Transformations
Section: Transformations and vectors | Syllabus: Cambridge IGCSE Mathematics (0580)
Reflection and Rotation
- A reflection is described fully by giving the equation of the mirror line, e.g. x=3, y=-2, y=x or y=-x
- A rotation is described fully by giving three things: the angle of turn, the direction (clockwise or anticlockwise, unless the angle is 180°), and the coordinates of the centre of rotation
- Reflecting in the line x=k maps (x,y)→(2k-x,\,y); reflecting in y=k maps (x,y)→(x,\,2k-y)
- Reflecting in y=x swaps the coordinates: (x,y)→(y,x); reflecting in y=-x gives (x,y)→(-y,-x)
- A rotation of 180° about centre (a,b) maps (x,y)→(2a-x,\,2b-y)
- Both reflection and rotation produce an image that is congruent to the object - the size and shape do not change
Worked Example: Describing a Rotation
- Question: Triangle P has vertices (1,1), (4,1) and (1,3). Triangle Q has vertices (3,-1), (3,-4) and (5,-1). Describe fully the single transformation that maps P onto Q.
- Step 1: Triangle Q is the same size and shape as P but turned through 90°, not slid or mirrored, so the transformation is a rotation
- Step 2: Testing the vertex (1,1)→(3,-1) against a 90° clockwise rotation about (1,-1) confirms the centre and direction; the other two vertices confirm it: (4,1)→(3,-4) and (1,3)→(5,-1)
- Answer: Rotation, 90° clockwise, centre (1,-1)
Common Mistakes
MistakeGiving only the mirror line without checking it is a full equation, e.g. writing "reflection, x" instead of "reflection in the line x = 3"
Fixalways write the mirror line as a complete equation such as x = 3, not just the axis it is parallel to
MistakeOmitting the direction of rotation when the angle is 90° or 270°
Fixa rotation description needs angle, direction AND centre - only a 180° turn can skip the direction, since clockwise and anticlockwise give the same result
Translation Using Vector Notation
- A translation slides every point of a shape by the same distance in the same direction, without turning or resizing it
- A translation is described fully by giving the column vector x. y , where the top number is the horizontal shift and the bottom number is the vertical shift
- A positive top value moves right, a negative top value moves left; a positive bottom value moves up, a negative bottom value moves down
- To translate a shape, add the translation vector to the coordinates of every vertex
Worked Example: Translating a Shape
- Question: Triangle A has vertices (-2,1), (0,1) and (-2,4). Find the coordinates of the vertices of triangle A after a translation by the vector -3. -5 .
- Step 1: Add the vector to each vertex: (-2,1)+(-3,-5)=(-5,-4)
- Step 2: (0,1)+(-3,-5)=(-3,-4)
- Step 3: (-2,4)+(-3,-5)=(-5,-1)
- Answer: (-5,-4), (-3,-4), (-5,-1)
Common Mistakes
MistakeSwapping the top and bottom numbers of the column vector, applying the vertical shift to the x-coordinate
Fixthe top number of a column vector is always horizontal (x), the bottom number is always vertical (y)
MistakeDescribing a translation using words like "left" and "down" instead of a column vector when full description is required
Fixa full description of a translation is the column vector itself - words alone are not an accepted description
Enlargement
- An enlargement changes the size of a shape but keeps it the same shape (mathematically similar); it is described fully by giving the scale factor and the centre of enlargement
- Scale factor =(image length/object length)
- A scale factor greater than 1 makes the image bigger; a scale factor between 0 and 1 (a fraction) makes the image smaller
- A negative scale factor produces an image on the opposite side of the centre of enlargement, rotated 180° from the object (E7.1)
- To find the centre of enlargement, draw straight lines through each pair of corresponding vertices on the object and image; these lines all meet at the centre
- To enlarge a shape by scale factor k about centre (a,b), each vertex (x,y) maps to (a+k(x-a),\; b+k(y-b))
Worked Example: Describing an Enlargement
- Question: Triangle A has vertices (2,6), (6,6) and (6,2). Triangle B has vertices (2,4), (4,4) and (4,2). Describe fully the single transformation that maps A onto B.
- Step 1: Triangle B is smaller than A but the same shape, so the transformation is an enlargement
- Step 2: Each side of B is half the length of the corresponding side of A, so the scale factor is (1/2)
- Step 3: Lines through corresponding vertices meet at (2,2), the centre of enlargement
- Answer: Enlargement, scale factor (1/2), centre (2,2)
Common Mistakes
MistakeWriting the scale factor upside down as object length ÷ image length
Fixscale factor is always image length divided by object length - check by comparing one matching pair of sides
MistakeAssuming an enlargement always makes a shape bigger
Fixa scale factor between 0 and 1 is still called an enlargement, even though the image is smaller than the object
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