Lesson 2

Transformations as Functions

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Problem 1

Match each coordinate rule to a description of its resulting transformation.

A:

\((x,y) \rightarrow (x+3,y)\)

B:

\((x,y) \rightarrow (2x,2y)\)

C:

\((x,y) \rightarrow (x,y+4)\)

D:

\((x,y) \rightarrow (x,y-4)\)

E:

\((x,y) \rightarrow (x-3,y+4)\)

1:

Translate by the directed line segment from \((0,0)\) to \((0,4)\).

2:

Translate by the directed line segment from \((0,0)\) to \((3,0)\).

3:

Dilate using the origin as the center and a scale factor of 2.

4:

Translate by the directed line segment from \((0,0)\) to \((0,\text-4)\).

5:

Translate by the directed line segment from \((0,0)\) to \((\text-3,4)\).

Solution

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Problem 2

  1. Draw the image of triangle \(ABC\) under the transformation \((x,y) \rightarrow (x-4,y+1)\). Label the result \(T\).
  2. Draw the image of triangle \(ABC\) under the transformation \((x,y) \rightarrow (\text- x,y)\). Label the result \(R\).
Triangle ABC. A at 2 comma 3, B at 4 comma 2, C at 3 comma 5.

Solution

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Problem 3

Here are some transformation rules. For each rule, describe whether the transformation is a rigid motion, a dilation, or neither.

  1. \((x,y) \rightarrow (x-2,y-3)\)
  2. \((x,y) \rightarrow (2x,3y)\)
  3. \((x,y) \rightarrow (3x,3y)\)
  4. \((x,y) \rightarrow (2-x,y)\)

Solution

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Problem 4

Reflect triangle \(ABC\) over the line \(x=0\). Call this new triangle \(A’B’C’\). Then reflect triangle \(A’B’C’\) over the line \(y=0\). Call the resulting triangle \(A''B''C''\).

Which single transformation takes \(ABC\) to \(A''B''C''\)?

Triangle ABC graphed on coordinate plane. A at 1 comma 1, B at 2 comma -1, C at 3 comma 0.
A:

Translate triangle \(ABC\) by the directed line segment from \((1,1)\) to \((\text-2,1)\).

B:

Reflect triangle \(ABC\) across the line \(y=\text-x\).

C:

Rotate triangle \(ABC\) counterclockwise using the origin as the center by 180 degrees.

D:

Dilate triangle \(ABC\) using the origin as the center and a scale factor of 2.

Solution

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(From Unit 6, Lesson 1.)

Problem 5

Reflect triangle \(ABC\) over the line \(y=2\).

Translate the image by the directed line segment from \((0,0)\) to \((3,2)\).

What are the coordinates of the vertices in the final image?

triangle ABC graphed. A = -5 comma 2, B = -6 comma -1, C = -3 comma 0.

Solution

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(From Unit 6, Lesson 1.)

Problem 6

The density of water is 1 gram per cm3. An object floats in water if its density is less than water’s density, and it sinks if its density is greater than water’s. Will a cylindrical log with radius 0.4 meters, height 5 meters, and mass 1,950 kilograms sink or float? Explain your reasoning.

Solution

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(From Unit 5, Lesson 17.)

Problem 7

These 3 congruent square pyramids can be assembled into a cube with side length 3 feet. What is the volume of each pyramid?

Rectangular pyramid 
Rectangular pyramid 
Rectangular pyramid 
A:

1 cubic foot

B:

3 cubic feet

C:

9 cubic feet

D:

27 cubic feet

Solution

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(From Unit 5, Lesson 12.)

Problem 8

Reflect square \(ABCD\) across line \(CD\). What is the ratio of the length of segment \(AA'\) to the length of segment \(AD\)? Explain or show your reasoning.

Square ABCD.

Solution

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(From Unit 2, Lesson 1.)