# 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 $$y = f(x) − 2$$.

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 $$y = f(x – 2)$$.

$$f(x + a)$$ represents a translation of the graph of $$f(x)$$ by the vector $$\begin{pmatrix} -a \\ 0 \end{pmatrix}$$.