Quadratic function in canonical form. Canonical form of the quadratic function

It is known that the quadratic function is determined by the following expression:

f(x)=ax2+bx+c

However, if we do some algebraic manipulations of the right side of this equality, through the process of completing squares.

(f(x)=ax2+bx+c (Placing the term The in evidence)

Note that the two highlighted parcels can be used for the square completion process:

So we just add and subtract the last term in our function f (x) (Process to complete squares).

Thus, completing the square in the function, we have:

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This expression can also be written as follows:

Calling from:

Note that:

So, another way to write the quadratic function canonically is:

f(x)=a(x-m)2+k

Let's do an example in which we should write any quadratic function:

f(x)=x2-3x-7

We must highlight the coefficients and determine the values ​​of m and k:


By Gabriel Alessandro de Oliveira
Graduated in Mathematics
Brazil School Team

Would you like to reference this text in a school or academic work? Look:

OLIVEIRA, Gabriel Alessandro de. "Quadratic function in canonical form"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/funcao-quadratica-na-forma-canonica.htm. Accessed on June 29, 2021.

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