Representing Transformations as Functions - Geometry

1. What is the coordinate rule for reflection across the line $y=-x$?

   $(x,y)\to(-y,-x)$. Both coordinates swap and change signs across this diagonal.

2. What does a dilation with scale factor $k$ do to all lengths?

   It multiplies every distance by $|k|$. The absolute value of the scale factor scales all distances.

3. What is the image of $(4,-7)$ under reflection across the $y$-axis?

   $(-4,-7)$. Reflection across $y$-axis negates the $x$-coordinate only.

4. Identify the transformation given by $T(x,y)=(y,x)$.

   Reflection across the line $y=x$. The rule swaps the coordinates without negation.

5. What is the coordinate rule for a $90^\circ$ counterclockwise rotation about the origin?

   $(x,y)\to(-y,x)$. Standard counterclockwise rotation matrix for $90°$ about origin.

6. What is the coordinate rule for a $180^\circ$ rotation about the origin?

   $(x,y)\to(-x,-y)$. Both coordinates change signs for a half-turn rotation.

7. What is the inverse rule for the transformation $T(x,y)=(x+5,y-2)$?

   $T^{-1}(x,y)=(x-5,y+2)$. Inverse translation subtracts the original translation vector.

8. What is the image of $(-3,5)$ under a dilation about the origin with $k=2$?

   $(-6,10)$. Multiply both coordinates by scale factor: $2(-3,5) = (-6,10)$.

9. What is the inverse of a dilation with scale factor $k\ne^0$ about the origin?

   A dilation with scale factor $\frac{1}{k}$ about the origin. Inverse dilation uses the reciprocal of the scale factor.

10. What is the inverse of a reflection across a line?

    The same reflection (it is its own inverse). Reflecting twice across the same line gives the identity.

11. What does a dilation with scale factor $k$ do to angle measures?

    It preserves all angle measures. Dilations are similarity transformations that preserve angles.

12. What is the coordinate rule for a vertical stretch by factor $k$?

    $(x,y)\to(x,ky)$. Only the $y$-coordinate is multiplied by the stretch factor.

13. What is the output of a transformation $T$ written as $T(x,y)$?

    An image point $(x',y')$ produced from the input point $(x,y)$. The function transforms the input coordinates to new coordinates.

14. Identify the transformation given by $T(x,y)=(-x,y)$.

    Reflection across the $y$-axis. The rule negates only the $x$-coordinate.

15. What is the image of $(7,-4)$ under a $180^\circ$ rotation about the origin?

    $(-7,4)$. Apply the rule $(x,y) \to (-x,-y)$ to get $(-7,4)$.

16. Which transformation preserves both distance and angle: translation or horizontal stretch?

    Translation. Translation is a rigid motion that preserves all measurements.

17. What is the coordinate rule for reflection across the $y$-axis?

    $(x,y)\to(-x,y)$. The $y$-coordinate stays the same, $x$-coordinate changes sign.

18. What is the coordinate rule for a $90^\circ$ clockwise rotation about the origin?

    $(x,y)\to(y,-x)$. Clockwise rotation is the inverse of counterclockwise rotation.

19. What property defines a rigid motion (isometry) in the plane?

    It preserves all distances and all angle measures. Rigid motions maintain all geometric measurements unchanged.

20. What is the coordinate rule for reflection across the line $y=x$?

    $(x,y)\to(y,x)$. The coordinates swap positions across the diagonal line.

21. Which classification fits $T(x,y)=(x+4,y-3)$: rigid motion, similarity, or neither?

    Rigid motion. Translation preserves both distance and angle measures.

22. What is the coordinate rule for a dilation centered at the origin with scale factor $k$?

    $(x,y)\to(kx,ky)$. Both coordinates are multiplied by the scale factor.

23. What is the image of $(4,-2)$ under a horizontal stretch by factor $3$?

    $(12,-2)$. Only the $x$-coordinate is multiplied by the stretch factor.

24. What is the image of a point under a transformation?

    The transformed point after the rule is applied. The resulting point after the transformation function is applied.

25. Identify the transformation given by $T(x,y)=(x-7,y+1)$.

    Translation by vector $\langle -7,1\rangle$. The rule adds constants to coordinates, indicating translation.

26. What is the coordinate rule for reflection across the $x$-axis?

    $(x,y)\to(x,-y)$. The $x$-coordinate stays the same, $y$-coordinate changes sign.

27. Identify the transformation given by $T(x,y)=(\frac{1}{2}x,y)$.

    Horizontal compression by factor $\frac{1}{2}$. Only the $x$-coordinate is scaled by $\frac{1}{2}$.

28. What is the image of $(3,-2)$ under translation by $\langle -5,4\rangle$?

    $(-2,2)$. Add vector components: $(3-5, -2+4) = (-2,2)$.

