Main Concept

Quadratic functions $f\left(x\right)$ can be written in three forms.

Factored form, the product of a constant and two linear terms:

$a\cdot \left(x-p\right)\cdot \left(x-q\right)$     or     $a\cdot {\left(x-p\right)}^2$

The parameters $p$ and $q$ are the roots of the function (the x-intercepts of the graph $y\=f\left(x\right)$). Converting a quadratic function to factored form is called factoring.

Expanded form, a sum of terms, each of which may be a product of a constant and some variables:

$a x^2\+b x\+c$

The parameter $c$ is the y-intercept, while the parameter $b$ is the slope of the tangent at 0. Converting a quadratic function to expanded form is called expanding.

Standard form, the sum of a constant term, $k$ and a constant, $a$ times the square of a linear term:

$a\cdot {\left(x-h\right)}^2\+k$

The vertex of the graph is located at the point $\left(h\, k\right)$. Converting a quadratic function to standard form is called completing the square.

In each case the parameter $a$ determines the vertical stretching of the graph.

a  =

p =

q =

Expanded form: 

Standard form: 

Factored form:

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