Full lesson notes
Everything covered in this lesson, in one place - useful for revision or printing.
\(\text{object} \rightarrow \text{image}\)
Lesson 1 gave you the four transformations and their rules. In this lesson you work on the grid: you describe each move, draw it, and check it with co-ordinates.
Translations: slide. Reflections: flip. Rotations: turn. Enlargements: resize.
Label every image with primes: point A becomes A′.
1. Translations
\(A(1\,;\,2) \rightarrow A'(3\,;\,-1)\)
Translate A(1 ; 2) two units to the right and three units down.
Right is positive and down is negative: the move is \((+2\,;\,-3)\).
\(x: 1 + 2 = 3 \qquad y: 2 - 3 = -1\)
To describe a translation, give the horizontal move first, then the vertical move.
A to A′: 2 right, 3 up: \((2\,;\,3)\) B to B′: 3 right, 2 down: \((3\,;\,-2)\)
C to C′: 4 left, 1 up: \((-4\,;\,1)\) D to D′: 4 left, 2 down: \((-4\,;\,-2)\)
P(2 ; 2) is translated to Q(4 ; 0). Subtract: image minus object.
\(x: 4 - 2 = 2 \qquad y: 0 - 2 = -2\)
The translation is \((2\,;\,-2)\): 2 right and 2 down.
Read each point's co-ordinates off the grid, then subtract.
B to C: \((-2\,;\,2) \rightarrow (1\,;\,3)\), translation \((3\,;\,1)\)
H to B: \((-2\,;\,-3) \rightarrow (-2\,;\,2)\), translation \((0\,;\,5)\)
J to D: \((-5\,;\,-5) \rightarrow (6\,;\,6)\), translation \((11\,;\,11)\)
To translate a shape, move every vertex by the same pair. To describe it, follow one corner.
A onto C: 2 left and 3 down, \((-2\,;\,-3)\)
D onto B: 1 left and 4 up, \((-1\,;\,4)\)
Key idea: the shape and size stay the same: object and image are congruent.
2. Reflections
A reflection flips the object over a mirror line.
Each image point is the same perpendicular distance from the mirror line as its object point.
Triangle A in m3 gives D; H in m1 gives A; C in m1 gives F; A in m2 gives B.
In the x-axis: keep x, change the sign of y. In the y-axis: change the sign of x, keep y.
C\((3\,;\,-4)\): in the x-axis \((3\,;\,4)\), in the y-axis \((-3\,;\,-4)\)
E\((-3\,;\,-2)\): in the x-axis \((-3\,;\,2)\), in the y-axis \((3\,;\,-2)\)
B\((2\,;\,0)\) is ON the x-axis, so in the x-axis it is its own image.
Change the sign of every x-co-ordinate; y stays the same.
\(M(-2\,;\,4) \rightarrow M'(2\,;\,4)\)
\(N(-4\,;\,1) \rightarrow N'(4\,;\,1)\) \(P(1\,;\,1) \rightarrow P'(-1\,;\,1)\)
Keep every x-co-ordinate; change the sign of every y-co-ordinate.
\(X(1\,;\,5) \rightarrow X'(1\,;\,-5)\) \(Y(-4\,;\,-2) \rightarrow Y'(-4\,;\,2)\)
\(Z(3\,;\,-1) \rightarrow Z'(3\,;\,1)\)
Key idea: name the mirror line, and check one point with the rule.
3. Rotations
A rotation turns the object about a centre, through an angle, in a direction (clockwise or anticlockwise).
90° anticlockwise: \((x\,;\,y) \rightarrow (-y\,;\,x)\)
180°: \((x\,;\,y) \rightarrow (-x\,;\,-y)\) 90° clockwise: \((x\,;\,y) \rightarrow (y\,;\,-x)\)
Rotate PA through 90° anticlockwise about P. From P to A you go 3 right and 2 up.
Turn the steps: 3 right becomes 3 up, and 2 up becomes 2 left.
From P to A′: 2 left and 3 up, so \((3\,;\,2) \rightarrow (-2\,;\,3)\)
Use the rule \((x\,;\,y) \rightarrow (-y\,;\,x)\) on every vertex.
\(A(4\,;\,-2) \rightarrow A'(2\,;\,4)\) \(B(4\,;\,3) \rightarrow B'(-3\,;\,4)\)
\(C(0\,;\,4) \rightarrow C'(-4\,;\,0)\) \(D(-4\,;\,3) \rightarrow D'(-3\,;\,-4)\)
Shape A turned about the origin: B is 90° anticlockwise, C is 180° and D is 90° clockwise.
The corner \((3\,;\,-2)\) goes to \((2\,;\,3)\), \((-3\,;\,2)\) and \((-2\,;\,-3)\).
Key idea: size and shape stay the same; only the position and the way it faces change.
4. Enlargements
Every length is multiplied by the same number: the scale factor \(k\).
Longest side of A: 4 units. Longest side of B: 12 units. \(k = 12 \div 4 = 3\)
\(k > 1\): bigger. \(0 < k < 1\): smaller. The angles never change.
Compare a length on the image with the matching length on the original.
Every length on the image car is twice as long: \(k = 2\)
\(k = \dfrac{\text{length on the image}}{\text{length on the object}}\)
The image robot is half the size: every length is halved.
\(k = \tfrac{1}{2}\): a scale factor between 0 and 1 gives a reduction.
Enlarge the staircase by \(k = 4\).
Perimeter: \(14 \rightarrow 56\), multiplied by 4.
Area: \(9 \rightarrow 144\), multiplied by \(4 \times 4 = 16\).
Perimeter is multiplied by \(k\); area by \(k \times k\).
Reduce the L-shape by \(k = \tfrac{1}{2}\).
Perimeter: \(20 \rightarrow 10\) (halved). Area: \(16 \rightarrow 4\), a quarter, because \(\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}\).
Key idea: lengths and perimeter scale by \(k\), area by \(k \times k\).
\(\begin{gathered} (x\,;\,y) \rightarrow (x+a\,;\,y+b) \\ \text{x-axis: } (x\,;\,-y) \qquad \text{y-axis: } (-x\,;\,y) \\ 180^{\circ}: (-x\,;\,-y) \qquad 90^{\circ} \text{ anticlockwise: } (-y\,;\,x) \\ \text{perimeter} \times k \qquad \text{area} \times k^{2} \end{gathered}\)
Translate by the same pair, image minus object. Reflect: change the sign of y (x-axis) or x (y-axis). Rotate about the origin with the rules. Enlarge: lengths by \(k\), area by \(k \times k\).
Always label the image with primes, and check one point with co-ordinates.