Translating graphs
The translation of graphs is explored
A graph can be translated horizontally, vertically or in both directions.
Translations parallel to the y-axis
\(y = x^2 + a\) represents a translation parallel to the \(y\)-axis of the graph of \(y = x^2\). If \(a\) is positive, the graph translates upwards. If \(a\) is negative, the graph translates downwards.
Example 1
\(y = x^2\)
\(y = x^2 + 3\)
Example 2
\(y = x^2\)
\(y = x^2 - 2\)
\(y=x^2 + a\) represents a translation of the graph of \(y = x^2\) by the vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).
This is also true for other graphs.
For example, \(y = x^3 - 2\) is a translation of \(y = x^3\)by the vector \(\begin{pmatrix} 0 \\ -2 \end{pmatrix}\) and \(y = sin x + 3\) is a translation of \(y = sin x\) by the vector \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\).
Translations parallel to the x-axis
\(y = (x + a)^2\) represents a translation parallel to the \(x\)-axis of the graph of \(y = x^2\).
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
\(y = x^2\)
\(y = (x + 3)^2\)
Example 2
\(y = x^2\)
\(y = (x - 2)^2\)
\(y = (x + a)^2\) represents a translation of the graph of \(y = x^2\) by the vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\)
This is also true for other graphs. For example, \(y = (x + 2)^3 \) is a translation of \(y = x^3\) by the vector \(\begin{pmatrix} -2 \\ 0 \end{pmatrix}\) and \(y = sin(x – 30)\) is a translation of \(y = sin x\) by the vector \(\begin{pmatrix} 30 \\ 0 \end{pmatrix}\).