How do we review transformations?

Most or almost all transformations fall under four categories: Reflection, dilation, translation, rotation.

To reflect an image you need a line to reflect over. You can reflect any image over any line.
For example: reflecting points (3,2) (5,1) and (4,5) over the y-axis would be (-3,2), (-5,1) and (-4,5).

To dilate an image or point you would need a scale factor. the scale factor is what you multiply the point's coordinates by. For example if the scale factor is 2 and the points are (5,4) the new point coordinate would be (10,8).


Translation, which is probably the easiest out of all the transformations, is when you move the point accordingly. For example if it says T (2,3) and the point is (4,6) then the new translated point is (6,9).


A Rotation is a transformation that turns a figure about a fixed point. If you nrotate any point it will probably be counter-clockwise unless requested to be rotated clockwise. If you have a point (1,6) and rotate it 90 degrees then it would be (-6,1). when rotating you could always follow this rule.
90 degrees: (A,B)--->(-B,A)
180 degrees: (A,B)--->(-A,-B)
270 degrees: (A,B)--->(B,-A)

Aim: How do we use the other definitions of transformations?

Other definitions of transformations would be things such as; Glide reflections, orientation, isometry and invarient.

Glide Reflection- the combination of a reflection in a line and a translation along that line.

Orientation- orientation refers to the arrangement of points, relative to one another, after a transformation has occured.

Isometry- an isometry is a transformation of the plane that preserves length.

Invarient- a figure or property that remains unchanged under a transformation of the plane is reffered to as invarient. No variations have occured.

All of these terms are other kinds of defintions of transformation.

One of the more important and more used terms here would be isometry. Every kind of transfromation falls under the definition of isometry except for dilation.

Aim: How do we identify composition of transformations?

Composition of Transformations- When two or more transfromations are combined to form a new transformation.

An example would be this:

Whichever transformation has the little circle in front of it is the one that is done first
because it is read as the reflection  over the x-axis after the translation of (3,4).


Try this: R y-axis after T (7,8) on point (2,3).

Aim: How do we graph dilations?

Dilation's- A dilation is a type of image that causes an image to stretch or in proportion to its original size.

When a image is dilated it has a scale factor in which it either shrinks or enlarges.if the scale factor is >1 then the image is enlarged. If the scale factor is <1 then the image will shrink. Whatever the scale factor is then you multiply it by each of the points.


 This image is an example of a dilation. 
The second triangle increased proportionally 
from the original shape.




Dilate these points if the scale factor is 2:
(4,3), (-1,2), (0,5)

Aim: How do we graph transformations that are reflections?

A reflection is a transformation that creates mirror images over the X or Y-axis:
Ex. This image is an expample of a refelction.



To reflect certain points on a graph you have to follow this rule:
Reflecting over the X-axis-- (x,y) ---> (x,-y)
Reflecting over the Y-axis-- (x,y) ---> (-x,y)


Try this: Reflect point (6,9) over the X-axis.