Tool to find the vertex form of a polynomial. The vertex form of a quadratic polynomial is an expressed form where the variable x appears only once.

Vertex Form of a Quadratic - dCode

Tag(s) : Symbolic Computation

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

The canonical form of a degree 2 polynomial (quadratic reduced form), is a simplified representation of this polynomial obtained by completing the square of the original polynomial (square completion).

A quadratic polynomial $ p(x)=ax^2+bx+c $ (with $ a $ not null) can be written in a canonical form $ p(x)=a(x−\alpha)^2+\beta $ with $ \alpha $ and $ \beta $ real numbers (the coefficient $ a $ is the same as in the first equation).

To find the canonical form of a polynomial of degree 2 of type $ p(x) = ax^2 + bx + c $ use the formula:

$$ p(x) = a \left( \left( x + \frac{b}{2a} \right)^2 \right) + \left( \frac{-b^2}{4a} + c \right) $$

Note: the polynomial is indeed in the format $ p(x) = a(x−\alpha)^2 + \beta $ with $ \alpha = \frac{-b}{2a} $ and $ \beta = c-\frac{b^2}{4a} $

The principle is to factorize the second degree coefficient to remove the first degree coefficient.

__Example:__ The polynomial of order two $ x^2-4x+6 $ can be written $ (x-2)^2+2 $

dCode converter to vertex form calculator uses multiple methods to find the canonical form of a polynomial function of second degree, including the completion of the square or Tschirnhaus transformation (both using mathematical expression factorization).

Among other uses, the canonical form makes it possible to determine the coordinates of the extremum of the polynomial function $ p(x) = ax^2 + bx + c = a(x−\alpha)^2 + \beta $. Indeed, $ \beta $ is an extremum reached when $ x = \alpha $. The extremum has coordinates $ ( \alpha, \beta ) $ i.e. $ \left( \frac{-b}{2a}, c-\frac{b^2}{4a} \right) $

It also makes it easier to determine the properties of the polynomial, such as the vertex of the associated parabola, the axis of symmetry, and the maximum or minimum values.

It is possible to generalize the approach to degrees $ n $ (superior to $ 2 $) by removing the term of degree $ n-1 $ using appropriate factors.

The Tschirnhaus method consists of performing a change of variable to eliminate the linear term in the polynomial. This then simplifies the process of completing the square and leads to the canonical form.

For a polynomial $$ p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0 $$ the Tschirnhaus transformation consists in writing it as $$ p(x) = k x^n + c $$

The result is called **depressed polynomial** and the technique is **polynomial depression**.

dCode retains ownership of the "Vertex Form of a Quadratic" source code. Except explicit open source licence (indicated Creative Commons / free), the "Vertex Form of a Quadratic" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Vertex Form of a Quadratic" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Vertex Form of a Quadratic" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Vertex Form of a Quadratic" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Vertex Form of a Quadratic* on dCode.fr [online website], retrieved on 2024-06-14,

vertex,form,polynomial,square,quadratic,factorization,factor,factorize,second,2,function,parabola

https://www.dcode.fr/vertex-form-quadratic

© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback