Translations

Translationsvideo

Translations

What are transformations in maths?

• There are four transformations to learn

  • translations, rotations, reflections and enlargements

• A transformation can change the position, orientation and/or size of a shape

  • The original shape is called the object

  • The transformed shape is called the image

• Vertices are labelled to show corresponding points

  • Vertices on the object are labelled A, B, C, etc.

  • Vertices on the image are labelled A’, B’, C’ etc.

What is a translation?

• A translation moves a shape

• The size and orientation (which way up it is) of the shape stays the same

  • The object and image are congruent

What is a translation vector?

• The movement of a translation is described by a vector

• You need to know how to write a translation using a vector (rather than words)

• Vectors are written as column vectors in the form  $\left(\begin{array}{c}\mathit{x}\\ \mathit{y}\end{array}\right)$  where:

  • $x$ is the distance moved horizontally

    • Negative means move to the left

    • Positive means move to the right

  • $y$ is the distance moved vertically

    • Negative means move down

    • Positive means move up

How do I translate a shape?

• STEP 1

  Interpret the translation vector

  • $\left(\begin{array}{c}3\\ -1\end{array}\right)$  means 3 to the right and 1 down

• STEP 2
  Move each vertex on the original object according to the vector

• STEP 3
  Connect the new vertices and label the translated image

  • It should look identical to the original object just in a different position

• In some cases the image can overlap the object

How do I describe a translation?

• To describe a translation, you must:

  • State that the transformation is a translation

  • Give the column vector that describes the movement

• To find the vector:

  • Pick a point on the original shape

  • Identify the corresponding point on the image

  • Count how far left or right ($x$) you need to go from the object to get to the image

    • If you go to the left then $x$ will be a negative number

  • Count how far up or down ($y$) you need to go from the object to get to the image

    • If you go down then $y$ will be a negative number

  • Write these numbers as a vector

    • $\left(\begin{array}{c}x\\ y\end{array}\right)$

How do I reverse a translation?

• To return a shape to its original position after a translation

  • the horizontal and vertical translations must both be reversed

• The column vector to reverse a translation is simply the same as the original vector, but with the sign of both values changed

  • E.g. For a translation described by the column vector $\left(\begin{array}{c}-2\\ 7\end{array}\right)$

  • The column vector for the reverse translation is $\left(\begin{array}{c}2\\ -7\end{array}\right)$