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## Transformations 2 - Sample Math Practice Problems

The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. In the main program, all problems are automatically graded and the difficulty adapts dynamically based on performance. Answers to these sample questions appear at the bottom of the page. This page does not grade your responses.

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### Complexity=1, Mode=mix

 1.   What type of transform takes ABCD to A'B'C'D'?        Translation        Rotation        Reflection 2.   What type of transform takes ABCD to A'B'C'D'?        Translation        Rotation        Reflection

### Complexity=2, Mode=rot

 1.   The transformation ABCD → A'B'C'D' is a rotation around (-1, 2) by ° Rotate P around (-1, 2) by the same angle. (You may need to sketch things out on paper.) P' = ( ,) 2.   The transformation ABCD → A'B'C'D' is a rotation around (-1, -3) by ° Rotate P around (-1, -3) by the same angle. (You may need to sketch things out on paper.) P' = ( ,)

### Complexity=2, Mode=refl

 1.   Transformation ABCD → A'B'C'D' is a reflection across the line e.g. y=-3, y=x. Reflect P over the same line. (You may need to sketch things out on paper.) P' = ( ,) 2.   Transformation ABCD → A'B'C'D' is a reflection across the line e.g. y=-3, y=x. Reflect P over the same line. (You may need to sketch things out on paper.) P' = ( ,)

### Complexity=3, Mode=trans

 1.   All points are to be transformed by a translation of (-7, 7) followed by another translation of (10, -3). Combined translation: ( ,) Let P = (8, 10). P'' = ( ,) 2.   All points are to be transformed by a translation of (2, -10) followed by another translation of (-3, 4). Combined translation: ( ,) Let P = (6, 6). P'' = ( ,)

### Complexity=3, Mode=rot

 1.   All points are transformed by a rotation of 15° around (1, 1) followed by another rotation of 75° around the same point. Combined angle of rotation around (1, 1): ° Rotate P around (1, 1) by the same angles. (You may need to sketch things out on paper.) P'' = ( ,) 2.   All points are transformed by a rotation of 60° around (3, 3) followed by another rotation of 30° around the same point. Combined angle of rotation around (3, 3): ° Rotate P around (3, 3) by the same angles. (You may need to sketch things out on paper.) P'' = ( ,)

### Complexity=3, Mode=refl

 1 Given a reflection across  x = -3 , followed by a reflection across  y = -x , the combined transformation is a        translation        rotation If this is a translation, give the translation components; if it is a rotation, give the coordinates of the pivot point: (),( ) 2 Given a reflection across  y = 1 , followed by a reflection across  y = -x , the combined transformation is a        translation        rotation If this is a translation, give the translation components; if it is a rotation, give the coordinates of the pivot point: (),( )

### Complexity=1, Mode=mix

1
What type of transform takes ABCD to A'B'C'D'?
Translation        Rotation        Reflection
Solution

Each point has been reflected across the y-axis.
2
What type of transform takes ABCD to A'B'C'D'?
Translation        Rotation        Reflection
Solution

Each point has been reflected across the y-axis.

### Complexity=2, Mode=rot

1
The transformation ABCD → A'B'C'D' is a rotation around (-1, 2) by °
Rotate P around (-1, 2) by the same angle. (You may need to sketch things out on paper.)
P' = ( ,)
Solution

Each point has been rotated around (-1, 2) by 180°.
A(-1, 9) → A'(-1, -5)
P(-6, -7) → P'(4, 11)
Since this is a rotation by an increment of 90°, we can figure out P' graphically by forming a right triangle with its hypotenuse connecting the pivot and point P. This should make it easy to draw the rotated P'. Your before and after triangles should both have legs of length 5 and 9, since P(-6, -7) is 5 from pivot (-1, 2) in the x direction and 9 from it in the y direction.
2
The transformation ABCD → A'B'C'D' is a rotation around (-1, -3) by °
Rotate P around (-1, -3) by the same angle. (You may need to sketch things out on paper.)
P' = ( ,)
Solution

Each point has been rotated around (-1, -3) by 180°.
A(7, 10) → A'(-9, -16)
P(0, 7) → P'(-2, -13)
Since this is a rotation by an increment of 90°, we can figure out P' graphically by forming a right triangle with its hypotenuse connecting the pivot and point P. This should make it easy to draw the rotated P'. Your before and after triangles should both have legs of length 1 and 10, since P(0, 7) is 1 from pivot (-1, -3) in the x direction and 10 from it in the y direction.

