Introduction to Transformations

Show your work for each problem.

1.
Which transformation slides every point the same distance in the same direction?
1. a)Reflection
2. b)Dilation
3. c)Rotation
4. d)Translation
Answer: Translation
A translation (slide) moves every point of a figure the same distance in the same direction. The figure does not rotate or change size.
2.
What is the original figure called before a transformation?
1. a)Image
2. b)Transformation
3. c)Reflection
4. d)Pre-image
Answer: Pre-image
The pre-image is the original figure before any transformation is applied. After the transformation, the result is called the image.
3.
Which transformation creates a mirror image of a figure?
1. a)Rotation
2. b)Reflection
3. c)Translation
4. d)Dilation
Answer: Reflection
A reflection flips a figure over a line (called the line of reflection) to create a mirror image. The image is the same distance from the line as the pre-image, but on the opposite side.
4.
Point is translated 4 units right. What is the new -coordinate?
Answer: 7
When translating 4 units right, we add 4 to the -coordinate: . The new point is .
5.
Point is translated 5 units down. What is the new -coordinate?
Answer: -3
When translating 5 units down, we subtract 5 from the -coordinate: . The new point is .
6.
Which transformation turns a figure around a fixed point?
1. a)Translation
2. b)Reflection
3. c)Dilation
4. d)Rotation
Answer: Rotation
A rotation turns a figure around a fixed point called the center of rotation. Each point stays the same distance from the center, but moves along a circular path.
7.
Point is reflected over the -axis. What is the new -coordinate?
Answer: -4
When reflecting over the -axis, the rule is . So becomes .
8.
Point is reflected over the -axis. What is the new -coordinate?
Answer: 5
When reflecting over the -axis, the rule is . So becomes .
9.
Which transformations preserve the size and shape of a figure? (Rigid transformations)
1. a)All four transformations
2. b)Rotation and Dilation
3. c)Dilation only
4. d)Translation, Reflection, Rotation
Answer: Translation, Reflection, Rotation
Translation, reflection, and rotation are rigid transformations - they preserve size and shape. Dilation changes the size of a figure (enlarges or reduces it), so it is NOT a rigid transformation.
10.
Point is rotated counterclockwise about the origin. Find the coordinates of .
Answer: (0, 3)
1. What is the rule for 90 degree counterclockwise rotation? (-y, x)
2. What is the x-coordinate of R? 3
3. What is the y-coordinate of R? 0
4. Apply the rule: what are the new coordinates? (0, 3)
11.
Point undergoes a dilation with center at the origin and scale factor . Find the coordinates of .
Answer: (6, 12)
1. What is the rule for dilation with center at the origin? (kx, ky)
2. What is the scale factor k? 3
3. Calculate the new x-coordinate: 3 times 2 6
4. Calculate the new y-coordinate: 3 times 4 12
12.
Triangle is transformed to triangle . If and , which transformation could NOT have been used?
1. a)Dilation with k not equal to 1
2. b)Reflection
3. c)Translation
4. d)Rotation
Answer: Dilation with k not equal to 1
Since , the side lengths are preserved. Dilation with would change the side lengths (multiply by k). Translation, reflection, and rotation all preserve lengths.
13.
Point is rotated about the origin. Find the coordinates of .
Answer: (2, -5)
1. What is the rule for 180 degree rotation about the origin? (-x, -y)
2. What is the x-coordinate of P? -2
3. What is the opposite of -2? 2
4. What is the opposite of 5? -5
14.
Point undergoes a dilation with center at the origin and scale factor . Find the coordinates of .
Answer: (3, 4)
1. What do you multiply each coordinate by? 0.5
2. Calculate 0.5 times 6 for the new x 3
3. Calculate 0.5 times 8 for the new y 4
4. Is this image larger or smaller than the original? smaller
15.
Point is translated 2 units left and 4 units up. Then it is reflected over the -axis. Find the final coordinates.
Answer: (-1, -7)
1. After moving 2 left, what is the new x-coordinate? -1
2. After moving 4 up, what is the new y-coordinate? 7
3. After translation, the point is at (-1, 7). Now reflect over the x-axis. What happens to y? changes sign
4. What are the final coordinates? (-1, -7)
16.
A triangle has area 12 square units. After a dilation with scale factor , what is the area of the image?
Answer: 48
When you dilate by scale factor , lengths multiply by but area multiplies by . So the new area is square units.
17.
Point is rotated clockwise about the origin. What are the coordinates of ?
1. a)
2. b)
3. c)
4. d)
Answer:
For clockwise rotation, the rule is . So .
18.
Point is translated 3 units left and 2 units down. What is the new -coordinate?
Answer: 2
Moving 3 units left: . The new -coordinate is 2. (The full point would be after also moving down.)