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

Graph showing translations parallel to the x-axis

Example 2

y = x^2

y = x^2 - 2

A graph showing translations parallel to the x-axis

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^3by 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

Graph showing  y = x^2 y = (x + 3)^2

Example 2

y = x^2

y = (x - 2)^2

Graph showing  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}.