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.
- True
- False
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.
Stretch along the x axis
If we have a function $$ \begin{equation}\begin{aligned} f(x)\\ \end{aligned}\end{equation} $$
We can stretch the graph along the $x$ axis via the transformation: $$ \begin{equation}\begin{aligned} f(nx)\\ \end{aligned}\end{equation} $$
This transformation takes each of the original $x$ values and multiplies them by a factor of $\frac{1}{n}$.
Stretch along the y axis
Given the function $$ \begin{equation}\begin{aligned} f(x)\\ \end{aligned}\end{equation} $$
We can consider the transformation: $$ \begin{equation}\begin{aligned} kf(x)\\ \end{aligned}\end{equation} $$
This takes each of the $y$ values of the original graph and multiplies them by a factor of $k$.
Research Questions
By completing this section, you will be able to answer the following:
- What transformations can we apply to functions?
- How can we use compositions to represent translations and stretches along the $x$ and $y$ axis?