Activity 2

To rotate a figure 90 ° 90 degrees  about the origin, trace the figure, one axis, and the origin. Then turn your tracing paper counterclockwise, keeping the origin in place and aligning the traced axis with the other original axis.

5. Use Δ A B C cap delta eh b c  from Activity 1 with A ( − 3 , 3 ) , B ( − 1 , 1 ) , and  C ( 1 , 4 ) . eh open negative 3 comma 3 close comma b open negative 1 comma 1 close comma , and c open 1 comma 4 close .  What is the image of Δ A B C cap delta eh b c  for a 90 ° 90 degrees  rotation about the origin?

   

6. Copy and complete the table. Use tracing paper to find point P ′ , p prime comma  the image of point P, for a rotation of 90 ° 90 degrees  about the origin.
7. Study the pattern in your table. Complete this rule for a rotation of 90 ° 90 degrees  about the origin: ( x , y ) → ( □ , □ ) . open x comma y close rightwards arrow open white square comma white square close .
8. Test your rule with tracing paper. For Δ T R N cap delta t r n  with T(0, 2), R(3, 0), and N(4, 5), a 90 ° 90 degrees  rotation about the origin should result in Δ T ′ R ′ N ′ cap delta , t prime , r prime , n prime  with T ′ ( − 2 , 0 ) , R ′ ( 0 , 3 ) , and  N ′ ( − 5 , 4 ) . t prime , open negative 2 comma 0 close comma , r prime , open 0 comma 3 close comma , and , n prime , open negative 5 comma 4 close .  Does it?
9. In parts (a) and (b) below, what should be the result of each composition?
   1. a 90 ° 90 degrees  rotation about the origin followed by a 90 ° 90 degrees  rotation about the origin
   2. ( x , y ) → ( − y , x ) open x comma y close rightwards arrow . open negative y comma x close  followed by ( x , y ) → ( − y , x ) open x comma y close rightwards arrow . open negative y comma x close
   3. Use tracing paper. Test your conjectures from parts (a) and (b) on Δ A B C cap delta eh b c  from Activity 1.

   P	P ′ p prime
   (3, 4)	□ white square
   ( − 3 , 4 ) open negative 3 comma 4 close	□ white square
   ( − 3 , − 4 ) open negative 3 comma . negative 4 close	□ white square
   ( 3 , − 4 ) open 3 comma negative 4 close	□ white square
   (3, 0)	□ white square
   (0, 4)	□ white square

Activity 3

To reflect a figure across an axis using tracing paper, trace the figure, the axis, and the origin. Then turn over your tracing paper, keeping the origin in place and aligning the traced axis with the original axis.

10. What is the reflection image of Δ A B C cap delta eh b c  across the x-axis? Across the y-axis?
11. Copy and complete the table. Use tracing paper to find point P x p sub x  and P y , p sub y , comma  the reflection images of point P across the x-axis and y-axis, respectively.
12. Study the table. Complete these rules for reflections across the axes.
    1. x - axis : ( x , y ) → ( □ , □ ) x minus , axis , colon open x comma y close rightwards arrow open white square comma white square close
    2. y - axis : ( x , y ) → ( □ , □ ) y minus , axis , colon open x comma y close rightwards arrow open white square comma white square close
13. Test your rules on Δ K L M cap delta k l m  with K ( − 2 , 3 ) , k open negative 2 comma 3 close comma  L(3, 1), and M ( 1 , − 2 ) . m open 1 comma negative 2 close .
    1. Across the x-axis, Δ K L M cap delta k l m  should map to ? _ . modified question mark with under bar below , .  Does it?
    2. Across the y-axis, Δ K L M cap delta k l m  should map to ? _ . modified question mark with under bar below , .  Does it?
14. In parts (a) and (b) below, what should be the result of each composition?
    1. ( x , y ) → ( x , − y ) open x comma y close rightwards arrow open x comma . negative y close  followed by ( x , y ) → ( − x , y ) open x comma y close rightwards arrow . open negative x comma y close
    2. a reflection across the x-axis followed by a reflection across the y-axis
    3. Use tracing paper. Test your conjectures from parts (a) and (b) on Δ A B C cap delta eh b c  from Activity 1.
15. Compare the results of Questions 9 and 14. Make a conjecture about the compositions suggested by each.
16. Use tracing paper to find a rule for a reflection across the line y = x . y equals x .  Test your rule on Δ A B C cap delta eh b c  from Activity 1.

    P	P x p sub x	P y p sub y
    (3, 4)	□ white square	□ white square
    ( − 3 , 4 ) open negative 3 comma 4 close	□ white square	□ white square
    ( − 3 , − 4 ) open negative 3 comma negative 4 close	□ white square	□ white square
    ( 3 , − 4 ) open 3 comma negative 4 close	□ white square	□ white square
    (3, 0)	□ white square	□ white square
    (0, 4)	□ white square	□ white square

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