4.1 Transformation of graphs using output

We modify output of a function in a couple of ways through arithmetic operations like addition, subtraction, multiplication, division and negation. These operations are similar to the one that we use to modify independent variable. The general symbolic representation for modification to output of a function is represented as :

$$af\left(x\right)+d;a,d\in R$$

These changes are called external or post-composition modifications. These modifications compliment modifications by input, but in slightly different manner. In the case of modification to output, all effects take place in y-direction i.e. vertical direction as against horizontal transformation arising from modifications affected to input. Second, these transformations are in the direction of operation on output. For example, if we multiply output by a positive constant greater than 1, then graph of core function is stretched along y-axis. This means change in the output is reflected in the same direction in which operation takes place.

Addition and subtraction operation with function

In order to understand this type of transformation, we need to explore how output of the function changes as we add constant value to the output. If we add 1 unit to the function, then each value of function is incremented by 1 unit. It is a straight forward situation. In notation, we would say that the graph of “f(x) + 1” is same as the graph of f(x), which has been moved up by 1 unit. Alternatively, we can also describe this transformation by saying that vertical reference of measurement i.e. x-axis has moved down by 1 unit.

Similarly, if we subtract 1 unit from the function, then each value of function is decremented by 1 unit. In notation, we would say that the graph of “f(x) - 1” is same as the graph of f(x), which has been moved down by 1 unit. Alternatively, we can also describe this transformation by saying that vertical reference of measurement i.e. x-axis has moved up by 1 unit. We conclude :

The plot of y=f(x) + |a|; |a|>0 is the plot of y=f(x) shifted up by unit “a”.

The plot of y=f(x) - |a|; |a|>0 is the plot of y=f(x) shifted down by unit “a”.

We use these facts to draw plot of transformed function f(x±|a|) by shifting plot f(x) by unit “|a|” along y-axis. Each point forming the plot is shifted parallel to x-axis. In the figure below, the plot depicts modulus function y=|x|. It is shifted “1” unit up and the function representing shifted plot is y=|x|+1. Note that corner of plot at x=0 is also shifted by 1 unit along y-axis. Further, the plot is shifted “2” units down and the function representing shifted plot is |x|-2. In this case, corner of plot is shifted by 2 units down along y-axis.

Multiplication and division of function

Multiplication and division scales core graph in accordance with the operation. Scaling, however, is limited to vertical i.e. y-direction. This means modification due to either of these two arithmetic operations has no scaling impact in x-direction. If we multiply output of the function by a positive constant greater than 1, then graph of core function is stretched vertically by the factor, which is equal to the constant being multiplied. The magnification of graph i.e. stretching in y-direction is more noticeable in non-linear graphs like sine and cosine graphs, whose values are bounded in the interval [-1,1]. Let us consider function,