| TRANSFORMATIONS: Making Changes |
Introduction
When we buy a CD player or other device that operates on either electrical or battery power, it comes with a transformer -- that black box attached to the wire. We plug it into a wall socket and it changes the alternating current that flows through the wires into direct current to run the device. Without the transformer, we could not run our CD players with the electrical current from a wall socket.
A Transformation is a change of state or position.
In the case of electricity, the transformer changes the state or quality of the current.
Geometric transformers change the position of the figure in question.
There are 3 kinds of geometric transformations that change the position of given shapes.
They are TRANSLATIONS, REFLECTIONS and ROTATIONS.
This lesson covers translations and reflections.
Rotations will be covered in a separate lesson.
Translations (also called slide or shift)
The simplest transformation is a TRANSLATION. All we need is the original figure and the translation instructions -- how far and in what direction to move or shift the figure.
Example: Say we have triangle ABC in the diagram and we want to translate it 3 units right
and 5 units down -- we would do exactly that to points A, B and C -- then we'd join the translated points to have the translated image of triangle ABC.
Note how the coordinates of A/ B / C / are simply the coordinates of A, B and C with 3 added to each x-value and 5 subtracted from each y-value.
A(5, 8) became A/ (8, 3), B(2, 6) became B / (5, 1), and C(7, 5) became C / (10, 0).
Generally, the image of point A is labeled A/ (read "A prime"), so A/ B/ C/ is the image of triangle ABC under the translation 3 units right and 5 units down.
Notice how a translation doesn't change anything but the position of the original figure. The orientation -- or direction in which it points or faces doesn't change. Neither does the size.
Let's study the coded message in the algebraic expression for a Translation:
T(x, y) = (x - 7, y) moves a point horizontally 7 units left.(see the x - 7, y stays the same.)
T(x, y) = (x, y + 9) moves the point vertically 9 units up. (see the y + 9, x stays the same.)
T(x, y) = (x + 1, y - 4) moves the point 1 unit to the right and 4 units down. (both x, y change.)
So, to perform a translation, slide the points (figure) according to the instructions in the words or the algebraic expression of the instructions.
Now get a pencil, an eraser and a note book (graph paper), copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
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Practice Exercise 1 (translations):
1) Plot these points on graph paper, then describe in words
the translation that changes point A into point B.
| a) A(-3, 2); B(-3, 7) | b) A(1, 6); B(9, 4) | c) A(3, -2); B(-5, 7) | d) A(0, 0); B(12, 0) |
2) Write the algebraic expression for the translations in question 1 then apply each to P(-10, 17).
3) a) Plot triangle ABC with A at (1, 4) B at (3, 7) and C at (2, 2).
b) With a RED pencil, plot A/ B/ C /, the image of triangle ABC under T(x, y) = (x + 1, y - 5)
c) What are the coordinates of triangle A/ B/ C / ?
d) With a BLUE pencil, plot A// B// C // the image of triangle ABC under T(x, y) = (x - 3, y + 2)
e) What are the coordinates of triangle A// B// C // ?
Reflections (also called flips)
When you see your REFLECTION in a mirror, the image seems to be positioned the same distance from the surface of the mirror as you are. Also, the right side of your face appears to be the left side of the reflection's face and vice versa. If you raise your right hand, the guy in the mirror raises the hand on the left of his body. So a reflection changes more than position. A reflection changes the orientation of the figure.
To perform a translation, all we need to know are the position of the original figure and the translation instructions. Since a reflection happens in a MIRROR -- we must also specify the mirror. In geometry, the mirror is called the line or axis of reflection.
Let's look at reflections of triangle ABC in both the x-axis (vertical reflection) and the y-axis
(horizontal reflection) to see what we can discover about the changes it undergoes.
It's pretty obvious that a REFLECTION IS A FLIP.
When we flip the triangle vertically through a horizontal axis (the x-axis) what was top becomes bottom, and what was pointing down like BC is now pointing up like B/ C /. In geometry, top and bottom, up and down are measured by the y-coordinate so a vertical flip changes the sign of the y-value or ordinate of any point.
A vertical Reflection in the x-axis transforms (x, y) into (x, - y)
Example: a vertical reflection changes (-3, 7) to (-3, - 7). (6, - 9) becomes (6, 9).
all we do is change the sign of the y-value of the point.
When we flip the triangle horizontally through a vertical axis (the y-axis) what was left becomes right. In geometry, left and right are measured by the x-coordinate so a horizontal flip changes the sign of the x-value or absisca of any point.
A horizontal Reflection in the y-axis transforms (x, y) into (- x, y)
Example: a horizontal reflection changes (-3, 7) to (3, 7). (6, - 9) becomes (- 6, - 9).
all we do is change the sign of the x-value of the point.
Note: We change the coordinate opposite to the axis of reflection because a reflection through the y-axis, changes horizontal placement -- whereas a reflection in the x-axis changes the vertical placement of a figure.
Now get a pencil, an eraser and a note book (graph paper), copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
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Practice Exercise 2 (reflections) :
1) Label the reflection from C to D as V for vertical or H for horizontal and
name the axis of reflection.
| a) C(-3, 2); D(-3, - 2) | b) C(1, 6); D(- 1, 6) | c) C(3, -2); D(-3, - 2) | d) C(1, 20); D(1, - 20) |
2) a) On graph paper, plot triangle ABC with A(1, 4) B(4, 5) and C(6, 7).
b) With a RED pencil, plot A/ B/ C /, the image of triangle ABC under a
vertical reflection in the x-axis.
c) Write the coordinates of A/, B/, and C / on your diagram.
d) With a BLUE pencil, plot A// B// C // the image of triangle ABC under a
horizontal reflection in the y-axis.
e) Write the coordinates of A// B// C // on your diagram.
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Solutions
Practice Exercise 1: (translations)
1)
| a) A(-3, 2); B(-3, 7) | b) A(1, 6); B(9, 4) | c) A(3, -2); B(-5, 7) | d) A(0, 0); B(12, 0) |
a) Move the point up by 5 units.
b) Move the point 8 units right, and 2 units down.
c) Move the point 8 units left, and 9 units up.
d) Move the point 12 units right.
2) Write the algebraic expression for the translations in question 1 then apply each to P(-10, 17).
| a) T(x, y) = (x, y + 5); P / = (- 10, 22) | b) T(x, y) = (x + 8, y - 2); P / = (- 2, 15) |
| c) T(x, y) = (x - 8, y + 9); P / = (- 18, 26) | d) T(x, y) = (x + 12, y); P / = (2, 17) |
3) a) Plot triangle ABC with A at (1, 4) B at (3, 7) and C at (2, 2).
c) A/ = (2, - 1); B/ = (4, 2); and C / = (3, - 3)
e) A// = (- 2, 6); B// = (0, 9); and C // = (-1, 4).
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Practice Exercise 2 (reflections) :
1) Label the reflection from C to D as V for vertical or H for horizontal and
name the axis of reflection.
| a) C(-3, 2); D(-3, - 2) V: x-axis |
b) C(1, 6); D(- 1, 6) H: y-axis |
c) C(3, -2); D(-3, - 2) H: y-axis |
d) C(1, 20); D(1, - 20) V: x-axis |
2) a) On graph paper, plot triangle ABC with A(1, 4) B(4, 5) and C(6, 7).
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