Function of the second degree

In Mathematical elementary, a function of the second degree is a function definite on $\backslash \; R$ by: $F\; (X)\; =\; ax^2\; +\; bx\; +\; C\; \backslash ,$ where has, B and C is realities ( has not no one) called the coefficients.

$ax^2$ is the term of the second degree, $bx$ is the term of the first degree and $c$ is the constant term.

After the functions closely connected, the functions of the second degree or trinomial of the second degree constitute the second field of study of the functions polynomials.

Canonical form

A function of the second degree has a reduced form or forms canonical which makes it possible to highlight its relation with the Fonction square:

$F\; (X)\; =\; has\; \backslash \; left\; (X\; +\; \backslash \; frac\; \{B\}\; \{2a\}\; \backslash \; right)\; ^2\; -\; \backslash \; frac\; \{b^2-4ac\}\; \{4a\}$ One can notice that $f\; \backslash \; left\; (\backslash \; frac\; \{-\; B\}\; \{2a\}\; \backslash \; right)\; =\; \backslash \; frac\; \{-\; b^2+4ac\}\; \{4a\}$

Example: if $F\; (X)\; =\; 2x\; ^2\; +\; 4x\; -\; 5\; \backslash ,$, one notice that $\backslash \; frac\; \{-\; B\}\; \{2a\}\; =\; -1$ and that $f\; (-\; 1)\; =\; -7\; \backslash ,$ thus $f\; (X)\; =\; 2\; (X\; +\; 1)\; ^2\; -\; 7\; \backslash ,$

Discriminant : One calls Discriminant the number $\backslash \; Delta\; =\; b^2-4ac$. One obtains then:

$F\; (X)\; =\; has\; \backslash \; left\; (X\; +\; \backslash \; frac\; \{B\}\; \{2a\}\; \backslash \; right)\; ^2\; -\; \backslash \; frac\; \{\backslash \; Delta\}\; \{4a^2\}$ From this canonical form all the results concerning the function result from the second degree.

Roots

See also: Quadratic equation

It is said that $r$ is a root of $f$ if $f\; (R)\; =\; 0$.

It is shown that

• if $\backslash \; Delta\; >\; 0$ then $f$ has two roots which are $r\_1\; =\; \backslash \; frac\; \{-\; B\; -\; \backslash \; sqrt\; \{\backslash \; Delta\}\}\; \{2a\}$ and $r\_2\; =\; \backslash \; frac\; \{-\; B\; +\; \backslash \; sqrt\; \{\backslash \; Delta\}\}\; \{2a\}$
• if $\backslash \; Delta\; =\; 0$ then $f$ has a double root which is $r\_0\; =\; \backslash \; frac\; \{-\; B\}\; \{2a\}$
• if then $f$ does not have a root in the together $\backslash \; R$ but it has of it in the together $\backslash \; mathbb\; \{C\}$.

Case of the obvious root

That is to say trinomial of the second degree, such as $f\; (X)\; =\; ax^2\; +\; bx\; +\; C\; \backslash ,$.
If $a+b+c=0\; \backslash ,$ then $f\; \backslash ,$ admits at least an obvious root equalizes with $1\; \backslash ,$.

Operations on the roots

One notes $S\; \backslash ,$ the sum of the roots, and $P\; \backslash ,$ the product of the roots of a polynomial of the second degree. One can thus write:

$S=\; \backslash \; frac\; \{-\; B\}\; \{has\}$ and $P=\; \backslash \; frac\; \{C\}\; \{has\}$

Factorization

If the discriminant is not negative, one can write the function of the second degree in the form of a product of functions of the first degree.

• if $\backslash \; Delta\; >\; 0\; \backslash ,$ then $f\; (X)\; =\; has\; (X\; -\; r\_1)\; (X\; -\; r\_2)\; \backslash ,$
• if $\backslash \; Delta\; =\; 0\; \backslash ,$ then $f\; (X)\; =\; has\; (X\; -\; r\_0)\; ^2\; \backslash ,$

Study of sign

See also: Inequation of the second degree

Preceding factorization (or the absence of factorization) makes it possible to build the table of sign of $f\; (X)\; \backslash ,$. Actually, there exist 6 cases of figure according to whether $a\; \backslash ,$ is positive or negative and according to whether $f\; \backslash ,$ has 2,1 or 0 roots. These six cases of figure are summarized in a method: “ the sign of trinomial coincides with that of $a\; \backslash ,$. except between the roots”

Chart

The canonical form of the function $f\; \backslash ,$ makes it possible to notice that its representative curve is the image of the curve of equation $y\; =\; ax^2\; \backslash ,$ by a translation of vector $\backslash \; vec\; U\; \backslash \; left\; (\backslash \; frac\; \{-\; B\}\; \{2a\},\; F\; \backslash \; left\; (\backslash \; frac\; \{-\; B\}\; \{2a\}\; \backslash \; right)\; \backslash \; right)$.

The representative curve is thus always a Parabole. Its top is the point $S\; \backslash \; left\; (\backslash \; frac\; \{-\; B\}\; \{2a\},\; F\; \backslash \; left\; (\backslash \; frac\; \{-\; B\}\; \{2a\}\; \backslash \; right)\; \backslash \; right)$ and its axis of symmetry is the line of equation $X\; =\; \backslash \; frac\; \{-\; B\}\; \{2a\}\; \backslash ,$.

The six parabolas below illustrate the six cases of figures of the study of sign, according to the sign of $a\; \backslash ,$ and that of $\backslash \; Delta$. It is pointed out that $f\; \backslash \; left\; (\backslash \; frac\; \{-\; B\}\; \{2a\}\; \backslash \; right)\; =\; -\; \backslash \; frac\; \{\backslash \; Delta\}\; \{4a\}\; \backslash ,$

\ hline X & - \ infty & & + \ infty \ \ \ hline F (X) & & + & \ \ \ hline \end{array} | align=center |
|- | valign=center | $\backslash \; Delta=0$ | align=center |
| align=center |
|- | valign=center | $\backslash \; Delta>0$ | align=center |
| align=center |
|}

Direction of variation

Lastly, one can deduce from this curve the direction of variation of $f\; \backslash ,$:

• If $a\; >\; 0\; \backslash ,$, the function is decreasing then increasing and reaches its minimum in $-\; b/2a\; \backslash ,$;
• If , the function is increasing then decreasing and reaches its maximum in $-\; b/2a\; \backslash ,$

This result is confirmed by the calculation of derived from $f\; \backslash ,$ which is $f\text{'}\; (X)\; =\; 2ax\; +\; B\; \backslash ,$.

Bonds

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