Quadratic Function

Quadratic function is a kind of function that is formed using the basic rules of arithmetic. The graph of a quadratic function is a curved line called parabola, which has properties that make it useful for representing certain types of complex mechanisms such as satellite dishes, car headlights, or radio telescopes.
The quadratic function can also be applied to solve such problems regarding the motion of projectiles, the path of a baseball, description of cables in suspension bridges and parabolic arches, as well as the maximization of revenues.

REMEMBER:

A quadratic function is defined by f(x) =ax2+bx+c, where a, b, and c are real numbers and a is not equal to 0.

Functions play a fundamental role in all areas of mathematics, as well as in other sciences and engineering. However, the intuition pertaining to functions, notation, and even the very meaning of the term "function" varies between the fields. More abstract areas of mathematics, such as set theory, consider very general types of functions, which may not be specified by a concrete rule and are not governed by any familiar principles. The characteristic property of a function in the most abstract sense is that it relates exactly one output to each of its admissible inputs. Such functions need not involve numbers and may, for example, associate to each nation the name of its capital, as discussed below.

Forms of a quadratic function:
A quadratic function can be expressed in three formats:
is called the general form or polynomial form,
Example: f(x)=x2+4x+3, g(x)=4x2-3x-5
is called the factored form, where r1 and r2 are the roots of the quadratic equation.
Examples: f(x)=-2(x-1)(x+1), g(x)=5(x-4)(x+5)
is called the standard form or vertex form.
Examples: f(x) =-2(x+6)2-3, where the opposite value of h is the x coordinate of the vertex and k is the y coordinate.
To convert the general form to factored form, one needs only the quadratic formula to determine the two roots r1 and r2. To convert the general form to standard form, one needs a process called completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute the factors.
Finding the Roots or Zeroes of The Quadratic Function:
The simplest form of finding the roots of the quadratic function is simply by factoring the given equation. But if the equation can not be factored, then the quadratic formula should be used.
This formula is called the quadratic formula.
The triangle represents the discriminants that determine what kind of roots or zeros are contained in the function.
Let
If 0\,\!" type="#_x0000_t75">, then there are two distinct roots since is a positive real number.
If , then the two roots are equal, since is zero.
If , then the two roots are complex conjugates, since is imaginary.
By letting and or vice versa, one can factor as .