day 1: transformations and translations · web viewmath 2 spring 2018 mrs. bello unit 1 anchor...

Unit 1

Transformations

Math 2

Spring 2018

Mrs. Bello

Unit 1 Anchor Standards:

· NC.M2.G-CO.2: Experiment with transformations in the plane; represent transformations in the plane; compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.g. stretches, dilations). Understand that rigid motions produce congruent figures while dilations produce similar figures.

· NC.M2.G-CO.5: Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image.

· NC.M2.G-CO.6: Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other.

Unit 1 Supporting Standards:

· NC.M2.F-IF.1: Extend the concept of a function to include geometric transformations in the plane by recognizing that:

· the domain and range of a transformation function f are sets of points in the plane;

· the image of a transformation is a function of its pre-image.

· NC.M2.F-IF.2: Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.

· NC.M2.G-CO.3: Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry. Represent transformations in the plane.

· NC.M2.G-CO.4: Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

· NC.M2.G-SRT.1: Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations with given center and scale factor:

1. When a line segment passes through the center of dilation, the line segment and its image lie on the same line. When a line segment does not pass through the center of dilation, the line segment and its image are parallel.

1. Verify experimentally the properties of dilations with given center and scale factor: The length of the image of a line segment is equal to the length of the line segment multiplied by the scale factor.

1. The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image.

1. Dilations preserve angle measure.

VOCABULARY

Transformation

Translations

Reflection

Rotation

Dilation

Rigid motion

Isometry

Image

Pre - Image

Composition

Function

Scale Factor

Similarity

Domain

Range

Notation

Congruence

Vector

INTRO TO TRANSFORMATIONS

Day 1: Transformations and Translations

Congruent figures _______________________________________________________________ .

When two figures are congruent, you can move one so that ________________

_______________________________________________.

Transformation of a geometric figure: change in its __________, _________, or ___________

Preimage – ________________ figure Notation: __________

Image – _______ or _______________ figureNotation: __________

Isometry – transformation in which preimage and image are the ____________ ___________ and _______________(also called: _________________________)

Examples:

___________________ , _____________________ , and _________________

Translation – an isometry that maps all points the ___________ ____________________ and the ___________ ___________________.

Activity 1: Patty Paper Translation

The translation T is defined by T(A) = B … meaning that it slides the figure the distance AB in the direction that goes.

1) Place the patty paper over this page. Trace the triangle and points A and B.

2)

Slide the patty paper along so that the A on the patty paper is on top of B on this sheet and B on the patty paper is still on on this sheet.

3) The position of the triangle on your patty paper now corresponds to the image of XYZ under the translation, T. If you press down hard with a sharp pencil, the image of the triangle can be seen on this page when you remove the patty paper.

A

B

X

Y

Z

Translation Vector – an arrow that indicates the distance and direction to translate a figure in a plane. in the activity above is an example of a translation vector.

The notation for a vector is: < -a, b > for a translation a units to the left and b units up.

Three ways to describe a transformation (using example shown right):

**Always be specific when completing any type of description!!

1) Words: Translation to the right 10 units and down 4 units.

2) Algebraic rule (motion rule): T: (x, y) -> (x + 10, y – 4)

3) Vector: < 10, - 4 >

Activity 2: Dot Paper Translations

1) Use the dots to help you draw the image of the first figure so that A maps to A’.

2) Use the dots to help you draw the image of the second figure so that B maps to B’.

3) Use the dots to help you draw the image of the third figure so that C maps to C’.

4) Complete each of the following translation rules using your mappings from 1 – 3 above.

a) For A, the translation rule is: T:(x, y) ( _______, _______ ) or <_____, _____>

b) For B, the translation rule is: T:(x, y) ( _______, _______ ) or <_____, _____>

c) For C, the translation rule is: T:(x, y) ( _______, _______ ) or <_____, _____>

.............

.............

B’

.............

.............

.............

B

A

.............

.............

.............

.............

C

A’

.............

.............

.............

C’

.............

.............

.............

Checkpoint: GEO has coordinates G(-2, 5), E(-4, 1) O(0, -2). A translation maps G to G’ (3, 1).

1. Find the coordinates of: a) E’ ( _____, _____)b) O’ ( _____, _____)

2. The translation rule is T: (x, y) ( _______, _______ )

3. The vector is <_____, _____>

4. Specifically describe the transformation: ________________________________________

40

9

Translations Practice (homework)

1. Graph and label with vertices

L(-3, -1), I(-1, 4), and P(2, 2)

Graph and label the image of under the translation

.

L’ _____

I’ _____

P’ _____

Write the rule in vector notation: ________

Write the shift using words:

2. Graph and label quadrilateral DUCK with vertices D(2,2), U(4, 1), C(3, -2), and K(0,-1) Graph and label the image of Quadrilateral DUCK when the Quadrilateral is shifted left 4 and up 3.

