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4.3 F) Translations
4.3 F) Translations
Translations move shapes from one position to another; they move shapes vertically and horizontally. Translations do not alter the shape in any way; they just move the shape from one position to another position. All of the points on a shape move by the same distance.
The shape below has been translated. The orange shape is the original shape and the green shape is the translated shape.
The shape below has been translated. The orange shape is the original shape and the green shape is the translated shape.
All of the points on the original shape have been moved 5 to the right and 3 down.
Column Vectors
We are able to use column vectors to describe translations. An example of a column vector is shown below:
We are able to use column vectors to describe translations. An example of a column vector is shown below:
The top value in a column vector is horizontal component of the translation; the movement in a leftwards or rightwards direction. A positive value for the top component is a movement rightwards and a negative value is a movement leftwards. The top value in the column vector above is a 2 and this represents a movement of 2 towards the right.
The bottom value in a column vector is the vertical component of the translation; the movement in an upwards or downwards direction. A positive value for the bottom component is a movement upwards and a negative value is a movement downwards. The value in the above column vector was -4 and this represents a movement of 4 in a downwards direction.
Let’s find the column vector that describes the translation in the first example that we looked at. The diagram is shown below:
The bottom value in a column vector is the vertical component of the translation; the movement in an upwards or downwards direction. A positive value for the bottom component is a movement upwards and a negative value is a movement downwards. The value in the above column vector was -4 and this represents a movement of 4 in a downwards direction.
Let’s find the column vector that describes the translation in the first example that we looked at. The diagram is shown below:
The above shape has been moved 5 to the right and 3 down. The top component in the column vector is the movement in a horizontal (right and left) direction with rightwards movements being positive and leftward movements being negative. The shape has been moved 5 to the right, which means that the top component in the column vector will be 5.
The bottom component of the column vector is the movement in a vertical direction (up and down) with upwards being a positive value and downwards being a negative value. The shape was moved 3 down and this means that the bottom component in the column vector is -3.
The column vector for the translation is shown below:
The bottom component of the column vector is the movement in a vertical direction (up and down) with upwards being a positive value and downwards being a negative value. The shape was moved 3 down and this means that the bottom component in the column vector is -3.
The column vector for the translation is shown below:
Translate the above shape by the column vector:
The first step in answering a translation question is to break the column vector down in to a horizontal and vertical component.
The top component in the column vector is the movement in a horizontal direction with positive values being rightwards and negative values being leftwards. The top component for the vector in this question is -2, which means that all of the points on the above shape move 2 to the left.
The bottom component in the column vector is the movement in a vertical direction with positive values being upwards and negative values being downwards. The bottom component for the vector in this question is 4, which means that all of the points on the above shape move upwards by 4.
Therefore, the overall translation is 2 towards the left and 4 up. The translated shape is shown below:
The top component in the column vector is the movement in a horizontal direction with positive values being rightwards and negative values being leftwards. The top component for the vector in this question is -2, which means that all of the points on the above shape move 2 to the left.
The bottom component in the column vector is the movement in a vertical direction with positive values being upwards and negative values being downwards. The bottom component for the vector in this question is 4, which means that all of the points on the above shape move upwards by 4.
Therefore, the overall translation is 2 towards the left and 4 up. The translated shape is shown below:
Describing a Translation
Describing a translation is fairly straight forwards; we just find the horizontal and vertical movement. We then put this movement into a column vector.
Example 2
Shape A had been translated to become shape B. Describe the transformation.
Describing a translation is fairly straight forwards; we just find the horizontal and vertical movement. We then put this movement into a column vector.
Example 2
Shape A had been translated to become shape B. Describe the transformation.
The first step in finding a transformation is to find the vertical and horizontal movement. The above shape has been moved 3 to the left and up by 4. I have shown this on the bottom left corner of the shape.
The top value in the column vector is the horizontal movement; the shape has moved 3 to the left, which means that the top value in the column vector is -3. The bottom value in the column vector is the vertical movement; the shape has moved 4 up, which means that the bottom value in the column vector is 4. The column vector for the translation is shown below: