Transformations

18.1   Introduction to Transformations

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Two figures are congruent if one can be moved so that it exactly overlaps the other. A figure cut out of paper can be turned, slid, and flipped over to see if it matches up with another figure. If the figure is put on a graph, then these movements will change the coordinates of the points. A transformation is a way to describe such a change of coordinates.

18.2   Transformation Notation

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A transformation begins with a general description of a point, such as P(x, y) which represents a point P with coordinates x and y. Following this is an arrow and then a description of the point’s image, the place where it ends up after the move. Usually the image of P is called P′, the image of A is called A′, and so on.

For example, the transformation P(x, y) P′(x – 5, 4 - y) means that the point A(2, 1) is moved to A′(2 – 5, 4 – 1) = A′ (– 3, 3), the point B(3, 5) is moved to B′(3 – 5, 4 – 5) = B′(–2, –1), and the point C(6, 1) is moved to C′(6 – 5, 4 – 1) = C′(1, 3). This transformation flips the triangle ΔABC over and slides it to the left, as shown in Fig. 18-1.

Fig. 18-1

SOLVED PROBLEMS

  18.1 Using transformation notation

Name the image of the points [&A(3, 1)&], [&B(3, 4)&], and [&C(5, 1)&] under the following transformations:

(a)  P(x, y) P′(x + 2, y – 1)

(b)  Q(x, y) Q′(x + 5, y)

(c)  R(x, y) R′(5x, 5y)

(d)  S(x, y) S′(–y, x)

(e)  T(x, y) T′(y, 5 – x)

Solutions

(a)  A′(3 + 2, 1-1) = A′(5, 0), B′(3 + 2, 4 – 1) = B′(5, 3), and C′(5 + 2, 1 – 1) = C′(7, 0)

(b)  A′(3 + 5, 1) = A′(8, 1), B′(3 + 5, 4) = B ′(8, 4), and C′ (5 + 5, 1) = C′ (10, 1)

(c)  A′(5 · 3, 5 · 1) = A′(15, 5), B′(5 · 3, 5 · 4) = B′(15, 20), and C′(5 · 5, 5 · 1) = C′(25, 5)

(d)  A′(–1, 3), B′(–4, 3), and C′(–1, 5)

(e)  A′(1, 5 – 3) = A′(1, 2), B′(4, 5 – 3) = B′(4, 2), and C′(1, 5 – 5) = C′(1, 0)

18.3   Translations

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A transformation that slides figures without flipping or rotating them is called a translation. The translation that slides everything to the right a units and up b units is [&P(x, y) P′~rom~(~normal~x+a, y+b)&].

SOLVED PROBLEMS

  18.2 Performing a translation

Let rectangle ABCD be formed by A(–1, 4), B(–1, 3), C(3, 3), and D(3, 4). Graph rectangle ABCD and its image under the following translations:

(a)  P(x, y) P′(x + 4, y + 3)

(b)  P(x, y) P″(x + 2, y – 5)

(c)  P(x, y) P(x – 6, y – 2)

Solutions

See Fig. 18-2.

Fig. 18-2

(a) A′(–1 + 4, 4 + 3) = A′(3, 7) B′(–1 + 4, 3 + 3) = B′(3, 6) C′(3 + 4, 3 + 3) = C′(7, 6), and D′(3 + 4, 4 + 3) = D′(7, 7)

(b) A″(–1 + 2, 4 – 5) = A″(1,–1) B″(–1 + 2, 3 – 5) = B″(1, –2), C″(3 + 2, 3 – 5) = C″(5, –2), and D″(3 + 2, 4 – 5) = D″(5, –1)

(c) A(–1 – 6, 4 – 2) = A(–7, 2) B(–1 – 6, 3 – 2) = B(–7, 1), C(3 – 6, 3 – 2) = C(–3, 1), and D(3 – 6, 4 – 2) = D(–3, 2)

  18.3 Recognizing a translation

Name the translation that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″, and (c) ΔABC as illustrated in Fig. 18-3.

