Summary: Analysis of Quadratic Functions

Key Equations

general form of a quadratic function	$f\left(x\right)=a{x}^{2}+bx+c$
standard form of a quadratic function	$f\left(x\right)=a{\left(x-h\right)}^{2}+k$

Vertex of a parabola (h,k)

h = - b / 2 a

$h=-b/2a$

k = f (- b / 2 a)

$k=f\left(-b/2a\right)$

The Quadratic Formula

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

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

The zeros, or $x$ -intercepts, are the points at which the parabola crosses the $x$ -axis. The $y$ -intercept is the point at which the parabola crosses the $y$ –axis.
The vertex can be found from an equation representing a quadratic function.
A quadratic function’s minimum or maximum value is given by the $y$ -value of the vertex.
The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.
Some quadratic equations must be solved by using the quadratic formula.
The vertex and the intercepts can be identified and interpreted to solve real-world problems.

vertex: the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

vertex form of a quadratic function: another name for the standard form of a quadratic function

zeros: in a given function, the values of $x$ at which $y=0$ , also called roots