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 f(x) = x^2 is the same as the graph of y = x^2. Writing graphs 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

f(x) + 3 = x^2 + 3

Graph showing plots of f(x)+3=x^2+3 & f(x)=x^2

Example 2

f(x) = x^2

f(x) - 2 = x^2 - 2

Graph showing plots of f(x)-2=x^2-2 & f(x)=x^2

f(x) + a represents a translation through 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

f(x + 3) = (x + 3)^2

Graph showing plots of f(x+3)=(x+3)^2 & f(x)=x^2

Example 2

f(x) = x^2

f(x - 2) = (x - 2)^2

Graph showing plots of f(x-2)=(x-2)^2 & f(x)=x^3

f(x + a) represents a translation through the vector \begin{pmatrix} -a \\ 0 \end{pmatrix}.