4.2 Transformation of graphs by modulus function

A function like y=f(x) has different elements. We can apply modulus operator to these elements of the function. There are following different possibilities :

1 : $y=f\left(\left|x\right|\right)$

2 : $y=\left|f\left(x\right)\right|$

3 : $\left|y\right|=f\left(x\right)$

4 : $x=\left|f\left(y\right)\right|$

Plotting concept

The most important point about plotting is to understand that application of modifying operator has different interpretation whether it is applied to independent variable “x” or function definition in x like “f(x)” or it is applied to dependent variable “y” or function definition like “f(y)”. There is a difference in the approach to interpretation.

Clearly, modulus operations have different implications for the graph of f(x). In general, every function can be interpreted to be an operator which operates on its argument, which in itself can be variable like “x”, expression like “ $x^2+2$ ” or other functions. This role is more visible for functions like modulus, greatest integer, fraction part and least integer function. For this reason, these functions are represented by symbolic notations like | |, [], {} and () as operators.

When operator is applied to independent variable or function definition, we evaluate operation of the operator on independent variable or function value. Here, interpretation is based on “evaluation” of the expression (independent variable or function definition) and application of operator thereafter. This applies to the transformations enumerated at (i) and (ii) above. Consider for example,

$$y=\left|f\left(x\right)\right|$$

The function of value at any value x=x is first evaluated. Then, modulus of value is calculated. Finally, it is assigned to y as its value.

This basis of interpretation changes when we apply operator to dependent variable “y” or function definition in “y”. Now the basis of interpretation is that of “assigning” a value to a function and then interpreting the assignment. Such is the case with transformations enumerated at (iii) and (iv) above. Consider for example,

$$\left|y\right|=f\left(x\right)$$

In this case, value of function evaluated at x=x is assigned to modulus function. We interpret equality of the modulus function [y] to a value in accordance with modulus definition. In this case, we know that :

$$\left|y\right|=a;a>0\Rightarrow y=\pm a$$

$$\left|y\right|=a;a=0\Rightarrow y=0$$

$$\left|y\right|=a;a<0\Rightarrow \text{Modulus can not be equated to negative value. No solution}$$

From the point of view of construction of plot, for a single positive value of f(x), say f(x)=4, we have two values of dependent variable i.e. -4 or 4. This needs to be considered while plotting |y|=f(x). In the plot, values of y are plotted against values of x. In this particular instant, there are two points (4,4) and (4,-4) on the graph corresponding to one value of independent variable (4).

Modulus function applied to the independent variable

The form of transformation is depicted as :

$$y=f\left(x\right)\Rightarrow y=f\left(\left|x\right|\right)$$

It can be seen that modulus operator here modifies independent varaible of the function. In other words, it is like changing input to the function in accordance with nature of modulus function. The input to the function is now either zero or positive number. This has the implication that part of the graph y=f(x) corresponding to negative value of x is not present in the graph of y=f(|x|). Rather, negative value of x is passed as positive value to the function. This means that negative value of independent variable x yields function value which is equal to function value obtained for corresponding positive x whose magnitude is same as that corresponding negative x. It implies that we can obtain function value for negative x by taking image of positive x across y-axis. This is image in y-axis.