### Section IN Linear Functions

**Linear Functions**

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

For example, the function defined by assigning the
value $f(x) = 2x+3$ to the number $x$, also represented
symbolically as $f: x \rightarrow 2x+3$ , or
verbally by saying "to find $f(x)$, multiply $x$ by
2 then add 3", is described as a linear function.
This function was visualized in Chapter 1, VF,
by the mapping diagram, **TMD1.**

Mapping Diagram TMD1

Mapping Diagram TMD1

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 linear functions.

#### Linear Function Definition

##### LF.LFILinear Functions are Important.

**Example LF.0**

**The First Linear Function Example.**

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

Treatment of linear functions and their graphical interpretation with lines and equations are familiar. [See wikipedia.org/wiki : Linear_function_(calculus)]

They appear in every textbook that deals at all with beginning algebra and coordinate geometry- from algebra I 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 linear functions with mapping diagrams. The end of this section includes some powerful and different ways to think about linear functions and the ways they are represented algebraically.

Here is a YouTube video (about 10 minutes) that covers some initial work on using mapping diagrams to visualize linear functions:

##### Subsection LF.SMR Slope, Magnification and Rate

##### Subsection LF.ID Increasing/Decreasing Linear Functions

##### Subsection LF.FP Focus Point on A Mapping Diagram

##### Subsection LF.COMP Composition of Linear Functions

##### Subsection LF.INV Inverse of a Linear Function

##### Subsection LF.FORM Forms of a Linear Function

##### Subsection LF.LEq Solving Linear Equations with Linear Functions

##### Subsection LF.LInEq Solving Linear Inequalities with Linear Functions

##### Subsection LF.APP Linear Function Applications (not yet ready)

Here is a link to a spreadsheet for exploring the effects of $m$ and $b$ on the mapping diagram for a linear function.