Quadratic Functions

A quadratic function is one that can be written in the form:

f(x) = y = ax2 + bx + c  where a ≠ 0    Standard Form

Any equation that has 2 as the largest exponent of x is a quadratic function. Standard Form is always written with the x2 term first, followed by the x term, and the constant term last. a, b, and c are called the coefficients of the equation.  It is possible for the b and/or c coefficient to equal zero. Examples of some quadratic functions in standard form are:

f(x) = 3x2 -2x + 5
y = -6x2 + 4x
f(x) = x2 + 7
y = -x2

The graph of a quadratic function is a parabola. A parabola is a smooth arc, the shape that occurs naturally when you throw a ball.

Solving a quadratic equation means finding the places where the parabola crosses the x-axis on a coordinate grid. 

Factored Form 

Quadratic equations result when you take two linear expressions and multiply them. For example:

y = (x + 3)(x - 2)
y = x2 + x - 6

The expression with the parentheses is called the factored form of the equation. It is the easiest form to use when solving quadratic equations.

Vertex Form

The vertex form of a quadratic function is probably the most useful when graphing a parabola because you can read off the location of the vertex of the parabola directly from the equation. It is the form you get when you solve a quadratic equation using the completing the squares method.

The vertex form of a quadratic function is:

f(x) = a(x - h)2 + v where the point (h,v) is the vertex of the parabola.

Graphing Parabolas

The easiest way to graph a parabola is to have it in vertex form. The vertex form not only tells you where the vertex is located, but also the shape of the parabola.

To start, let's graph the simplest quadratic function there is: y = x2
This is an upward opening parabola with its vertex at the origin (0,0). To see the shape, lets look at a Table of Values for this equation:


y= x2

-2 4
-1 1
0 0
1 1
2 4

To graph a quadratic function from vertex form

Solving Quadratic Equations


Completing the Square

Quadratic Formula

Quadratic formula if ax2 + bx + c = 0
a ≠ 0
x = 

-b ±  (b2 - 4ac)


The Discriminant

the term under the square root sign in the quadratic formula: b2 - 4ac

Imaginary Solutions

Any time the discriminant (the part under the square root sign) is negative, you will have two imaginary solutions. If you are working only in the real number system then there is no solution. To find the imaginary solutions, find the positive square root and add on the letter i.

For example:  - 36 = 6i

It is also possible to set some calculators in complex mode, and they will do the work for you.

Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer Last Updated: 02/04/2011
K. A. Shearer 2005
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