Transformations

Rules for transformations of geometric shapes on a coordinate plane
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Transformation
To move a figure (or object).
Congruent
Figures that have the same size and shape. Figures that are reflected, rotated, or translated are congruent
Pre-Image (original/First)
The original object given. The FIRST or OLD figure.
Prime Notation
Symbolic representation given to images as a result of a transformation. If P is the original, P' (P prime) is the object after one transformation
Image (prime notation/Second/Copy)
A copy of the original: Example: A' is a copy of A
The SECOND or NEW figure.
TranSLation
To SLide a figure.
The SLIDE does not change the size, orientation (the way it faces), or shape of the image.
The image is congruent to the original.
Translation (slide)
(x, y) → (x + k, y + k)
K is a real (any) number

A translation can either effect only one value, for example the point can only slide up 2 units and only affect the y value (x, y+2) or the translation can affect both the x and y value, for example the point can slide up 2 units and to the left 3 units (x-3, y+2).
ReFLection (Flip)
To FLip a figure.
When a figure is facing itself. The distance from the reflection line is equal in the pre-image (original) and the image (new) when a reflection has taken place.

The FLIP does not change the size, or shape of the image, but does change the orientation.
The image is congruent to the original.
Reflection over the x-axis
(x, y) → (x, - y)
x stays same; y changes sign
Reflection over the y-axis
(x, y) → (- x, y)
x changes sign; y stays the same
RoTAion (Turn Around)
To Turn a figure Around a point.
We will use the origin (0, 0) as our point of rotation.

The TURN AROUND does not change the size, or shape of the image, but does change the orientation.
The image is congruent to the original.
Rotation 90° about the origin
(x, y) → (y, - x)
x change signs; y stays the same and then SWITCH the x and y values
Rotation 180° about the origin
(x, y) → (- x, - y)
x change signs; y change signs
Here they both change their signs, it is like a double reflection.
Rotation 270° about the origin
(x, y) → (- y, x)
x stays the same; y changes it's sign, and then SWITCH the x and y values
Scale Factor
A scale (k) written as a ratio in simplest form. The scale factor is a number used to multiply the lengths of a figure to stretch (enlarge) or shrink (reduce) it to a similar image.
Always multiply by the Scale Factor when preforming a Dilation.
To find the scale factor of 2 objects use Second/First.
S ÷ F
Enlargement
A dilation in which the image is larger than the pre-image.
Reduction
A dilation in which the image is smaller that the pre-image.
Dilation (Size Change)
Enlargement or reduction of the original figure.

The size change of course changes the size but the shape and orientation stay the same.
The image is no longer congruent but it is similar to the original.
Dilation (use Scale Factor)
(x, y) → (kx, ky)
where k is a real number
Here you multiply the original x and y value by the scale factor.