D’ _____

U’ _____

C’ _____

K’ _____

Write the rule in vector notation: ________

Write the rule in algebraic notation: ______________________________

3. Graph and label quadrilateral MATH with vertices M(4, 1), A(2, 4), T(0,6), and H(1,2). Graph and label the image of Quadrilateral MATH when the Quadrilateral is shifted according to the vector <-3, -4>

M’ _____

A’ _____

T’ _____

H’ _____


Describe in words the shift:

4. Write the rule mapping the pre-image to the image.

Write the rule in vector notation: _______


Describe in words the shift:

Day 2: Reflections with Polygons

A reflection is a transformation in which the image is a mirror image of the preimage.

· A point on the line of reflection maps to ______________________ .

· Other points map to the _________________________ side of the reflection line so that the reflection line is the ________________________________ of the segment joining the preimage and the image.

Activity: Reflections in the coordinate plane. Given:

1) On the first grid, draw the reflection of in the x-axis.

Record the new coordinates: R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

2) On the second grid, draw the reflection of in the y-axis.

Record the new coordinates: R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

3) Draw in the line on the fourth coordinate grid. Reflect over this line and list the coordinates of the image.

R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

4) Draw in the line on the fourth coordinate grid. Reflect over this line and list the coordinates of the image.

R’( _____ , _____ ), E’( _____ , _____ ), F’( _____ , _____ )

Checkpoint: Look at the patterns and complete the rule.

1. Reflection in the x-axis maps (x, y) ( _______, _______ )

2. Reflection in the y-axis maps (x, y) ( _______, _______ )

3. Reflection in the line maps (x, y) ( _______, _______ )

4. Reflection in the line maps (x, y) ( _______, _______ )

For each of the following problems reflect the point E (3, -7) over the given line. Then,

a) the coordinates for the image and b) write a rule for the transformation

5. Reflect over x – axis

a.

b. (___,___) b. Rule:

6.

7. Reflect over the line y = -x

c.

d. (___,___) b. Rule:

8.

9. Reflect over the line x = -2

e.

f. (___,___)

For each of the following problems reflect the point F (-5, -2) over the given line. Then,

a) find the coordinates for the image and b) write a rule for the transformation

10. Reflect over y – axis

g.

h. (___,___) b. Rule:

11.

12. Reflect over the line y = x

i.

j. (___,___) b. Rule:

13.

14. Reflect over the line y = 3

k.

l. (___,___)

Practice – Translations and Reflections (homework #2-16)

Graph and label each figure and the image under the given translation. Name the new coordinates.

11

1. with vertices L(-3, -1), I(-1, 4), and P(2, 2) under the translation

.

L’ _____

I’ _____

P’ _____

3. Quadrilateral DUCK with vertices D(2,2), U(4, 1), C(3, -2), and K(0,-1) under the translation .

D’ _____

U’ _____

C’ _____

K’ _____

5. Quadrilateral MATH with vertices M(4, 1), A(2, 4), T(0,6), and H(1,2) under the translation

.

M’ _____

A’ _____

T’ _____

H’ _____

2. Quadrilateral BAND with vertices B(-3, -3), A(2, -3), N(5, 1), and D(1, 1) under the translation .

B’ _____

A’ _____

N’ _____

D’ _____

4. ΔEFG if E(-1, 2), F(2, 4) and G(2, -4) reflected over the y-axis.

E’ _____

F’ _____

G’ _____

6. Quadrilateral VWXY if V(0, -1), W(1, 1), X(4, -1), and Y(1, -5) reflected over the line .

V’ _____

W’ _____

X’ _____

Y’ ______

7. ΔPQR if P(-3, 4), Q(4, 4) and R(2, -3) reflected over the x-axis.

P’ _____

Q’ _____

R’ _____

9. ΔBEL if B(-2, 3), E(2, 4), and L(3, 1) reflected over the line .

B’ _____

E’ _____

L’ _____

8. Square SQUR if S(1, 2), Q(2, 0), U(0, -1), andR(-1, 1) reflected over the line .

S’ _____

Q’ _____

U’ _____

R’ _____

10. Quadrilateral MATH if M(1, 4), A(-1, 2) T(2, 0) and H(4, 0) reflected over .

M’ _____

A’ _____

T’ _____

H’ _____

The point A (3, -6) is translated left 4 units and up 6 units

11. What are the coordinates of A’? __________

12. Write a rule for the translation ___________________

The point B (-5, 4) is translated down 5 units and right 2 units.

