up, the range of the corresponding quadratic function is $[k, \infty)$ , while if it opens down, the range is $(- \infty, k]$ .