The form of transformation is depicted as :
The graph of y=f(x) is transformed in y=f({x}) by virtue of changes in the argument values. The independent variable is subjected to fraction part operator. This changes the normal real value input to function. Instead of real numbers, independent variable to function is rendered to be fractions irrespective of values of x. A value like x = - 2.3 is passed to the function as 0.7 in the interval [0,1).
Clearly, real values of “x” are truncated to fraction values in all intervals. It means that same set of values of the function y=f(|x|) corresponding to interval of x defined by [0,1] will repeat in other intervals along x-axis. The FPF is a periodic function with a period of 1. Taking advantage of this fact, we obtain graph of y=f({x}) by repeating part of graph for x in [0,1) to other intervals along x-axis. Clearly, transformed function y=f({x}) is periodic with a period of 1.
From the point of construction of the graph of y=f({x}), we need to modify the graph of y=f(x) as :
1 : Draw lines parallel to y-axis (vertical lines) at integral values along x-axis to cover the graph of y=f(x).
2 : Identify part of the graph for values of x in [0,1). Include end point corresponding to x=0 and exclude end point corresponding to x=1.
3 : Repeat the part of the graph identified in step 2 for other intervals of x
The lines drawn in step 3 is the graph of y=f({x}).
Example 1
Problem : Draw the graph of sin{x}.
Solution : Following the construction steps, graph of y=sin{x} is drawn by transforming y = sinx as shown here.
Graph of y=sin{x} |
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Example 2
Problem :
Draw the graph of
Solution : Rearranging, we have :
Following the construction steps, graph of
Transformation of exponential graph |
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