Fig. 18-3

Solutions

(a)  P(x, y) P′(x – 4, y + 2)

(b)  P(x, y) P″(x + 2, y – 5)

(c)  P(x, y) P(x – 6, y – 3)

  18.4 Naming translations

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Name the translation that moves everything:

(a)  Up 6 spaces

(b)  Down 1 space

(c)  To the right 2 spaces

(d)  To the left 10 spaces

(e)  Up 5 spaces and to the right 3 spaces

(f)  Down 7 spaces and to the right 4 spaces

(g)  6 spaces to the left and 4 spaces up

Solutions

(a)  P(x, y) P′(x, y + 6)

(b)  P(x, y) P′(x, y – 1)

(c)  P(x, y) P′(x + 2, y)

(d)  P(x, y) P′(x – 10, y)

(e)  P(x, y) P′(x + 3, y + 5)

(f)  P(x, y) P′(x + 4, y – 7)

(g)  P(x, y) P′(x – 6, y + 4)

18.4   Reflections

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A transformation that flips everything over is called a reflection. This is because the image of an object in a mirror looks flipped over, as illustrated in Fig. 18-4.

Fig. 18-4

The reflection in Fig. 18-4 is a reflection across the y-axis because the edge of the mirror is pressed against the y-axis. The line where the mirror meets the plane is called the axis of symmetry.

The reflection across the vertical line x = a is given by P(x, y) P′(2a – x, y).

The reflection across the horizontal line y = a is given by P(x, y) P′(x, 2a – y).

SOLVED PROBLEMS

  18.5 Performing reflections

Let triangle ABC be formed by A(– 1, 1), B(0, 3), and C(3, 1). Graph ΔABC and its image under:

(a)  Reflection across the x axis (y = 0), P(x, y) P′(x,–y)

(b)  Reflection across the line x = 4, P(x, y) P″(8 – x, y)

(c)  Reflection across the line y = 5, P(x, y) P(x, 10 – y)

Solutions

See Fig. 18-5.

(a)  A′(–1, –1), B′(0, –3), and C′(3, –1)

(b)  A″(8 – (–1), 1) = A″(9, 1), B″(8 – 0, 3) = B″(8, 3), and C″(8 – 3, 1) = C″(5, 1)

(c)  A(–1, 10 – 1) = A(–1, 9), B(0, 10 – 3) = B(0, 7), and C(3, 10 – 1) = C(3, 9)

Fig. 18-5

  18.6 Recognizing reflections

Name the reflection that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″, and (c) ΔABC as illustrated in Fig. 18-6.

Fig. 18-6

Solutions

(a)  Reflection across the y axis, P(x, y) P′(–x, y)

(b)  Reflection across the line y = –1, P(x, y) P″(x,–2 – y)

(c)  Reflection across the line x = 4, P(x, y) P(8 – x, y)

  18.7 Naming reflections

Name the transformation that

(a)  Reflects across x = 2

(b)  Reflects across y = 6

(c)  Reflects across x = –10

(d)  Reflects across

Solutions

(a)  P(x, y) P′(4 – x, y)

(b)  P(x, y) P′(x, 12 – y)

(c)  P(x, y) P′(–20 – x, y)

(d)  P(x, y) P′(x, 1 – y)

18.4A   Reflectional Symmetry

A figure has reflectional symmetry if it looks the same after being flipped across an axis of symmetry that runs through its center. As illustrated in Fig. 18-7, a figure can have (a) one, (b) several, or (c) no axes of symmetry.

Fig. 18-7

SOLVED PROBLEMS

  18.8 Recognizing reflectional symmetry

Which of the figures in Fig. 18-8 have reflectional symmetry?

Fig. 18-8

Solutions

Only (b), (e), and (f) have reflectional symmetry. Note that when (d) is flipped, it will look like Fig. 18-9, which is different from the original in that the upper-left-hand crossing is horizontal instead of vertical.

Fig. 18-9

18.5   Rotations

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If a pin were pushed through the origin on a graph and the paper were to be turned, the result would be a rotation about the origin. A rotation is described by the number of degrees by which the paper is turned.