13. What are the coordinates of B’? __________

14. Write a rule for the translation ___________________

15. The point C’ ( 0, 7) is the result of translating a point up five units and left 2 units. What were the coordinates of the original point? __________

16. The point D (11, 3) is translated to become the point D’ (6, 9). What is the rule for the translation between these two points? ____________________

Day 3: Rotations

Rotation – a ______________ in a given _____________________ a given number of

____________________ about a fixed _____________

Activity: Rotations on the Coordinate Plane

has coordinates . Trace the triangle and the x- and y-axes on patty paper.

1) Rotate 90 counter-clockwise, using the axes you traced to help you line it back up correctly. Record the new coordinates.

A’( ____ , ____ ), B’( ____ , ____ ), C’( ____ , ____ )

2) Rotate 270 clockwise, using the axes you traced to help you line it back up correctly. Record the new coordinates.

A’( _____ , _____ ), B’( ____ , ____ ), C’( ____ , ____ )

3) Rotate 90 clockwise, using the axes you traced to help you line it back up correctly. Record the new coordinates.

A’( _____ , _____ ), B’( _____ , _____ ), C’( _____ , _____ )

4) Rotate 270 counter-clockwise, using the axes you traced to help you line it back up correctly. Record the new coordinates.

A’( _____ , _____ ), B’( _____ , _____ ), C’( _____ , _____ )

5) Rotate 180, using the axes you traced to help you line it back up correctly. Record the new coordinates.

A’( _____ , _____ ), B’( _____ , _____ ), C’( _____ , _____ )

Checkpoint: Look at the patterns and complete the rule.

1. A 90 counter-clockwise rotation maps (x, y) ( _______, _______ ).

2. A 90 clockwise rotation maps (x, y) ( _______, _______ ).

3. A 180 rotation maps (x, y) ( _______, _______ ).

4. A rotation of 270 clockwise is equivalent to a rotation of _________________________________.

5. A rotation of 270° counterclockwise is equivalent to a rotation of ____________________________.

Practice-Rotations (homework)

Graph the preimage and image. List the coordinates of the image.

1) ΔRST: R(2, -1), S(4, 0), and T(1, 3)2) ΔFUN: F(-4, -1), U(-1, 3), and N(-1, 1)

90° counter clockwise about the origin.180° clockwise about the origin.

R’ (___,___) S’(___,___) T’(___,___)F’ (___,___) U’(___,___) N’(___,___)

3) ΔTRL: T(2, -1), R(4, 0), and L(1, 3)4) ΔCDY: C(-4,2), D(-1, 2), and Y(-1, -1)

90° clockwise about the origin.180° counter clockwise about the origin.

T’ (___,___) R’(___,___) L’(___,___)C’ (___,___) D’(___,___) Y’(___,___)

5) ΔSCR: S(-3,1), C(-1,3), and R(-1,-1)6) ΔSCR: S(-3,1), C(-1,3), and R(-1,-1)

90° clockwise about the origin90° counter clockwise about the origin

S’ (___,___) C’(___,___) R’(___,___)S’ (___,___) C’(___,___) R’(___,___)

Day 4: Practice – Rotations

Transformation Graphic Organizer

Watch Video Clip and take notes

Definition

Reflection

Real – Life Examples

Picture

Additional

Information

Definition


Rotation

Picture

Additional

Information

Definition


Translationn

Picture

Additional

Information

Reflect, Translate & Rotate your initials below:

5. Gerald is rearranging the furniture in his living room. He has to leave before he is finished, so he draws the diagram at right for his wife to place the end table. Draw the new position of the end table. Include the answers to the following questions in your explanation. Use complete sentences!

What method did you use?

Is there only one possible answer?

What does the arrow tell you?

What do you call this motion?

What could you call the table before it moved? After?

Practice with Translations using Algebra

Given the translation from ABC to A’B’C’, find the specified values.

1. Find x, y, AB, and BC given the diagram .

Use the following diagram for questions 2 – 4.

2. Find x, y, mC, and mA’ given mA = y, mA’ = 2y - 16, mC = 3x - 2, mC’ = 40 .

3. Find x, y, BC, and AC given BC = 3x, B’C’ = 12, AC = 2y - 5, and A’C’ = 7 .

4. Find r , s , mC, and mA given mA = 4r, mC = 2s + 12, mA’ = 80, and mC’ = 60 .

Practice – Rotations, Reflections and Translations (homework)

1)

20

2)

Translate QRS if Q(4,1), R(1,-2), S(2,3) by the rule (x , y) (x –3, y -4).

3)

Reflect Q’R’S’ if Q’(1,-3), R’(-2,-6), and S’(-1,-1) over the x-axis.

4)

Translate CAR if C(-1,-4), A(2,6), R(-4,-2) down 2 units and right 3 units.

5)

Reflect C’A’R’ if C’(1,4), A’(-2,-3), and R’(3,2) over the line y = x.

