4.1 Transformation of graphs using output  (Page 2/2)

$$y=4\sin x$$

The amplitude of function "4sinx" is 4 times that of core graph "sinx". In the same fashion, a division by a positive constant greater than 1 results in shrinking of core graph by the factor, which is equal to constant being multiplies. Let us consider division of function :

$$y=\frac{1}{2}\sin x$$

The amplitude of the graph "sinx" changes from 1 to 1/2 in the graph of "1/2 sinx".

Negation of function

What would happen if we negate output of a function? Answer is easy. All positive values will turn negative and all negative values will turn positive. It means that graph of core function which is being negated will be swapped across x-axis in the transformation. The graph of “f(-x)”, therefore, is mirror image in x-axis. In other words, we would need to flip the graph f(x) across x-axis to draw graph “-f(x)”.

Combined output operations

Certain functions are derived from core function as a result of multiple arithmetic operations on the output of core function. Consider an example :

$$f\left(x\right)=-2\sin x-1$$

We can consider this as a function composition which is based on sine function f(x) = sinx as core function. Here, sequence of operations on the function is important. Difference in interpreting input and output composition is that input composition is evaluated such that defining input transitions are valid. This results in a order of evaluation which gives precedence to addition/subtraction over multiplication/division. This evaluation order is clearly opposite to normal composition order of arithmetic operations in which multiplication/division is given precedence over addition/subtraction. We, therefore, say that decomposition of function for input operation is opposite to that of composition order. In the case of output operation, however, composition order of arithmetic operations is maintained during decomposition. It is logical also. After all, we are operating on a value – not something that goes into function to generate values in accordance with function rule as is the case with independent variable. It is, therefore, expected that we carry out arithmetic operations on the function just the way we evaluate algebraic expressions. In the nutshell, we shall give precedence to multiplication/division over addition/subtraction. In the example abvoe, we subtract "-1" to "-2sinx" - not to core function "sinx".

Keeping above in mind, the correct sequence of operation for graphing is :

(i) 2f(x) i.e. multiply function f(x) by 2 i.e. stretch the graph vertically by 2.

(ii) -2f(x) i.e. negate function f(x) i.e. flip the graph across x-axis.

(iii) -2f(x) – 1 i.e. subtract 1 from -2f(x) i.e. shift the graph down by 1 units.

Combined input and output operations

The combined input and output operation is symbolically represented as :

$$af\left(bx+c\right)+d;a,b,c,d\in R$$ Carrying out output operation before input operation does not make sense. There will be two different outputs which are not connected to each other. Hence, logical order is that we first carry out input operations then follow it with output operations.

Acknowledgment

Author wishes to thank Ms. Aditi Singh, New Delhi for her editorial suggestions.