The 90° clockwise rotation (or 270° counter-clockwise) about the origin is given by P(x, y) P′(y, – x).

The 180° rotation about the origin is given by P(x, y) P′(–x, – y).

The 270° clockwise (or 90° counter-clockwise) rotation about the origin is given by P(x, y) P′(–y, x).

In general, the clockwise rotation about the origin of u° is given by P(x, y) P′(x cos θ + y sin θ, y cos θ – x sin θ)

SOLVED PROBLEMS

  18.9 Performing rotations

Let triangle ABC be given by A(2, 1), B(3, 1), and C(3, 4). Graph the image of ΔABC as rotated about the origin by (a) 90° clockwise, (b) 180°, and (c) 270° clockwise.

Solutions

See Fig. 18-10.

Fig. 18-10

(a)  A′(1,–2), B′(1,–3), and C′(4,–3)

(b)  A″(–2,–1), B″(–3,–1), C″(–3,–4)

(c)  A(–1, 2), B(–1, 3), C(–4, 3)

18.10 Recognizing rotations

Name the rotation that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″&], and (c) ΔABC as illustrated in Fig. 18-11.

Fig. 18-11

Solutions

(a)  270° clockwise or 90° counter-clockwise about the origin, P(x, y) P′(– y, x)

(b)  180° about the origin (either clockwise or counter-clockwise), P(x, y) P″(–x, –y)

(c)  45° clockwise, P(x, y) P(xcos 45° + y sin 45°, y cos 45°– x sin 45°)

18.11 Naming rotations

Name the transformation that rotates clockwise about the origin:

(a)  20°

(b)  30°

(c)  60°

(d)  75°

Solutions

(a)  P(x, y) P′(x cos 20° + y sin 20°, y cos 20° – x sin 20°) = P′(0.9397x + 0.3420y, 0.9397y – 0.3420x)

(b)  P(x, y) P′(x cos 30° + y sin 30°, y cos 30° – x sin 30°) = P′(0.866x + 0.5y, 0.866y – 0.5x)

(c)  P(x, y) P′(x cos 60° + y sin 60°, y cos 60° – x sin 60°) = P′(0.5x + 0.866y, 0.5y – 0.866)

(d)  P(x, y) P′(x cos 75° + y cos 75°, y sin 75° – x sin 75°) = P′(0.2588x + 0.9659y, 0.2588y – 0.9659x)

18.5A   Rotational Symmetry

A figure has rotational symmetry if it can be rotated around its center by fewer than 360° and look the same as it did originally. In Fig. 18-12, there is (a) a figure that looks the same under a 72° rotation, (b) a figure that looks the same under a 120° rotation, (c) a figure that looks the same under a 180°, and (d) a figure without rotational symmetry.

Fig. 18-12

SOLVED PROBLEMS

18.12 Recognizing rotational symmetry

For each figure in Fig. 18-13, give the smallest angle by which the figure could be rotated around its center and still look the same.

Fig. 18-13

Solutions

(a)  90°

(b)  120°

(c)  360° (no rotational symmetry)

(d)  180°

(e)  360° (no rotational symmetry)

(f)  90°

18.6   Rigid Motions

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Any combination of translations, reflections, and rotations is called a rigid motion because figures are moved without changing angles, lengths, or shapes. The image of a figure under a rigid motion will always be congruent to the original.

SOLVED PROBLEMS

18.13 Graphing rigid motions

Let triangle ABC be formed by A(–4, 2), B(–4, 1), and C(–1, 1). Graph ΔABC and its image under the following combinations of transformations:

(a)  Reflect across the y axis and then move to the right 4 spaces.

(b)  Rotate 90° clockwise around the origin then move up 3 spaces.

(c)  Reflect across y = 2 then move up 2 spaces and to the left 3 spaces.

(d)  Reflect across the x axis and then reflect across the y axis.

(e)  Rotate 90° counter-clockwise around the origin and then reflect across x = –3.

Solutions

See Fig. 18-14.