6)

7)

What did you notice in problems 1&2 and problems 3&4? How were the shapes related? Explain how you could transform QRS by translating it left 3 and down 4 and then reflecting the image over the x-axis. Where does the final image end up?

8)

How would you rotate CAR about the origin and then reflect it over the line y = x?

Day 5: Combinations of Motion

Investigation: Reflections over two lines

1. a. Lines l and k below are a pair of parallel lines. Label the vertices of the triangle A, B, C, etc.

k

l

b. Use patty paper to reflect ABC over line l. Label the corresponding vertices of the image A’, B’, and C’.

c. Use patty paper to reflect A’B’C’ over line k. Label the corresponding vertices of the image A’’, B’’, and C’’.

d. Draw the segment connecting the vertices A and A’’. Draw the segment connecting B and B’’. Draw the segment connecting C and C’’. Compare the lengths of each segment.

e. Write a statement comparing the size of the figures and positions relative to the lines of reflection.

f. What single transformation will map triangle ABC to triangle A”B”C”? Be specific in your description. Explain how you know.

Investigation: Reflections over Two Lines (continued)

2. a. Lines l and k intersect at point P. Label the vertices of the triangle A, B, C, etc.

b. Use patty paper to reflect ABC over line l. Label the corresponding vertices of the image A’, B’, and C’.

c. Use patty paper to reflect A’B’C’ over line k. Label the corresponding vertices of the image A’’, B’’, and C’’.

d.Draw segments connecting A to P, and A” to P. Compare the measure of the angle you just drew to the measure of the acute angle formed by lines l and k.

e.Write a statement comparing the size of the figures and positions relative to the lines of reflection.

f. What single transformation will map triangle ABC to triangle A”B”C”? Be specific in your description. Explain how you know.

Checkpoint and Summary

A ____________________ is a sequence of _________________________________________.

Two reflections across _____________ lines is the same as a __________________________.

A _________________ is the same as a double reflection around ____________________ lines.

The point of rotation is the ______________ of the ________ __________________ _____.

Reflection with a Translation

a. On a coordinate grid, draw a triangle using A(-9, -2), B(-6, -1), C(-6, -3) to represent a duck foot.

b. Reflect ABC across the x-axis, then translate the image horizontally 5 units to the right. Label the final image A’B’C’.

c. How are the coordinates of A’B’C’ related to those of ABC?Write a coordinate rule for this composite transformation.

d. Apply the same combination of the two transformations to ABC but in the opposite order. Does the order in which you apply the translation and reflection matter?

e. Now apply the coordinate rule you gave in Part c at least three more times to A’B’C’. Describe how alternate images such as images one and three, or two and four, are related.

f. The combination of a reflection across a line and a translation in a direction parallel to the line is called a glide reflection.

g. Start with a new triangle. Then apply a glide reflection in which the reflection line is the y-axis. Write a coordinate rule for this glide reflection.

Summary

A glide reflection is the composition of a _________________ and a _______________________ where the _______________ motion is _________________ to the _______________________.

Practice – Combination of Motion (homework)

Use the graph of the square to the right to answer questions 1-3.

1. Perform a glide reflection over the x-axis and a translation to the right 6 units. Write the new coordinates.

2. What is the rule for this glide reflection?

3. What glide reflection would move the image back to the pre-image?

Use the graph of the triangle to the right to answer questions 4-6.

4. Perform a glide reflection over the y-axis and down 5 units. Write the new coordinates.

5. What is the rule for this glide reflection?

6. What glide reflection would move the image back to the pre-image?


7. Reflect the pre-image over y = -1 followed by y = -7. Draw the new triangle.

8. What one transformation is this double reflection the same as?

9. Write the rule.


10. Reflect the pre-image over y = -7 followed by y = -1. Draw the new triangle.

11. What one transformation is this double reflection the same as?

12. Write the rule.

13. How do the final triangles in #7 and #10 differ?

Use the trapezoid in the graph to the right to answer questions 14-16.

14. Reflect the pre-image over the x-axis then the y-axis. Draw the new trapezoid

15. Now, start over. Reflect the trapezoid over the y-axis then the x-axis. Draw this trapezoid.

16. Are the final trapezoids from #14 & #15 different? Why do you think that is?

Day 6: Practice Combinations of Motions

For # 1-4, there is a composition of motions given. Using the rules transformations, find the image of the point A (-1, 3) and come up with a new rule after both transformations have taken place.

1) Translate point A 4 units right and 2 units up, and then reflect that point over the line y = x.

A’ ______________ A‘’ _______________Rule (x,y) ________________________

2) Rotate point A 90 degrees counter clockwise, and then translate the point right 5 and up 2.

