# 9.3 Transformations

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## Transformations - enrichment, not in caps

In this section you will learn about how the co-ordinates of a point change when the point is moved horizontally and vertically on the Cartesian plane. You will also learn about what happens to the co-ordinates of a point when it is reflected on the $x$ -axis, $y$ -axis and the line $y=x$ .

## Translation of a point

When something is moved in a straight line, we say that it is translated . You will recall that the term 'translation' was also mentioned in the section on functions to refer to what happens when an entire function is shifted in a straight line. What happens to the co-ordinates of a point that is translated horizontally or vertically?

## Discussion : translation of a point vertically

Complete the table, by filling in the co-ordinates of the points shown in the figure.

 Point $x$ co-ordinate $y$ co-ordinate A B C D E F G

What do you notice about the $x$ co-ordinates? What do you notice about the $y$ co-ordinates? What would happen to the co-ordinates of point A, if it was moved to the position of point G?

When a point is moved vertically up or down on the Cartesian plane, the $x$ co-ordinate of the point remains the same, but the $y$ co-ordinate changes by the amount that the point was moved up or down.

For example, in [link] Point A is moved 4 units upwards to the position marked by G. The new $x$ co-ordinate of point A is the same ( $x$ =1), but the new $y$ co-ordinate is shifted in the positive $y$ direction 4 units and becomes $y$ =-2 +4 =2. The new co-ordinates of point A are therefore G(1;2). Similarly, for point B that is moved downwards by 5 units, the $x$ co-ordinate is the same ( $x=-2,5$ ), but the $y$ co-ordinate is shifted in the negative $y$ -direction by 5 units. The new $y$ co-ordinate is therefore $y$ =2,5 -5 =-2,5.

If a point is shifted upwards, the new $y$ co-ordinate is given by adding the shift to the old $y$ co-ordinate. If a point is shifted downwards, the new $y$ co-ordinate is given by subtracting the shift from the old $y$ co-ordinate.

## Discussion : translation of a point horizontally

Complete the table, by filling in the co-ordinates of the points shown in the figure.

 Point $x$ co-ordinate $y$ co-ordinate A B C D E F G

What do you notice about the $x$ co-ordinates? What do you notice about the $y$ co-ordinates?

What would happen to the co-ordinates of point A, if it was moved to the position of point G?

When a point is moved horizontally left or right on the Cartesian plane, the $y$ co-ordinate of the point remains the same, but the $x$ co-ordinate changes by the amount that the point was moved left or right.

For example, in [link] Point A is moved 4 units right to the position marked by G. The new $y$ co-ordinate of point A is the same ( $y$ =1), but the new $x$ co-ordinate is shifted in the positive $x$ direction 4 units and becomes $x$ =-2 +4 =2. The new co-ordinate of point A at G is therefore (2;1). Similarly, for point B that is moved left by 5 units, the $y$ co-ordinate is the same ( $y=-2,5$ ), but the $x$ co-ordinate is shifted in the negative $x$ -direction by 5 units. The new $x$ co-ordinate is therefore $x$ =2,5 -5 =-2,5. The new co-ordinates of point B at H is therefore (-2,5;1).

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