Fig. 18-14

18.14 Combining transformations

Name the single transformation that does the same thing as the combination of:

(a)  P(x, y) P′(x + 7, y – 2) and then Q′(x, y) Q″(x + 3, y + 5)

(b)  P(x, y) P′(–x, y) and then Q′(x, y) Q″(x + 3, y + 2)

(c)  P(x, y) P′(–x,–y) and then Q′(x, y) Q″(4 – x, y)

(d)  P(x, y) P′(y,–x) and then Q′(x, y) Q″(x – 5, y + 1)

Solutions

(a)  R(x, y) R″((x + 7) + 3, (y – 2) + 5) = R″(x + 10, y + 3)

(b)  R(x, y) R″(–x + 3, y + 2)

(c)  R(x, y) R″(4 – (–x),–y) = R″(4 + x,–y)

(d)  R(x, y) R″(y – 5, –x + 1)

18.15 Recognizing rigid motions

Name the transformation that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″, and (c) ΔABC as illustrated in Fig. 18-15.

Fig. 18-15

Solutions

(a)  The triangle has been reflected across the y axis and then moved up 1 space, so P(x, y)P′(–x, y + 1).

(b) The triangle has been reflected across the line y = 3, then moved up 1 space and to the left 2 spaces, so P(x, y) P′(x – 2, 7 – y).

(c) The triangle has been rotated clockwise around the origin 90° and then moved up 8 spaces and to the right 6 spaces, so P(x, y) P′(y + 6, – x + 8).

18.16 Naming rigid motions

Name the transformation that

(a)  Rotates everything around the origin 180°, then moves everything up 3 spaces

(b)  Reflects across x = 4, then slides everything down 2 spaces

(c)  Rotates everything 90° clockwise around the origin, then reflects across the y axis

(d)  Rotates around the origin 90° counter-clockwise, then slides to the left 5 spaces

Solutions

(a)  P(x, y) P′(–x, –y + 3)

(b)  P(x, y) P′(8 – x, y – 2)

(c)  P(x, y) P′(–y, –x)

(d)  P(x, y) P′(–y – 5, x)

18.7   Dihilations

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A dihilation (also called a scaling or an enlargement) is not a rigid motion because it multiplies all lengths by a single scale factor. The image of a figure under a dihilation will always be similar to the original.

The dihilation that enlarges everything by a scale factor of k is P(x, y) P′(kx, ky).

SOLVED PROBLEMS

18.17 Performing dihilations

Let triangle ABC be formed by A(2, 2), B(4, 2), and C(4, 3). Graph the image of ΔABC under (a) magnification by 2 and (b) scaling by .

Solutions

See Fig. 18-16.

Fig. 18-16

(a)  A′(4, 4), B′(8, 4), and C′(8, 6)

(b)  A″(1, 1), B″(2, 1), and C″(2, 1.5)

18.18 Naming dihilations

Name the transformation that

(a)  Scales everything 5 times larger

(b)  Shrinks every length to half size

(c)  Triples all linear dimensions

(d)  Depicts everything at scale

(e)  Dihilates by a scale factor of 12

Solutions

(a)  P(x, y) P′(5x, 5y)

(b)  

(c)  P(x, y) P′(3x, 3y)

(d)  

(e)  P(x, y) P′(12x, 12y)

SUPPLEMENTARY PROBLEMS

18.1.   Name the image of points A(6, 2), B(–1, 4), and C(2, 7) under the transformation

(a)  P(x, y) P′(x + 3, y)

(b)  P(x, y) P′(2 – x, y)

(c)  P(x, y) P′(–x + 1, –y +3)

(d)  P(x, y) P′(–y, x + 8)

(e)  P(x, y) P′(2x, 2y)

(f)  P(x, y) P′(4 – 3x, 3y)

(g)  P(x, y) P′(2 – y, 5 – x)

18.2.   Let triangle ABC be defined by A(1, –1), B(2, 2), and C(3, –1). Graph the image of ΔABC under

(a)  P(x, y) P′(x + 5, y)

(b)  P(x, y) P′(x, y – 4)

(c)  P(x, y) P′(x – 3, y + 2)

18.3.   Name the translation that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″, and (c) ΔABC as illustrated in Fig. 18-17.

