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$$af\left(x\right)+d;\phantom{\rule{1em}{0ex}}a,d\in R$$
These changes are called external or post-composition modifications.
Addition and subtraction to independent variable x is represented as :
$$x+c;\phantom{\rule{1em}{0ex}}c\in R$$
The notation represents addition operation when c is positive and subtraction when c is negative. In particular, we should underline that notation “bx+c” does not represent addition to independent variable. Rather it represents addition/ subtraction to “bx”. We shall develop proper algorithm to handle such operations subsequently. Similarly, addition and subtraction operation on function is represented as :
$$\mathrm{f(x)}+d;\phantom{\rule{1em}{0ex}}d\in R$$
Again, “af(x) + d” is addition/ subtraction to “af(x)” not to “f(x)”.
Product and division operations are defined with a positive constant for both independent variable and function. It is because negation i.e. multiplication or division with -1 is a separate operation from the point of graphical effect. In the case of product operation, the magnitude of constants (a or b) is greater than 1 such that resulting value is greater than the original value.
$$bx;\phantom{\rule{1em}{0ex}}\left|b\right|>1\phantom{\rule{1em}{0ex}}\text{for independent variable}$$ $$af\left(x\right);\phantom{\rule{1em}{0ex}}\left|a\right|>1\phantom{\rule{1em}{0ex}}\text{for function}$$
The division operation is eqivalent to product operation when value of multiplier is less than 1. In this case, magnitude of constants (a or b) is less than 1 such that resulting value is less than the original value.
$$bx;\phantom{\rule{1em}{0ex}}0<\left|b\right|<1\phantom{\rule{1em}{0ex}}\text{for independent variable}$$ $$af\left(x\right);\phantom{\rule{1em}{0ex}}0<\left|a\right|<1\phantom{\rule{1em}{0ex}}\text{for function}$$
Negation means multiplication or division by -1.
Addition/ subtraction operation on independent variable results in shifting of core graph along x-axis i..e horizontally. Similarly, product/division operations results in scaling (shrinking or stretching) of core graph horizontally. The change in graphs due to negation is reflected as mirroring (across y–axis) horizontally. Clearly, modifications resulting from modification to input modifies core graph horizontally. Another important aspect of these modification is that changes takes place opposite to that of operation on independent variable. For example, when “2” is added to independent variable, then core graph shifts left which is opposite to the direction of increasing x. A multiplication by 2 shrinks the graph horizontally by a factor 2, whereas division by 2 stretches the graph by a factor of 2.
On the other hand, modification in the output of function is reflected in change in graphs along y-axis i.e. vertically. Effects such as shifting, scaling (shrinking or stretching) or mirroring across x-axis takes place in vertical direction. Also, the effect of modification in output is in the direction of modification as against effects due to modifications to input. A multiplication of function by a positive constant greater than 1, for example, stretches the graph in y-direction as expected. These aspects will be clear as we study each of the modifications mentioned here.
There is a bit of ambiguity about the nature of constants in symbolic representation of transformation. Consider the representation,
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