The vertex form of a parabola is $y\&equals; a{\left(x-h\right)}^{2 } \&plus;k$  where $\left(h\&comma;k\right)$ is the vertex. The variable $a$ has the same value and function as the variable in the standard form. If $a \&gt; 0$ (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 $y\&equals;{\mathrm{ax}}^2 \&plus; \mathrm{bx} \&plus; c$ to vertex form, you must complete the square.

Example of completing the square:

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