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Matrices Transformations
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3 (a)
–6
–4
–5
–4
–3
–2
–3 –2 –1 1 2 3 4 5 6 7x
y
A
–1
0
1
2
3
4
5
On the grid, draw the image of
(i) shape A after a reflection in the line x = 1, [2]
(ii) shape A after an enlargement with scale factor –2, centre (0, 1), [2]
(iii) shape A after the transformation represented by the matrix 01
10-c m . [3]
(b) Describe fully the single transformation represented by the matrix 3003
c m.
..............................................................................................................................................................
.............................................................................................................................................................. [3]
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6
–4
–2
2
4
6
8
–6
–2 86420–4–6–8x
y
T
U
V
(a) (i) Draw the image of triangle T after a reflection in the line x = 0. [2]
(ii) Draw the image of triangle T after a rotation through 90° clockwise about (–2, –1). [2]
(iii) Describe fully the single transformation that maps triangle T onto triangle U.
......................................................................................................................................................
...................................................................................................................................................... [2]
(iv) Describe fully the single transformation that maps triangle T onto triangle V.
......................................................................................................................................................
...................................................................................................................................................... [3]
(b) (i) Find the matrix that represents the transformation in part(a)(i).
f p [2]
(ii) Describe fully the single transformation represented by the inverse of the matrix in part(b)(i).
......................................................................................................................................................
...................................................................................................................................................... [2]
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5y
x0–1–2–3–4–5–6–7–8
–8
–7
–6
–5
–4
–3
–2
–1
8
7
6
5
4
3
2
1
1 2
B
3 4 5 6 7 8
A
(a) v48=--c m
(i) Draw the image of triangle A after the translation by vector v. [2]
(ii) Calculate v .
................................................. [2]
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(b) (i) Describe fully the singletransformation that maps triangle A onto triangle B.
......................................................................................................................................................
...................................................................................................................................................... [3]
(ii) Find the matrix that represents the transformation that maps triangle A onto triangle B.
f p [2]
(iii) Calculate the determinant of the matrix in part(b)(ii).
................................................. [1]
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2
T
W
U
y
x
8
7
6
5
4
3
2
1
–1
–2
–3
–4
–5
–6
0–1 1 2 3 4 5 6–2–3–4–5–6–7–8
(a) On the grid, draw the image of
(i) triangle T after a translation by the vector 44-e o, [2]
(ii) triangle T after a reflection in the line y = – 1. [2]
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(b) Describe fully the single transformation that maps triangle T onto triangle U.
Answer(b) ...........................................................................................................................................
............................................................................................................................................................. [3]
(c) (i) Describe fully the single transformation that maps triangle T onto triangle W.
Answer(c)(i) ................................................................................................................................
..................................................................................................................................................... [2]
(ii) Find the 2 × 2 matrix that represents the transformation in part(c)(i).
Answer(c)(ii) f p [2]
__________________________________________________________________________________________
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ForExaminer′sUse
2 (a)
Q
P
y
x
6
5
4
3
2
1
–1
–2
–3
–4
–5
–6
–7
–8
0–1 1 2 3 4 5 6 7 8–2–3–4–5–6–7
(i) Describe fully the single transformation which maps shape P onto shape Q.
Answer(a)(i) ...................................................................................................................... [2]
(ii) On the grid above, draw the image of shape P after refl ection in the line y = –1. [2]
(iii) On the grid above, draw the image of shape P under the transformation represented by the
matrix 101 0
-e o. [3]
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(ii) Find the ratio AL : AN in its simplest form.
Answer(b)(ii) ................ : ................ [3]
(c)
A
B
4
3
2
1
–1
–2
–3
–4
–5
–6–7 –5 –4 –3 –2 –1 10 2 3 4 5
y
x
(i) On the grid, draw the image of triangle A after the transformation represented by the
matrix 1.5
01.5
0 -
-f p. [3]
(ii) Find the 2 × 2 matrix which represents the transformation that maps triangle A onto triangle B.
Answer(c)(ii) f p [2]
__________________________________________________________________________________________
Question11isprintedonthenextpage.
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19 (a) N = 1
0 10-
f p Describe fully the single transformation represented by N.
Answer(a) ...........................................................................................................................................
............................................................................................................................................................. [3]
(b) Find the matrix which represents the single transformation that maps triangle A onto triangle B.
A
B
2 4 6 8 101 3 5 7 9
7
6
5
4
3
2
1
0
y
x
Answer(b) f p [2]
(c) On the grid, draw the image of triangle A under a stretch, factor 3, with the y-axis invariant. [2]
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3
7
6
5
4
3
2
1
–1
–2
–3
–4
–5
–6
–7
–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 60x
y
A
C
B
(a) Draw the image of
(i) shape A after a translation by 1
3
-f p , [2]
(ii) shape A after a rotation through 180° about the point (0, 0), [2]
(iii) shape A after the transformation represented by the matrix 1
0
0
1-f p. [3]
(b) Describe fully the single transformation that maps shape A onto shape B.
Answer(b) .................................................................................................................................................
.............................................................................................................................................................. [3]
(c) Find the matrix which represents the transformation that maps shape A onto shape C.
Answer(c) f p [2]
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4
A
B
y
x
8
7
6
5
4
3
2
1
–1
–2
–3
–4
–5
–6
–7
–8
0–2 2 4 6 8–4–6–7 –5 –3 –1 1 3 5 7–8
(a) Describe fully the single transformation that maps triangle A onto triangle B.
Answer(a) ...........................................................................................................................................
............................................................................................................................................................. [3]
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(b) On the grid, draw the image of
(i) triangle A after a refl ection in the line x = –3, [2]
(ii) triangle A after a rotation about the origin through 270° anticlockwise, [2]
(iii) triangle A after a translation by the vector 15-
-e o. [2]
(c) M is the matrix that represents the transformation in part (b)(ii).
(i) Find M.
Answer(c)(i) M = f p [2]
(ii) Describe fully the single transformation represented by M–1, the inverse of M.
Answer(c)(ii) ...............................................................................................................................
..................................................................................................................................................... [2]__________________________________________________________________________________________
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ForExaminer′s
Use
17 (p, q) is the image of the point (x, y) under this combined transformation.
pqf p =
10
01
-f p xy
f p + 32f p
(a) Draw the image of the triangle under the combined transformation.
y
x1 2 3 4–4 –3 –2 –1 0
4
3
2
1
–1
–2
–3
–4[3]
(b) Describe fully the single transformation represented by 10
01
-f p .
Answer (b) ................................................................................................................................ [2]_____________________________________________________________________________________