29. What is the image of $(12,-9)$ under a dilation about the origin with $k=\frac{1}{3}$?

    $(4,-3)$. Multiply both coordinates by $\frac{1}{3}$: $(4,-3)$.

30. What is the image of $(-2,6)$ under a $90^\circ$ clockwise rotation about the origin?

    $(6,2)$. Apply the rule $(x,y) \to (y,-x)$ to get $(6,2)$.

31. Identify the transformation given by $T(x,y)=(-y,x)$.

    $90^\circ$ counterclockwise rotation about the origin. The rule matches the standard $90°$ counterclockwise rotation formula.

32. What is the image of $(5,-1)$ under reflection across the line $y=-x$?

    $(1,-5)$. Reflection across $y=-x$ swaps and negates both coordinates.

33. Which classification fits $T(x,y)=(3x,3y)$: rigid motion, similarity, or neither?

    Similarity (not rigid unless $|3|=1$). Uniform scaling preserves angles but changes distances by factor $3$.

34. Which transformations are the basic rigid motions in the plane?

    Translations, rotations, reflections, and compositions of these. These are the three fundamental rigid transformations and their combinations.

35. Identify whether a translation changes orientation of a figure.

    It preserves orientation. Translation slides without rotating or flipping the figure.

36. What is an invariant point under a transformation?

    A point that maps to itself, so $T(P)=P$. Fixed points remain unchanged under the transformation.

37. What is the inverse of a rotation by $\theta$ about the origin?

    A rotation by $-\theta$ about the origin. Inverse rotation uses the negative of the original angle.

38. What is the coordinate rule for a horizontal stretch by factor $k$?

    $(x,y)\to(kx,y)$. Only the $x$-coordinate is multiplied by the stretch factor.

39. What is the image of $(-6,1)$ under reflection across the $x$-axis?

    $(-6,-1)$. Reflection across $x$-axis negates the $y$-coordinate only.

40. Identify the transformation given by $T(x,y)=(2x,2y)$.

    Dilation about the origin with scale factor $2$. Both coordinates are multiplied by the same factor $2$.

41. Which property distinguishes similarity transformations from rigid motions?

    Similarity preserves angles but not necessarily distances. Rigid motions preserve both, similarity only preserves angles.

42. What is the fixed point of a rotation about the origin?

    The origin, since it remains unchanged. The center of rotation is always invariant under rotation.

43. What does it mean to describe a transformation as a function on points?

    A rule mapping each input point $(x,y)$ to exactly one output point $(x',y')$. Each input point has exactly one corresponding output point.

44. What is the image of $(3,8)$ under a $90^\circ$ counterclockwise rotation about the origin?

    $(-8,3)$. Apply the rule $(x,y) \to (-y,x)$ to get $(-8,3)$.

45. Identify whether a rotation changes orientation of a figure.

    It preserves orientation. Rotation turns without flipping, maintaining orientation direction.

46. What is a non-rigid transformation example that does not preserve angles?

    A horizontal or vertical stretch (non-uniform scaling). Non-uniform scaling changes angles by stretching in one direction only.

47. What is the inverse of a translation $(x,y)\to(x+a,y+b)$?

    $(x,y)\to(x-a,y-b)$. Inverse translation uses opposite direction vector components.

48. Which transformation typically changes angle measures: dilation or horizontal stretch?

    Horizontal stretch. Horizontal stretch distorts angles while dilation preserves them.

49. What is the coordinate rule for a glide reflection (in words, not a formula)?

    A reflection followed by a translation parallel to the line of reflection. A composition of reflection and translation in that specific order.

50. What is the coordinate rule for a translation by vector $\langle a,b\rangle$?

    $(x,y)\to(x+a,y+b)$. Add the translation vector components to each coordinate.

51. What is the fixed line of a reflection?

    The line of reflection; all points on it satisfy $T(P)=P$. Every point on the reflection line maps to itself.

52. What is the image of $(-5,6)$ under a vertical stretch by factor $-2$?

    $(-5,-12)$. Only the $y$-coordinate is multiplied by the stretch factor.

53. What is the image of $(2,9)$ under reflection across the line $y=x$?

    $(9,2)$. Reflection across $y=x$ swaps the coordinates.

54. Which classification fits $T(x,y)=(2x,y)$: rigid motion, similarity, or neither?

    Neither (it does not preserve angle measures). Non-uniform scaling distorts angles and is not a similarity.

55. What is the preimage of a point under a transformation?

    The original point before the transformation is applied. The starting point before any transformation is applied.

56. Identify whether a reflection changes orientation of a figure.

    It reverses orientation. Reflection creates a mirror image, reversing orientation.

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