A’ ______________ A‘’ _______________Rule (x,y) ________________________

3) Translate point A 4 units left and 2 units down, and then reflect that point over the y-axis.

A’ ______________ A‘’ ______________Rule (x,y) →_______________________

4) Rotate point A 90 degrees clockwise, and then reflect over the line y = -x.

A’ ______________ A‘’ ______________Rule (x,y) →________________________

You Try: Use Point B (2, -4)

5) Translate point B 4 units right and 2 units down, and reflect that point over the x-axis.

B’ ______________ B‘’ _______________Rule (x,y) →_______________________

6) Rotate point B 180 degrees counter clockwise, and then reflect over the x-axis.

B’ ______________ B‘’ ______________Rule (x,y) →_______________________

7) Translate point B 4 units left and 2 units up, and then reflect the point over y = x.

B’ ______________ B‘’ _______________Rule (x,y) →_______________________

8) Rotate point B 180 degrees clockwise, then translate down 6 units and right 4 units.

B’ ______________ B‘’ _______________Rule (x,y) →_______________________

Now you are going to try some multiple transformations:

1) Translate ALT if A(-5,-1), L(-3,-2), T(-3,2)

by moving it right 6 and down 3, then reflect the

image over the y-axis.

2) Reflect TAB if T(2,3), A(1,1), and B(4,-3) over the x-axis, then reflect the image over the y-axis.

33

3) Rotate ALT if A(-5,-1), L(-3,-2),

T(-3,2) clockwise around the origin, then reflect the image over the x-axis.

4) Reflect TAB if T(2,3), A(1,1), and B(4,-3) over the y-axis, then translate the image by moving right 2 and down 1

5) Rotate ALT if A(-5,-1), L(-3,-2),

T(-3,2) clockwise around the origin, then reflect the image over the y-axis.

Day 7: Dilations

Alice in Wonderland

In the story, Alice’s Adventures in Wonderland, Alice changes size many times during her adventures. The changes occur when she drinks a potion or eats a cake. Problems occur throughout her adventures because Alice does not know when she will grow larger or smaller.

Part 1

As Alice goes through her adventure, she encounters the following potions and cakes:

Red potion – shrink by Chocolate cake – grow by 12 times

Blue potion – shrink by Red velvet cake – grow by 18 times

Green potion – shrink by Carrot cake – grow by 9 times

Yellow potion – shrink by Lemon cake – grow by 10 times

Find Alice’s height after she drinks each potion or eats each bite of cake. If everything goes correctly, Alice will return to her normal height by the end.

Starting Height

Alice Eats or Drinks

Scale factor from above

New Height

54 inches

Red potion

6 inches

6 inches

Chocolate cake

Yellow potion

Carrot cake

Blue potion

Lemon cake

Green potion

Red velvet cake

54 inches

Part 2

A) The graph below shows Alice at her normal height.

B) Place a ruler so that it goes through the origin and point A. Plot point A’ such that it is twice as far from the origin as point A. Do the same with all of the other points, starting with points B, C, D, E, and F. Connect the points to show Alice after she has grown. (Hint: measure with centimeters so that you can use decimal values.)

B

D

E

C

F

C) Label some of the corresponding preimage and image coordinate pairs. Compare their values to complete the questions below.

1. How many times larger is the new Alice? ______________________

2. How much farther away from the origin is the new Alice? _____________________

3. What are the coordinates for point A? ___________ Point A’? ___________

4. What arithmetic operation do you think happened to the coordinates of A?

5.

Write your conclusion as an Algebraic Rule

D) What arithmetic operation on the coordinates do you think would shrink Alice in half?

E) Write your conclusion as an Algebraic rule.

F) If Alice shrinks in half, how far away from the origin will her image be from her preimage?

G) On the grid above, graph the image of Alice if she is shrunk by a scale factor of ½ from her original height.

Checkpoint:

A dilation stretches or shrinks the original figure.

The description of a dilation should include the ___________ _______________, the ________________ of the dilation, and whether the dilation is an ___________________________ or a _______________________.

The amount by which the image grows or shrinks is called the "____________ ____________."

The _____________ of dilation is a fixed point in the plane about which all points are expanded or contracted.

A dilation is

· An enlargement of the pre-image if the _______ _________ is ____________________.

· A reduction of the pre-image if the _______ _________ is ______________________.

If the scale factor is 1, then the pre-image and image are _______________________.

Algebraic Rule: (x, y) → (ax, ay) if a > 1 then the dilation is __ _____________

if 0 < a < 1 then the dilation is __ ____________

The distance between the center of a dilation and any point on the image is equal to the __________ __________ multiplied by the distance between the dilation center and the corresponding point on the image.

Circle the appropriate choice for the following characteristic/property:

A dilation is SOMETIMES / ALWAYS / NEVER an ‘Isometry’ unless the scale factor is 1.

