Scaling, translation and reflection of functions

Often in mathematics, we have a given elementary function (for example, $e^x$ , $\sin(x)$ ) and new functions are derived by addition or multiplication of a constant to the independent variable or to the functional value. For example, the cumulative distribution function of an exponential random variable with parameter $\lambda$ is $1 - \lambda e^{-\lambda x}$ . In cases like this, while the reader may have the picture of the elementary function ( $e^x$ ) in mind, visualizing the derived function quickly is of tremendous value and is often taken for granted. In this article, we will discuss how the behavior of derived functions relate to the given functions.

In all the following, we assume $f$ is a real-valued function defined on $\mathbb{R}$ . And, $a \in \mathbb{R}$ and $a > 0$ .

The plot of $f(x+a)$ will be that of $f(x)$ left shifted by $a$ units
The plot of $f(x-a)$ will be that of $f(x)$ right shifted by $a$ units
For $a > 1$ , the plot of $f(ax)$ will be that of $f(x)$ “shrunk” along $X$ -axis by a factor of $a$
For $a < 1$ , the plot of $f(ax)$ will be that of $f(x)$ “elongated” along $X$ -axis by a factor of $a$
For a periodic function $f(x)$ of period $T$ , $f(ax)$ will be a periodic function of period $\frac{T}{a}$ . In other words, for a periodic function $f(x)$ of frequency $\omega$ , $f(ax)$ will be a periodic function of frequency $a\omega$
The plot of $f(-x)$ will be that of $f(x)$ reflected on the $Y$ -axis
The plot of $f(x)+a$ will be that of $f(x)$ up shifted by $a$ units
The plot of $f(x)-a$ will be that of $f(x)$ down shifted by $a$ units
For $a > 1$ , the plot of $af(x)$ will be that of $f(x)$ “elongated” along $Y$ -axis by a factor of $a$
For $a < 1$ , the plot of $af(x)$ will be that of $f(x)$ “shrunk” along $Y$ -axis by a factor of $a$
The plot of $-f(x)$ will be that of $f(x)$ reflected on the $X$ -axis
If $f$ is an invertible function, the plot of $f^{-1}(x)$ will be that of $f(x)$ reflected on the line $y=x$

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