Precalculus Review

Calculus I > Precalculus Review

Quadratic Functions

Quadratic equations. Provided $a \neq 0$ , the solutions of a quadratic equation

are given by the quadratic formula

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

Parabola. The graph of an equation with leading coefficient $a \neq 0$ represents a parabola. It opens upward if , or downward if . 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 is called a quadratic function if

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

with leading coefficient $a \neq 0$ . The quadratic function

represents a parabola; it opens upward if

, or downward if

. When $(b^2 - 4ac) \ge 0$ , the graph of

has

-intercepts; the exact values of

-intercepts are obtained from the quadratic formula. There is no

-intercept if

.

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

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

The above formula is called a standard equation of a parabola with the vertex

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

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