Visualizing Functions *
To the Student:
"The time has come," the Walrus said, " to talk
of many things." And in mathematics, the further you study the subject,
the more those many things are connected to the concept of a function. But it
not only by talking about functions that we come to understand these things.
To bring a deeper understanding to these "things" we try to give them
visual significance. Seeing the information of a function presented in a variety
of forms helps us recognize in a more concrete fashion some of the fundamental
features that functions can capture.
This module has been developed to help you understand functions
more deeply using four common ways to present the relational information of
a function. You may have encountered some of these methods previously, but in
this module you will be asked to see how these methods give us different ways
to view the same underlying function relation. The four ways that connect an
understanding of functions are:
 Tables
of paired data for two "variable" quantities.
 Algebraic equations
that determine a method for determining the value of one (controlled or dependent)
variable quantity uniquely from the value of a second (controlling or independent)
varaible quantity.
 The visualization of pairs of data
for two variable quantities using the cartesian coordinate
system for the plane.
 The visualization of pairs of data
for two variable quantities using transformation figures,
also called mapping diagrams.
Transformation Figures: The last of these methods may not
be as familiar to you as the others. It is given very little time in most introductory
presentations of the function concept. The key idea in visualizing functions with
mapping diagrams or transformation figures is to have two parallel number lines
(or axes) representing the source (controlling or independent) variable values
and the target (controlled or dependent) variable values. The function can
be thought of as a process that relates the points (numbers) on these parallel
axes.
Here is an illustration that should help you see some of the
transformation figure's features along with the presentation of a function using
an algebraic formula and a table of data. This module will provide you with
more examples that will help you see some of the power of this visualization.
x

f(x)=2x+3

5

13

4

11

3

9

2

7

1

5

0

3

1

1

2

1

3

3

4

5

5

7

Table 1

Figure 1

Example 1: Suppose y is a function of x given
by the equation y = f (x) = 2x+ 3. Table 1 shows a selection
of the values this function relates, while this same information is visualized
in Figure 1. Notice that larger numbers in the source column of the table
correspond to larger values in the target column. On the transformation figure
this feature can be seen by the fact that the lines connecting the corresponding
points on the source and target lines do not cross. This is evidence of a
function with increasing values.
So how is a tranformation figure formed? A point on the
source line is chosen which corresponds to a number. The function is applied
to that number, and the resulting value is found represented on the target line.
An arrow drawn from the point on the source line to the corresponding point
on the target line visualizes the relation between the corresponding numbers.
In a sense, the transformation figure is a visualization of a
function table. The numbers in the two columns of the table are represented
by points on the two lines in the figure. The function relation that the table
displays implicitly by having corresponding numbers in the same row is visualized
in the transformation figure by the arrow. While the relative size of
the numbers in the target column of the table is not represented in the display,
the transformation figure uses the number line order to represent this aspect
of the function's values.
Graphs of
Functions and Other Relations: In your previous
work with functions and equations you have worked extensively with the graphical
visualization using Cartesian coordinates for the plane to identify the function
pairing of numbers.
In the graph
of a function f we identify the pair of numbers a and f(a)
with the point in the plane with coordinates (a,f(a)).
We can plot marks at many of these points but when the domain of the function
is an interval or as is more common all real numbers, we cannot hope to plot
all the points. Instead we try to give a sense of how the points are related
by drawing a curve that passes through some points that are known to be on the
graph of the function. In doing this we are drawing figures much as students
in elementary school draw figures by connecting the dots in order, or as economists
graph the hour to hour price of some stock on the stock market or as a chemist
would visualize the minute by minute temperature reading on a laboratory thermometer
during an experiment.
As you continue to work through the activities in this module,
your understanding of the interaction between these four methods of visualizing
a function; tables, equations, cartesian graphs and transformation figures,
should increase so that you can use each of these methods to gain a deeper
understanding of functions in your future studies of mathematics.
*This
work is the collaboration of Professors Martin Flashman, Yoon Kim, and Ken Yanosko,
Department of Mathematics, Humboldt State University, Arcata, CA, 95521.
Some of the text and many of the figures are taken from The Sensible Calculus.
© 2002 by Martin Flashman
Comments may be sent to Martin Flashman: flashman@humboldt.edu.