# Graph transformations

## In a nutshell

Points on a coordinate grid can be transformed (moved) under various operations. The new points are called images of the original points. Translations (shifting by some distance) and reflections are two types of transformations. They each use particular notation or phrasing as their instructions.

## Translations

Translations can be presented in a column vector form, where the top number represents the horizontal change (the displacement in the $x$-direction) and the bottom number represents the vertical change (the displacement in the $y$-direction). For example, the following vector describes a translation three places to the left and seven places up:

$\begin{pmatrix}-3\\7\end{pmatrix}$

Translations can be applied to shapes or points on a coordinate grid. They move the shape or point accordingly. The new point is called the image of the original point under the translation.

##### Example 1

*The translation $\begin{pmatrix}-3\\7\end{pmatrix}$ is applied to the point $(8,-10)$. What are the new coordinates of the point after the translation?*

*The translation says the point has move $-3$ places in the $x$-direction (three places to the left) and $7$ places in the $y$-direction (seven places up). Hence the new coordinates are*

*$(8-3,-10+7)=\underline{(5,-3)}$*

When translating a shape, simply apply the translation to each point on the shape. The easier way is to apply the translation to the corners or any key points on the shape, before joining them up in their new positions.

## Reflections

Shapes and points can also be reflected in lines. You'll see reflections in vertical or horizontal lines here. Recall that these lines respectively have the form

$x=d$

and

$y=c$

where $c$ and $d$ are constants.

### Reflecting in the axes

Reflecting in the $x$-axis changes a point's $y$-coordinate to the negative of its original value. So $(a,b)$ becomes $(a,-b)$. Reflecting in the $y$-axis changes a point's $x$-coordinate to the negative of its original value. So $(a,b)$ becomes $(-a,b)$.

### Reflecting in a general vertical line

The equation of a vertical line is $x=d$. To reflect a point in this line, find the $x$-distance to the line (the difference between the point's $x$-coordinate and $d$), and go that lot again past the line. Keep the $y$-coordinate the same.

##### Example 2

*Reflect the point $(3,9)$ in the line $x=8$.*

*First you find the $x$-distance to the line. This is from $3$ to $8$. This is a distance of $5$. Thus, you go an additional $5$ in the same direction. This takes you to $13$. Hence, the image of $(3,9)$ under the reflection in the line $x=8$ is $\underline{(13,9)}$.*

### Reflecting in a general horizontal line

The equation of a horizontal line is $y=c$. The process here is identical as when reflecting in a vertical line, except you swap $x$ actions with $y$ actions: to reflect a point in a horizontal line, find the $y$-distance to the line (the difference between the point's $y$-coordinate and c), and go that lot again past the line. Keep the $x$-coordinate the same.

##### Example 3

*Reflect the point $(3,9)$ in the line $y=-4$.*

*Find the $y$-distance from the point to the line: this is from $9$ to $-4$, which is $-13$. Now travel this $-13$ from the line: *

*$-2-13=-15$*

*This is the $y$-coordinate of the image. So the coordinates of the image of $(3,9)$ under the reflection in $y=-2$ are *

*$\underline{(3,-15)}$*