Practice Dilations (homework)

1. Graph and connect these points

(2, 2) (3, 4) (5, 2) (5, 4).

2. Graph the image on the same coordinate plane by applying a scale factor of 2.

What is the Algebraic Rule for this transformation? ___________________________

How do the preimage and image compare?

What are the coordinate pairs of the image?

3. Graph the image on the same coordinate plane by applying a scale factor of 1/2.

What is the Algebraic Rule for this transformation? _________________________

How do the preimage and image compare?

What are the coordinate pairs of the image?

4. What happens when you apply a scale factor greater than 1 to a set of coordinates?

5. What happens when you apply a scale factor less than 1 to a set of coordinates?

6. What happens when you apply a scale factor of 1 to a set of coordinates?

Day 8: Dilations with Coordinates

For each problem, graph the image points, and describe the transformation that occurred. Specify if the transformation is an enlargement or reduction and by what scale factor. Then, examine the coordinates to create an Algebraic Rule.

1) The coordinates of ABC are

A(2, -1), B(3, 2) and C(-3, 1). The coordinates of A’B’C’ are A’(1, -1/2), B’(3/2, 1), and C’(-3/2, 1/2).

Transformation:

Algebraic Rule:

2) The coordinates of ABC are

A(2, -1), B(3, 2) and C(-3, 1). The coordinates of A’B’C’ are A’(4, -2), B’(6, 4), and C’(-6, 2).

Transformation:

Algebraic Rule:

3) The coordinates of ABC are A(2, -1),

B(3, 2) and C(-3, 1). The coordinates of A’B’C’ are A’(3, -3/2), B’(9/2, 3), and

C’(-9/2, 3/2).

Transformation:

Algebraic Rule:

To find the Scale Factor: pick a pair of corresponding sides and calculate ________________________.

#6 Find the scale factor.



Day 9: Functions Review

Class work: Interpreting Functions

Kim and Jim are twins and live at the same home. They each walk to school along the same path at exactly the same speed. However, Jim likes to arrive at school early and Kim is happy to arrive 7 minutes later, just as the bell rings. Pictured at right is a graph of Jim’s distance from school over time.

1. Use a dotted line to sketch Kim’s graph of distance from school over time

(once she leaves for school).

1. How many minutes after 7AM does Jim leave for school? _____________

1. How many minutes after 7AM does Jim arrive at school? _____________

1. How many minutes after 7AM does Kim leave for school? _____________

1. How many minutes after 7AM does Kim arrive at school? _____________

1. What is Jim’s farthest distance from school? _____________

1. What is Jim’s closest distance to school? _____________

1. What is Kim’s farthest distance from school? _____________

1. What is Kim’s closest distance to school? _____________

1. Use your answers to the above questions to fill in the following:

Jim’s domain: _______ ≤ x ≤ _________

(where x represents time after 7AM)

Jim’s range: _______ ≤ y ≤ _________

(where y represents distance from school)

Kim’s domain: _______ ≤ x ≤ _________

(where x represents time after 7AM)

Kim’s range: _______ ≤ y ≤ _________

(where y represents distance from school)

Extend: Kim’s graph is a horizontal translation of Jim’s graph. When a graph translates horizontally, how do the domain and range change?

Domain and Range in translations

Quick review: The domain is the set of all possible x-values on the graph. The range is the set of all possible y-values on the graph.

1. Describe the translation(s) from the pre-image to the image.

1. Given the following graph, state the domain and range of the pre-image:

Domain: ____________

Range: _____________.

1. State the domain and range of the image:

Domain: ____________Range: _____________.

1.

Draw and label the image of translated left 2 and down 3.

1. State the domain and range of the

pre-image:

Domain: ____________

Range: _____________.


Domain: ____________Range: _____________.

1.

Draw and label the image of reflected over the x-axis.

1. State the domain and range of the pre-image:

Domain: ____________

Range: _____________.


1. Domain: ____________Range: _____________.

1.

Draw and label the image of reflected over the y-axis


Domain: ____________

Range: _____________.


Domain: ____________Range: _____________.

1.

Draw and label the image of reflected over the line y = x.


Domain: ____________

Range: _____________.


Domain: ____________Range: _____________.

1.

Draw and label the image of rotated 90°.


Domain: ____________

Range: _____________.


Domain: ____________Range: _____________.

1.

Draw and label the image of dilated by a factor of 3 with a center of (0,0)


Domain: ____________Range: _____________.


Domain: ____________Range: _____________.

Day 10: Unit 1 – Math 2 Test Review

For each transformation, state the coordinates of the image of the point and the general rule for the image of the point .

