### Section VF Introduction to Mapping Diagrams: Visualizing Functions

**Visualizing Functions: An Overview**

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**Introduction: **The more one studies mathematics,
the more one encounters the concept of a function. By
talking about functions we come to understand how
they encode information. To bring a deeper understanding
to functions we give them visual expression. Seeing the
information of a function presented in a variety of forms
helps us recognize more concretely some of its fundamental
features.

These resource materials have been developed to help
students understand functions more sensibly and
deeply using four common ways to present the encoded
information of a function. Students may have encountered
some of these methods previously, but in this resource you
will see how each of these methods give us different
ways to view the same underlying relation.

The four ways that connect an understanding of functions
are:

**Tables**of data for "variable" quantities.-
**Algebraic equations**that determine a method for determining the value of (controlled or dependent) variable quantities uniquely from the value of other (controlling or independent) variable quantities. - The visualization of data for variable
quantities using the
**Cartesian coordinate system.** - The visualization of data for variable quantities
using
**mapping diagrams.**

Mapping Diagram TMD1

Mapping Diagram TMD1

**Other Names:**Mapping diagrams are also described as

**or***function diagrams, dynagraphs, parallel coordinate graphs,***cograph****s**.
See the **Bibliography** for other historical and current relevant resources.

##### VF.MDI Mapping Diagrams are Important.

**The key idea**in visualizing real valued functions of one variable with mapping diagrams is to have

**two parallel number lines (or axes), one line representing the source (input, controlling or independent) variable values and the other line representing the target (output, controlled or dependent) variable values**. The function is visualized by arrows that relate points (numbers) on these parallel axes.

#### Mapping Diagram Definition

Using some dynamic technology with GeoGebra we can see connections between the graph and mapping diagram visualizations.

#### Connecting Mapping Diagrams and Graphs

**Example TMD.0**

**The First Example.**

This example presents a single function using an algebraic formula, a table of data, a graph and a mapping diagram.

*You can download and try the worksheet for this section now*: Worksheet.VF1.pdf.