### Section IN Quadratic Functions

**Quadratic Functions**

**Introduction: **Of all the functions
that are studied, after linear functions. the
simplest and most common class of functions is the class
of** ***quadratic
functions***.** These functions are
characterized by the operations: squaring a number.
multiplication by a constant and addition of a constant.

For example, the function defined by assigning the
value $f(x) =x^2 - 2x - 3$ to the number $x$, also
represented symbolically as $f: x \rightarrow x^2 - 2x -
3$ , or verbally by saying "to find
$f(x)$, **square $x$**, subtract $2$ times $x$ and
finally subtract $3$", is described as a quadratic
function. This function is visualized by
the mapping diagram, **QF1.**

Mapping Diagram QF1

Mapping Diagram QF1

This section will provide examples, explanations, exercises and problems that will help students use the power of the mapping diagram along with the three other tools (equations, tables , and graphs) to understand quadratic functions.

#### Quadratic Function Definition

##### QF.QFIQuadratic Functions are Important.

**Example QF.0**

**The First Quadratic Function Example.**

This example presents the quadratic function $f(x) = x^2 - 2x - 3$ with a table of data, a graph and a mapping diagram. The "quadratic coefficient" is $1$, the "linear coefficient"i s $-2$, and the "constant " is $-3$.

Treatment of quadratic functions and their graphical interpretation with parabolas and equations are familiar. [See wikipedia.org/wiki : Quadratic_function]

They appear in every textbook that deals at all with intermediate algebra and coordinate geometry- from algebra II to beginning calculus. What is missing is a balanced treatment using mapping diagrams to reinforce the function aspect of visualization. That will be emphasis of this section.

Comparisons will be made when appropriate to graphs- but we will develop the basic concepts for quadratic functions with mapping diagrams. The end of this section includes some powerful and different ways to think about quadratic functions and the ways they are represented algebraically.