Translation along the x axis
Given the function $$ \begin{equation}\begin{aligned} f(x)\\ \end{aligned}\end{equation} $$
A translation of this function along the $x$ axis can be represented by $$ \begin{equation}\begin{aligned} f(x+a)\\ \end{aligned}\end{equation} $$
Here the function’s graph is left-shifted by $a$ units. When we have the form: $$ \begin{equation}\begin{aligned} f(x-a)\\ \end{aligned}\end{equation} $$
This represents a right-shift of the graph of the function by $a$ units.
Q1: The function $g(x)=\sin(x+3)$ is translation of the function $f(x)=\sin x$ along the $x$ axis in the positive (right) direction by $3$ units.
- False
- True
The $+3$ indicates that there is a left shift by $3$ units
Translation along the y axis
Consider the function $$ \begin{equation}\begin{aligned} f(x)\\ \end{aligned}\end{equation} $$
A translation of this function along the $y$ axis can be given as $$ \begin{equation}\begin{aligned} f(x)+b\\ \end{aligned}\end{equation} $$
Here the function’s graph is being shifted upwards by $b$ units. For the translation: $$ \begin{equation}\begin{aligned} f(x)-b\\ \end{aligned}\end{equation} $$
The graph of the function is shifted downwards by $b$ units.