Name:______Date:_____Period:____

Learning Packet: B/F/GMs. Anderle

There will be an exam the Tuesday (F/G) or Wednesday (B) that we return on transformations (sorry ). I will be available for extra help in the morning – at 7 so even students who have H period can come. Relax. It is not difficult, as long as your memorize those formulas! The exam has 10 multiple choice questions, 2 questions where you have to draw in the axis of symmetry, 9 transformation questions, and 2 Part II transformation questions (graphing). If there are any questions, I will be checking my e-mail all week. So please do not hesitate to e-mail me.

F Period: I know that we did not go over Rotations; however, if you follow the rules, then you will be fine. Below I have listed all of the rules for all of the transformations that we need to know for the exam.

Please complete this packet NEATLY and on a SEPARATE SHEET OF PAPER! I WILL BE COLLECTING THIS!!! I will not look at any sloppy work. So print out TWO copies. One that you will hand in and one that you will have in class so we can go over it on Monday!

Transformations Definitions & Formula Sheet

Translation:

Slides a figure a certain distance. The shape does not change size.

Mapping Rule: Ta, b(x, y) = (x + a, y + b).

Dilation:

Enlarges or shrinks an image.

Mapping Rule: Dk(x,y) = (kx,ky)

Line Reflection:

An image is “flipped” over a line.

Mapping Rules:

-Reflection over the x-axis

rx-axis(x, y) = (x, -y)

-Reflection over the y-axis

ry-axis(x, y) = (-x, y)

-Reflection over the y=x line

ry = x(x, y) = (y, x)

-Reflection over the y=-x line

ry=-x(x, y) = (-y, -x)

-Reflection over the line y = k (where k is a NUMBER)

ry = k(x, y) = (x, 2k – y)

-Reflection over the line x = k (where k is a NUMBER)

rx = k(x, y) = (2k – x, y)

Point Reflection:

An image is “flipped” over a point.

-Point Reflection in the Origin

rorigin(x, y) = (-x, -y) or rO(x, y) = (-x, -y

-Point Reflection over a given point (h,k) (where h and k are constants)

r(h, k)(x, y) = (2h – x, 2k – y)

Rotation:

An image is “turned” over a point (usually the origin). Rotations are usually counterclockwise unless otherwise noted. If the image is rotated CLOCKWISE the degree value becomes negative.

Rotation of 90° and -270°

R90° = R-270°(x,y) = (-y,x)

Rotation of 180° and -180°

R180° = R-180˚(x,y) = (-x,-y)

Rotation of 270 or -90

R270˚ = R-90˚(x,y) = (y,-x)

Properties of Transformations:

All transformations except for reflection preserve collinearity. Collinearity is when the letter order of the vertices are in the same order. In a reflection the order changes from clockwise to counterclockwise. All transformations except for a dilation preserve distance. Isometry is when a transformation preserves distance. Therefore, all transformations except for a dilation are isometry!

(we will review this some more when we return).

Part II: Transformation Questions

To complete, these examples we just need to follow the steps below. First do part a, then the results that you get in part a, use those for part b, and so on.

When a problem asks what is one single transformation that would map A onto A” all that you have to do is add the two translations together.

For example, if you performed T2,3 on A and then T-3,1 on A’ to get from A to A” all that you have to do is add 2,3 to -3,1  Your new translation is T-1,4.

Please complete all work on a separate sheet of paper. Make sure you GRAPH!!!! To get graph paper, go to the link on my website.

Example: Please complete this EXACTLY how I do below. If it is not that EXACT way you WILL LOSE POINTS!!!

  1. The coordinates of the vertices of ΔLEO are L(1,2), E(2,0), and O(0,3). Graph and label ΔLEO.
  2. Graph and state the coordinates of ΔL’E’O’, the image of ΔLEO after the transformation D2.
  3. Graph and state the coordinates of ΔL”E”O”, the image of ΔL’E’O’ after the transformation T2,-3.
  4. Graph and state the coordinates of ΔL’’’E’’’O’’’, the image of ΔL”E”O” after the transformation ry=x.

