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.
xf(x)=2x+3
5
134113927
1
503
-1
1-2-1
-3
-3
-4
-5-5-7Table 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.
