The Transformation Simulation above is summarized in the notes below.

Transformation

Example

original = ΔABC
image = ΔA'B'C'

Mapping Notation

original = ΔABC
image = ΔA'B'C'
Translation (Slide)
  • The object and image have the same orientation.
  • Corresponding sides of the object and image are parallel -->>--.
  • The object and image are congruent (corresponding angles and sides are congruent).

 

mapping: (x, y) --> (x+3, y+2)

ordered pair: [3, 2] means 3 right and 2 up

slidearrow1.gif (1292 bytes)

slide.gif (2285 bytes)

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Additional Mapping Example

(x, y) --> (x-4, y-5)

ordered pair: [-4, -5]

means left 4 down 5 as shown by the slide arrow:

slidearrow2.gif (1564 bytes)

(x, y) --> (x+3, y+2)

 

ΔABC is moved 3 units right and 2 units up as shown by the slide arrow to the left.

ΔABC is located at:

A(7,4), B(5,2), C(3, 4)

Image ΔA'B'C' is

A'(7+3,4+2)
B'(5+3,2+2)
C'(3+3,4+2)

or

A'(10,6), B'(8,4), C'(6, 6)

Reflection through the x-axis (Flip)
  • The object and image are equidistant from a line of reflection.
  • Segments joining corresponding sides of an object and image are   perpendicular to the reflection line.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (x, -y)

flipx.gif (2542 bytes)

(x, y) --> (x, -y)

ΔABC is being flipped across the x-axis (y = 0).

ΔABC is located at:

A(4,6), B(4,2), C(2,2)

Image ΔA'B'C' is

A'(4,-6), B'(4,-2), C'(2,-2)

Reflection through the y-axis (Flip)
  • The object and image are equidistant from a line of reflection.
  • Segments joining corresponding sides of an object and image are   perpendicular to the reflection line.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (-x, y)

flip.gif (2564 bytes)

(x, y) --> (-x, y)

ΔABC is being flipped across the y-axis (x = 0).

ΔABC is located at:

A(4,6), B(4,2), C(2,2)

Image ΔA'B'C' is

A'(-4,6), B'(-4,2), C'(-2,2)

Reflection through y = x (Flip)
  • The object and image are equidistant from a line of reflection.
  • Segments joining corresponding sides of an object and image are   perpendicular to the reflection line.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (y, x)

flip_y_equals_x.gif (2811 bytes)

(x, y) --> (y, x)

ΔABC is being flipped across the line y = x.

ΔABC is located at:

A(4,6), B(4,2), C(2,2)

Image ΔA'B'C' is

A'(6, 4), B'(2, 4), C'(2, 2)

Reflection through y = - x (Flip)
  • The object and image are equidistant from a line of reflection.
  • Segments joining corresponding sides of an object and image are   perpendicular to the reflection line.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (-y, -x)

flip_y_equals_neg_x.gif (3086 bytes)

(x, y) --> (-y, -x)

ΔABC is being flipped across the line y = -x.

ΔABC is located at:

A(4,6), B(4,2), C(2,2)

Image ΔA'B'C' is:

A'(-6,-4), B'(-2,-4), C'(-2,-2)

Reflection through x = 1 (Flip)
  • The object and image are equidistant from a line of reflection.
  • Segments joining corresponding sides of an object and image are   perpendicular to the reflection line.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (2-x, y)

flip_y_equals_1.gif (2721 bytes)

(x, y) --> (2-x, y)

ΔABC is being flipped across the line x = 1.

ΔABC is located at:

A(4,6), B(4,2), C(2,2)

Image ΔA'B'C' is:

A'(-2,6), B'(-2,2), C'(0,2)

Rotation about the origin

tc_halfcw.gif (1000 bytes)half turn clockwise (cw) or tc_halfccw.gif (1001 bytes)half turn counterclockwise (ccw)

  • The object and image are equidistant from a given point.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping (x, y) --> (-x, -y)

halfcw.gif (2395 bytes)

turn center = origin

(x, y) --> (-x, -y)

ΔABC is rotated 1/2 turn around the origin (clockwise or counterclockwse.

ΔABC is located at:

A(-3,-4), B(-2,-1), C(-5,-5)

Image ΔA'B'C' is:

A'(3, 4), B'(2, 1), C'(5, 5)

Rotation about the origin

tc_quartercw.gif (982 bytes)quarter turn clockwise (cw) or tc_3quarterccw.gif (1031 bytes) three quarter turn counterclockwise (ccw)

  • The object and image are equidistant from a given point (turn centre).
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (y, -x)

quartercw.gif (2362 bytes)

turn center = origin

(x, y) --> (y, -x)

ΔABC is rotated 1/4 turn around the origin.

ΔABC is located at:

A(-3,-4), B(-2,-1), C(-5,-5)

Image ΔA'B'C' is:

A'(-4, 3), B'(-1, 2), C'(-5, 5)

Rotation about the origin

tc_quarterccw.gif (987 bytes)quarter turn counterclockwise (ccw) or tc_3quartercw.gif (1030 bytes)three quarter turn clockwise (cw)

  • The object and image are equidistant from a given point.
  • The object and image are congruent (corresponding angles and sides are congruent).

mapping: (x, y) --> (-y, x)

quartercw.gif (2494 bytes)

turn center = origin

(x, y) --> (-y, x)

A(-3,-4), B(-2,-1), C(-5,-5)

ΔABC is rotated 1/4 turn around the origin.

Image ΔA'B'C' is:

A'(4, -3), B'(1, -2), C'(5, -5)

Dilatation about the origin (Enlargement or Reduction)
  • The object and image are similar when the dilatation factor 1. (corresponding angles are congruent).
  • The object and image are congruent when the dilatation factor = 1. (corresponding angles and sides are congruent).

mapping: (x, y) --> (3x, 3y)

dilatation.gif (2609 bytes)

dilatation centre = origin

(x, y) --> (3x, 3y)

ΔABC is being enlarged around the origin.

A(2,3), B(2,1), C(1,1)

Image ΔA'B'C' is:

A'(6, 9), B'(6, 3), C'(3, 3)

Comments to:  Jim Reed
Started September, 1998. Copyright 1999, 2000, 2001, 2002, 2003, 2004

 

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