1. a. D2L(1,2)  L’(2,4)

D2E(2,0)  E’(4,0)

D2O(0,3)  O’(0,6)

b. T2,-3L’(2,4)  L”(4,1)

T2,-3E’(4,0)  E”(6,-3)

T2,-3O’(0,6)  O”(2,3)

c. ry=xL”(4,1)  L”’(1,4)

ry=xE”(4,0)  E”’(0,4)

ry=xO”(2,3)  O”’(3,2)

Now Try:

  1. The coordinates of the vertices of ΔIAN are I(8,5), A(6,3), and N(5,-2). Graph and label ΔIAN.
  2. Graph and state the coordinates of ΔI’A’N’, the image of ΔIAN after the transformation T-7,1.
  3. Graph and state the coordinate of ΔI”A”N”, the image of ΔI’A’N’ after the transformation T2,-3
  4. Name a single transformation that would map ΔIAN onto ΔI”A”N”
  5. The coordinates of the vertices of ΔMAT are M(6,0), A(0,3), and T(3,9). Graph and label ΔMAT.
  6. Graph and state the coordinates of ΔM’A’T’, the image of ΔMAT after the transformation (x,y)  (-1/3x, -1/3y)
  7. Graph and state the coordinates of ΔM”A”T”, the image of ΔM’A’T’ after a counterclockwise rotation of 270˚.
  8. Graph and state the coordinates of ΔM’’’A’’’T’’’, the image of ΔM”A”T” after a reflection in the origin.

Composition of Transformations

When two or more transformations are combined to form a new transformation, the result is called a composition of transformations. In a composition, the first transformation produces an image upon which the second transformation is then performed.

The symbol for composition is an open circle.

The notation rx-axis ◦ T3,4 is read as a reflection in the x-axis FOLLOWING a translation of (x+3,y+4). Basically, this process is done in REVERSE. So whenever you see a composition of a transformation, you work BACKWARDS!!!

Example:

What is R180˚ ◦ T-2,3A(5,14)?

Step 1: We need to complete the composition in reverse.

So we must first do T-2,3

T-2,3A(5,14) A’(3,17)

Step 2:We need to do the next transformation: R180˚

R180˚A’(3,17) A”(-3,-17)

Now Try: For #3-5 please GRAPH the problems on GRAPH paper. Do NOT make your OWN graph. Complete those examples, the same way that you completed the Part II questions. You cannot simply go from the original point to the ending point. I must see ALL of the steps in between.

  1. What is T2,1 ◦ ry=-x ◦ D3(2,1) ?
  2. What is T7,-2 ◦ D1/2 ◦ RO (8,10)?
  3. The coordinates of triangle FUN are F(-5,1), U(-1,1), and N(-1,7). On a coordinate plane, draw and label ΔFUN. Perform the composition transformation ry-axis ◦ rx-axis.
  4. Graph ΔABC with vertices A(1,4), B(3,7), and C(5,1). Graph and label the composition T-5,-2 ◦ rx-axis.
  5. Graph ΔTOM with vertices T(-2,3), O(4,5), and M(0,5). Graph and label the composition R90˚ ◦ ry-axis.

Glide Reflection

When a translation (a slide) and a reflection are performed one after the other, a transformation called a glide reflection is produced. In a glide reflection, the line of reflection is parallel to the direction of the translation.It does not matter whether you glide first and then reflect, or reflect first and then glide.This transformation is commutative.

***Complete the GLIDE REFLECTION problems the same way that you completed the Part II questions AND the Composition of Transformation examples***

Now Try:

  1. ΔJKL has vertices J(6,-1), K(10,-2), and L(5,-3). Graph ΔJKL and its image after a T0,4 and a reflection in the y-axis.
  2. ΔPQR has vertices P(1,1), Q(2,5), and R(4,2). Graph ΔPQR and its image after the glide reflection rx-axis ◦ T-3,-3.
  3. Quadrilateral LISA has vertices L(-3,4), I(-1,3), S(-1,1), and A(-4,2). Graph LISA and its image after rx-axis ◦ T5,0
  4. ΔTUV has vertices T(2,-1), U(5,-2), and V(3,-4). Graph ΔTUV and its image after a transformation (x,y)(x – 1, y + 5) and a rotation of 180˚. Is this a glide reflection?
  5. ΔDFG has vertices D(2,8), F(1,2), and G(4,6). Graph ΔDFG and its image after ry=x ◦ T3,3. Is this a glide reflection?
  6. Examine the graph below. Is this an example of a glide reflection?