Image of

Image of

1. Reflect over y-axis

2. Reflect over x-axis

3. Reflect over

4. Reflect over

5. Reflect over

6. Reflect over

7. Rotate clockwise about the origin

8. Rotate counter-clockwise about the origin

9. Rotate about the origin

37

For each of the following, graph and label the image for each transformation described.

10. Reflect over the y-axis11. Rotate 180 about the origin 12. Translate right 4

units & down 3 units

A

C

B

D

A

C

B

D

A

C

B

D

Perform each of the transformations using ∆ABC below for #13 – 16.

A (7, –4), B (0, 6), C (–2, 3)

7.

13. Reflect over the x-axis

A’__________

B’__________

C’__________

15. Reflect over the line y = x

A’__________

B’__________

C’__________

14. Rotate 90◦ counter-clockwise about the origin

A’__________

B’__________

C’__________

16. Dilate about the origin by a magnitude of ½

A’__________

B’__________

C’__________

State whether the specified pentagon is mapped to the other pentagon by a reflection, translation, or rotation

2

1

17. Pentagon 1 to Pentagon 3_______________________

4

3



6

5



Answer each of the following.

22. If translation , then

23. , if F’ find F. ________________

24. M is reflected over the y-axis. If M’ is find M. ________________

25. C is rotated about the origin 90°. If C’ is , find C. ________________

26. Y is rotated about the origin 180. If the image of Y is (0, -3) find Y. ____________

27. A figure is reflected over the line . If the preimage is (2, 7), find the image. ____________

28. has vertices . Find the coordinates of the image of the triangle if it is dilated about the origin by a magnitude of 3.

A’( _____, _____ ), B’( _____, _____ ), C’( _____, _____ )

29. ABCD is dilated about point O by a magnitude of 2 to produce A’B’C’D’.

The lengths of the segments of the preimage are as follows:

AB = 6, BC = 5, CD = 3, AD = 4

a. What is the length of ?

b. What is the length of ?

30. For each problem, there is a composition of motions. Using your algebraic rules, come up with a new rule after both transformations have taken place.

a. Translate a triangle 5 units left and 3 units up, and then reflect the triangle over the x-axis.

b. Rotate a triangle 90 degrees counter clockwise, and then reflect in the line y = x.

c. Reflect in the y-axis, and then translate right 4 units and down 2 units.

34. Quad. I Quad. II. Complete each part using the similar quadrilaterals.

a. The scale factor of Quad. I to Quad. II is ________.

15

6

18

y

x

25

40

b. Quad. STAR Quad. ________.

c. x = ________

d. y = ________

35.

a. What is the scale factor of to ? _________

b. Find AC. _________

c. Find DE. _________

Find the value of x.

36. x = _________37. x = _________

Unit 1 – Review 2

1. Which of these transformations are isometries?

a. I onlyb. II and III onlyc. I and III onlyd. I, II, and III

(I) parallelogram EFGH parallelogram XWVU

(II) hexagon CDEFGH hexagon YXWVUT

(III) triangle EFG triangle VWU

2.Which of the following transformations creates a figure that is not congruent to the original figure?

I. translationII. rotationIII. dilation

a. III onlyb. II and III onlyc. II onlyd. I, II, and III

3.Name the translation image of after a

reflection over line t and then a reflection over line r.

a.

b.

c.

d.

3. ABC with vertices A(-2, -1), B(1, 4) and C(5, 2) is stretched vertically by a factor of 3.

What are the coordinates of A’ ?

a.

(-6,-3)

b.

(1, 2)

c.

(-6, -1)

d.

(-2, -3)

5. Which type of isometry is the equivalent of two reflections across intersecting lines?

a.

glide reflection

c.

rotation

b.

Reflection

d.

none of these

6. Which translation from solid-lined figure to dashed-lined figure is given by the vector

<-3, 3>?

a.

4

8

–4

–8

x

4

8

–4

–8

y

c.

4

8

–4

–8

x

4

8

–4

–8

y

b.

4

8

–4

–8

x

4

8

–4

–8

y

d.

4

8

–4

–8

x

4

8

–4

–8

y

7. DEF has vertices D(-5, 0), E(6, -7) and F(7, 2) and D’E’F’ has vertices D’(-15, 0),

E’(18, -21) and F’(21, 6). What is the scale factor of the dilation?

a.

b.

c.

0

d.

8.

Write a rule to describe the transformation that is a reflection in the y-axis.

a.

(x, y) (y, x)

c.

(x, y) (–x, y)

b.

(x, y) (x, –y)

d.

(x, y) (–x, –y)

9. The vertices of a triangle are P(–7, –1), Q(2, 1), and R(–5, 3). Name the algebraic rule for the composition of reflecting in the line y = x, then translating left 7 and down 4.

a.

(x, y) -> (y – 7, x – 4)

c.