Fig. 18-17

18.4.   Name the translation that moves everything

(a)  Down 5 spaces

(b)  To the right 6 spaces

(c)  Up 3 spaces and 7 spaces to the left

(d)  Down 2 spaces and 8 spaces to the right

(e)  Up 4 spaces and to the left 1 space

18.5.   Let trapezoid ABCD be formed by A(1, 3), B(5, 3), C(4, 1), and D(2, 1). Graph trapezoid ABCD and its image under (a) reflection across the y axis P(x, y) P′(–x, y), (b) reflection across the line y = –1, P(x, y) P″(x, –2 – y), and (c) reflection across the line x = 8, P(x, y) P(16 – x, y).

18.6.   Name the reflection that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″, and (c) ΔABC as illustrated in Fig. 18-18.

Fig. 18-18

18.7.   Name the transformation that

(a)  Reflects across y = 5

(b)  Reflects across x = –2

(c)  Reflects across y = –1

(d)  Reflects across

18.8.   Which of the figures in Fig. 18-19 has reflectional symmetry?

Fig. 18-19

18.9.   Let parallelogram ABCD be defined by A(1, 2), B(4, 2), C(5, 1), and D(2, 1). Graph parallelogram ABCD and its image under (a) a 90° clockwise rotation about the origin, (b) a 180° rotation about the origin, and (c) a 270° clockwise rotation about the origin.

18.10.   Name the rotation that takes ΔABC to (a) ΔA′B′C′, (b) ΔA″B″C″, and (c) ΔABC as illustrated in Fig. 18-20.

Fig. 18-20

18.11.   Name the transformation that rotates clockwise about the origin:

(a)  40°

(b)  50°

(c)  80°

18.12.   For each figure in Fig. 18-21, give the smallest angle by which the figure could be rotated around its center and still look the same.

Fig. 18-21

18.13.   Let triangle ABC be defined by A(2, 1), B(3, 2), and C(3,–1). Graph ΔABC and its image under the following combinations of transformations:

(a)  Reflect across the line y = 3 and then move to the right 2 spaces.

(b)  Rotate about the origin 90° clockwise and then move to the right 1 space and down 3 spaces.

(c)  Rotate about the origin 270° clockwise and then reflect across the x axis.

(d)  Reflect across the line x = –1 and then move up 2 spaces.

18.14.   Name the single transformation that does the same thing as

(a)  P(x, y) P′(x + 5, y – 3) and then Q′(x, y) Q″(x + 1, y + 2)

(b)  P(x, y) P′(5 – x, y) and then Q′(x, y) Q″(x – 4, y + 2)

(c)  P(x, y) P′(y,–x) and then Q′(x, y) Q″(x + 3, y – 6)

(d)  P(x, y) P′(–y, x) and then Q′(x, y) Q″(x, 4 – y)

(e)  P(x, y) P′(x,–3 – y) and then Q′(x, y) Q″(6 – x, y)

18.15.   Name the transformation that takes ΔABC to (a) [&ΔA′B′C′&], (b) [&ΔA″B″C″&], and (c) [&ΔABC&] as illustrated in Fig. 18-22.

Fig. 18-22

18.16.   Name the transformation that

(a)  Reflects across the x axis and then moves everything down 3 spaces

(b)  Rotates around the origin clockwise 90° and then moves everything to the right 2 spaces

(c)  Reflects across the line y = 2 and then rotates 180° around the origin

(d)  Rotates 180° around the origin and then reflects across the line y = 2

(e)  Moves everything up 3 spaces and to the left 1 space, then reflects across the line x = –4

18.17.   Let rectangle ABCD be formed by A(–1, 2), B(1, 2), C(1, 1), and D(–1, 1). Graph this rectangle and also its image under the transformation P(x, y) P′(3x, 3y).

18.18.   Name the transformation that

(a)  Scales everything to be twice as large

(b)  Scales everything by scale factor 8

(c)  Dihilates everything by a scale factor of