Quadratic function: Difference between revisions

Revision as of 14:37, 11 May 2014

Definition

A quadratic function is a function of the form:

$x \mapsto a x^{2} + b x + c$

where $a, b, c$ are real numbers and $a \neq 0$ . In other words, a quadratic function is a polynomial function of degree two.

Key invariants

Expression	Name	Significance in the case $a, b, c \in R$
$b^{2} - 4 a c$	(unnormalized) discriminant	The discriminant is positive (i.e., $b^{2} - 4 a c > 0$ ) iff the quadratic has two distinct real roots The discriminant is zero (i.e., $b^{2} - 4 a c = 0$ ) iff the quadratic has a real root of multiplicity two The discriminant is negative (i.e., $b^{2} - 4 a c < 0$ ) iff the quadratic has no real roots
$a$	leading coefficient	Leading coefficient is positive (i.e., $a > 0$ ) iff that the function approaches infinity as $x \to \infty$ and as $x \to - i n f t y$ Leading coefficient is negative (i.e., $a < 0$ ) iff that the function approaches infinity as $x \to \infty$ and as $x \to - i n f t y$
$- b / a$	sum of roots	If the roots are $α, β$ (counted with multiplicity, and they need not be real roots), then the polynomial is $a (x - α) (x - β)$ and the sum of roots $α + β$ is $- b / a$ .
$c / a$	product of rots	If the roots are $α, β$ (counted with multiplicity, and they need not be real roots), then the polynomial is $a (x - α) (x - β)$ and the product of roots $α β$ is $c / a$ .

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 ==Key invariants==
+{| class="sortable" border="1"
-The [[discriminant]] of a quadratic function is the quantity <math>b^2 - 4ac</math>. The sign of the discriminant determines the nature of the factors and zeros of the quadratic function.
+! Expression !! Name !! Significance in the case <math>a,b,c \in \R</math>
+|-
+| <math>b^2 - 4ac</math> || (unnormalized) [[discriminant]] || The discriminant is positive (i.e., <math>b^2 - 4ac > 0</math>) iff the quadratic has two distinct real roots<br>The discriminant is zero (i.e., <math>b^2 - 4ac = 0</math>) iff the quadratic has a real root of multiplicity two<br>The discriminant is negative (i.e., <math>b^2 - 4ac < 0</math>) iff the quadratic has no real roots
+|-
+| <math>a</math> || leading coefficient || Leading coefficient is positive (i.e., <math>a > 0</math>) iff that the function approaches infinity as <math>x \to \infty</math> and as <math>x \to -infty</math><br>Leading coefficient is negative (i.e., <math>a < 0</math>) iff that the function approaches infinity as <math>x \to \infty</math> and as <math>x \to -infty</math>
+|-
+| <math>-b/a</math> || sum of roots || If the roots are <math>\alpha,\beta</math> (counted with multiplicity, and they need not be real roots), then the polynomial is <math>a(x - \alpha)(x - \beta)</math> and the sum of roots <math>\alpha + \beta</math> is <math>-b/a</math>.
+|-
+| <math>c/a</math> || product of rots || If the roots are <math>\alpha,\beta</math> (counted with multiplicity, and they need not be real roots), then the polynomial is <math>a(x - \alpha)(x - \beta)</math> and the product  of roots <math>\alpha\beta</math> is <math>c/a</math>.
+|}