### Complexity=2, Mode=refl

1
Transformation ABCD → A'B'C'D' is a reflection across the line
e.g. y=-3, y=x.
Reflect P over the same line. (You may need to sketch things out on paper.)
P' = ( ,)
Solution

P is at (2, 5). Since the line of reflection is y = x, reflecting a point comes down to switching the x and y coordinates.
2
Transformation ABCD → A'B'C'D' is a reflection across the line
e.g. y=-3, y=x.
Reflect P over the same line. (You may need to sketch things out on paper.)
P' = ( ,)
Solution

P is at (0, -2). Since the line of reflection is y = x, reflecting a point comes down to switching the x and y coordinates.

### Complexity=3, Mode=trans

1All points are to be transformed by a translation of (-7, 7)
followed by another translation of (10, -3).
Combined translation: ( ,)
Let P = (8, 10).
P'' = ( ,)
Solution

To figure out the combined translation vector, add the individual vectors by component:
```  (-7, 7)
+ (10, -3)
-------------
(3, 4)

P'' = P(8, 10)
+ (3, 4)
---------------
(11, 14)```
2All points are to be transformed by a translation of (2, -10)
followed by another translation of (-3, 4).
Combined translation: ( ,)
Let P = (6, 6).
P'' = ( ,)
Solution

To figure out the combined translation vector, add the individual vectors by component:
```  (2, -10)
+ (-3, 4)
-------------
(-1, -6)

P'' = P(6, 6)
+ (-1, -6)
---------------
(5, 0)```

### Complexity=3, Mode=rot

1
All points are transformed by a rotation of 15° around (1, 1)
followed by another rotation of 75° around the same point.
Combined angle of rotation around (1, 1): °
Rotate P around (1, 1) by the same angles. (You may need to sketch things out on paper.)
P'' = ( ,)
Solution
15° + 75° = 90
Since this is a rotation by an increment of 90°, we can figure out P'' graphically by forming a right triangle with its hypotenuse connecting the pivot and point P. This should make it easy to draw the rotated P''. Your before and after triangles should both have legs of length 3 and 5, since P(4, -4) is 3 from pivot (1, 1) in the x direction and 5 from it in the y direction.
2
All points are transformed by a rotation of 60° around (3, 3)
followed by another rotation of 30° around the same point.
Combined angle of rotation around (3, 3): °
Rotate P around (3, 3) by the same angles. (You may need to sketch things out on paper.)
P'' = ( ,)
Solution
60° + 30° = 90
Since this is a rotation by an increment of 90°, we can figure out P'' graphically by forming a right triangle with its hypotenuse connecting the pivot and point P. This should make it easy to draw the rotated P''. Your before and after triangles should both have legs of length 2 and 11, since P(5, -8) is 2 from pivot (3, 3) in the x direction and 11 from it in the y direction.

### Complexity=3, Mode=refl

1Given a reflection across  x = -3 , followed by a reflection across  y = -x , the combined transformation is a
translation        rotation
If this is a translation, give the translation components; if it is a rotation, give the coordinates of the pivot point:
(),( )
Solution
Two reflections along intersection lines results in a rotation around the intersection point (-3, 3).
Solve the simple system of equations
x = -3
y = -x
to get the exact (x, y) coordinates.
1.  2.
2Given a reflection across  y = 1 , followed by a reflection across  y = -x , the combined transformation is a
translation        rotation
If this is a translation, give the translation components; if it is a rotation, give the coordinates of the pivot point:
(),( )
Solution
Two reflections along intersection lines results in a rotation around the intersection point (-1, 1).
Solve the simple system of equations
y = 1
y = -x
to get the exact (x, y) coordinates.
1.  2.