(x, y) -> (x – 7, y – 4)

b.

(x, y) -> (y – 4, x – 7)

d.

(x, y) -> (x – 4 , y – 7)

10. Identify as a reflection, translation,

rotation, or glide reflection. Find the reflection line,

translation vector, center and angle of rotation,

or glide vector and reflection line.

a)

rotation ; 180° about (–0.5, 0)

b)

reflection; x = 5

c)

glide vector <8, 0> and reflection line y = 4

d)

rotation; 180° about (1, 4)

11. ABC with vertices A(-5, -3), B(-1, 4) and C(5, -1) is rotated 270. What are the

coordinates of A’, B’, and C’ ?

a.

A’(3, 5)

B’(-4, 1)

C’(1, -5)

b.

A’(3, -5)

B’(-4, -1)

C’(1, 5)

c.

A’(-3, 5)

B’(4, 1)

C’(-1, -5)

d.

A’(-3, -5)

B’(4, -1)

C’(-1, 5)

12.A figure is translated <3, -3>. Which translation will move the image back to its original position?

a. <3, -3>b. <-3, 3>c. <-3, 0>d. <0, 3>

13.A reflection in a line is a type of transformation.

a. trueb. false

14.Isometries preserve angle measures and parallel lines.

a. trueb. false

15.A reflection in a line is a type of transformation.

a. trueb. false

16.In a glide reflection, the order in which the two transformations are performed _____ matters.

a. sometimesb. alwaysc. never

17.A composition of isometries is _____ an isometry.

a. sometimesb. alwaysc. never

18.The vertices of a triangle are P(–3, 8), Q(–6, –4), and R(1, 1). Name the vertices of the image reflected in the x-axis.

a.

c.

b.

d.

19.The vertices of a rectangle are R(–5, –5), S(–1, –5), T(–1, 1), and U(–5, 1). After

translation, is the point (0, –13). Find the translation vector and coordinates of .

a. b. c. d.

20. DEF has vertices D(-5, 0), E(6, -7) and F(7, 2) and D’E’F’ has vertices D’(-15, 0),

E’(18, -21) and F’(21, 6). What is the scale factor of the dilation?

a.

b.

c.

0

d.

21.Describe the transformation shown.

a.

270 Rotation

c.

Reflection in y = -x

b.

Reflection in y = x

d.

180 Rotation

22.

Solve the proportion:

1. x = -3

1. x = 6

1. x = 12/7

1. x = 3

23.

If M(-5, 3) is reflected in the line x=-2, then M’ is (3, 1).

a. trueb. false

24.If N(3,4) is reflected in the line y=-1, then N’

is (3, -6).

a. trueb. false

25.Use a vector to describe the translation that is 4 units left and 8 units up.

a. <4, 8> b. <4, -8> c. <-4, 8> d. <-4, -8>

26.Write a rule to describe the transformation that is a reflection in the

y-axis.

a.

(x, y) (–x, y)

c.

(x, y) (y, x)

b.

(x, y) (–x, –y)

d.

(x, y) (x, –y)

27.Identify the dilation shown as an enlargement or reduction and find its scale factor, k.

F

7.5

C

D

E

6

E’

F’

D’

2.5

2

a. reduction; k=1/2

b. reduction; k=1/3

c. enlargement; k=1/3

d. enlargement; k=2

28.

The hexagon GIKMPR and FJN are regular (equal angles & equal sides).

The dashed line segments form 30° angles.

Find the image of after a rotation of 240° about point O.

a.

b.

c.

d.

29.Find the angle of rotation about O that maps Q to F.

a.

330°

b.

60°

c.

270°

d.

300°

30.Identify as a reflection, translation,

rotation, or glide reflection. Find the reflection line,

translation vector, center and angle of rotation,

or glide vector and reflection line.

a)

reflection; x = 5

b)

glide vector <8, 0> and reflection line y = 4

c)

rotation; 180° about (1, 4)

d)

rotation ; 180° about (–0.5, 0)

40

AB

D'

C'

A'

B'

O

A

D

C

B

Quad. II

Quad. I

A

R

T

SF

M

L

I

9

15

21

28

C

A

B

E

D

x

16

12

14

20

x

30

30

12

12

20

4

1

3

1

1

3

CA

B

HJ

G

I

F

E

D

12–12x

4

8

12

–4

–8

–12

y

3

1

4

1

3

84

xx

+

=

123456–1–2–3–4–5–6x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

F

L

G

M

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Image

Preimage

B'

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AB

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AB

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AB

uuur

AB

y

x

=

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=

1

x

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2

=

y

24

5x + y

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y

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A

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B'

A'

C'

C'

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(

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xy

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day 1: transformations and translations · web viewmath 2 spring 2018 mrs. bello unit 1 anchor...

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