Quadratic functions

Quadratic equations. Provided $a \neq 0$, the solutions of a quadratic equation $a x^2 + b x + c = 0$ are given by

$\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (3)

Parabola. The graph of an equation  $y = ax^2 + c$ with leading coefficient $a \neq 0$ represents a parabola. It opens upward if $a > 0$, or downward if $a < 0$. The lowest point of an upward parabola is called the vertex of the parabola. Likewise, the vertex of a downward parabola is the highest point.

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Quadratic functions. A function $f$ is called a quadratic function if

$\displaystyle f(x) = a x^2 + b x + c$ (4)

with leading coefficient $a \neq 0$. The quadratic function $f$ represents a parabola; it opens upward if $a > 0$, or downward if $a < 0$. When $(b^2 - 4ac) \ge 0$, the graph of $f$ has $x$-intercepts; the exact values of $x$-intercepts are given by (3). There is no $x$-intercept if $(b^2 - 4ac) < 0$.

Standard equations of parabola. The quadratic function (4) is also written in the form

$\displaystyle f(x) = a \left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$ (5)

The formula (5) is called a standard equation of a parabola with vertex

$\displaystyle \left(-\frac{b}{2a},\: c - \frac{b^2}{4a}\right).
$

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Department of Mathematics
Last modified: 2005-09-29