Translating graphs
The translation of graphs is explored
A translation is a movement of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis.
Functions
The graph of \(y = f(x)\) where \(f(x) = x^2\) is the same as the graph of \(y = x^2\). Writing equations as functions in the form \(f(x)\) is useful when applying translations and reflections to graphs.
Translations parallel to the y-axis
If \(f(x) = x^2\), then \(f(x) + a = x^2 + a\). Here we are adding \(a\) to the whole function.
The addition of the value \(a\) represents a vertical translation in the graph. If \(a\) is positive, the graph translates upwards. If \(a\) is negative, the graph translates downwards.
Example 1
\(f (x) = x^2\)
Draw the graphs of \(y = f(x)\) and \(y = f(x) + 3\).
This is a translation of \(y = f(x)\) by 3 units in the positive \(y\) direction.
Example 2
\(f(x) = x^2\)
Draw the graphs of \(y = f(x)\) and \(y = f(x) 鈭 2\).
This is a translation of \(y = f(x)\) by 2 units in the negative \(y\) direction.
\(f(x) + a\) represents a translation of the graph of \(f(x)\) by the vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).
Translations parallel to the x-axis
If \(f(x) = x^2\) then \(f(x + a) = (x + a)^2\)
Here we add \(a\) to \(x\), not to the whole function. This time we will get a horizontal translation. If \(a\) is positive then the graph will translate to the left. If the value of \(a\) is negative, then the graph will translate to the right.
Example 1
\(f(x) = x^2\)
Draw the graphs of \(y = f(x)\) and \(y = f(x + 3)\).
Example 2
\(f(x) = x^2\)
Draw the graphs of \(y = f(x)\) and \(y = f(x 鈥 2)\).
\(f(x + a)\) represents a translation of the graph of \(f(x)\) by the vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).