6.5 Transformation of graphs

Transformation of graphs means changing graphs. This generally allows us to draw graphs of more complicated functions from graphs of basic or simpler functions by applying different transformation techniques. It is important to emphasize here that plotting a graph is an extremely powerful technique and method to know properties of a function such as domain, range, periodicity, polarity and other features which involve differentiability of a function. Subsequently, we shall see that plotting enables us to know these properties more elegantly and easily as compared to other analytical methods.

Graphing of a given function involves modifying graph of a core function. We modify core function and its graph, applying various mathematical operations on the core function. There are two fundamental ways in which we operate on core function and hence its graph. We can either modify input to the function or modify output of function.

Broad categories of transformation

• Transformation applied by modification to input
• Transformation applied by modification to output
• Transformation applied by modulus function
• Transformation applied by greatest integer function
• Transformation applied by fraction part function
• Transformation applied by least integer function

We shall cover first transformation in this module. Others will be taken up in other modules.

Important concepts

Graph of a function

It is a plot of values of function against independent variable x. The value of function changes in accordance with function rule as x changes. Graph depicts these changes pictorially. In the current context, both core function and modified function graphs are plotted against same independent variable x.

Input to the function

What is input to the function? How do we change input to the function? Values are passed to the function through argument of the function. The argument itself is a function in x i.e. independent variable. The simplest form of argument is "x" like in function f(x). The modified arguments are "2x" in function f(2x) or "2x-1" in function f(2x-1). This changes input to the function. Important to underline is that independent variable x remains what it is, but argument of the function changes due to mathematical operation on independent variable. Thus, we modify argument though mathematical operation on independent variable x. Basic possibilities of modifying argument i.e. input by using arithmetic operations on x are addition, subtraction, multiplication, division and negation. In notation, we write modification to the input of the function as :

$$\text{Argument/input}=bx+c;b,c\in R$$

These changes are called internal or pre-composition modifications.

Output of the function

A modification in input to the graph is reflected in the values of the function. This is one way of modifying output and hence corresponding graph. Yet another approach of changing output is by applying arithmetic operations on the function itself. We shall represent such arithmetic operations on the function as :