Describe Transformation Effects Using Coordinates - 8th Grade Math

1. What is the image of $(x,y)$ after a dilation with scale factor $k$ centered at $(a,b)$?

   $(a+k(x-a),\,b+k(y-b))$. Scale distance from center $(a,b)$ by factor $k$.

2. Which transformations always preserve distance (are rigid motions): translation, rotation, reflection, dilation?

   Translation, rotation, and reflection. Only dilation changes distances; the others preserve them.

3. Identify the image of $P(-6,8)$ after dilation about the origin with $k=\frac{1}{2}$.

   $(-3,4)$. Multiply by $\frac{1}{2}$: $($-6 \cdot \frac{1}{2}, 8 \cdot \frac{1}{2}$) = $(-3,4)$.

4. What is the image of $(x,y)$ after a $180^\circ$ rotation about the origin?

   $(-x,-y)$. Rotate $180°$ by negating both coordinates.

5. Identify the image of $P(-2,5)$ after reflection across the $y$-axis.

   $(2,5)$. Reflecting across $y$-axis negates $x$: $(-2,5) \rightarrow (2,5)$.

6. What are the coordinates of the image of $(x,y)$ after a translation by $(a,b)$?

   $(x+a,y+b)$. Add translation vector components to original coordinates.

7. Identify the image of $P(5,-2)$ after dilation about the origin with $k=-3$.

   $(-15,6)$. Multiply by $-3$: $(5 \cdot (-3), -2 \cdot (-3)) = (-15,6)$.

8. What happens to lengths and angle measures under a translation, rotation, or reflection?

   Lengths and angles stay the same. These are rigid transformations that preserve size and shape.

9. What is the image of $(x,y)$ after a reflection across the line $y=x$?

   $(y,x)$. Swap coordinates to reflect across the line $y=x$.

10. What is the coordinate rule for reflecting a point across the $y$-axis?

    $(x,y)\rightarrow(-x,y)$. Negate the $x$-coordinate while keeping $y$ unchanged.

11. What is the coordinate rule for dilating about the origin by scale factor $k$?

    $(x,y)\rightarrow(kx,ky)$. Multiply both coordinates by scale factor $k$.

12. Identify the image of $(-6,3)$ after a $180^\circ$ rotation about the origin.

    $(6,-3)$. Negate both: $(-6,3) → (6,-3)$ for $180°$ rotation.

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

    $(x,y)\rightarrow(y,-x)$. Swap coordinates and negate new $y$ for $90°$ CW rotation.

14. Identify the image of $(-2,5)$ after reflection across the line $y=x$.

    $(5,-2)$. Swap coordinates: $(x,y)$ becomes $(y,x)$ across $y=x$.

15. Identify the image of $P(-4,1)$ after a $90^\circ$ clockwise rotation about the origin.

    $(1,4)$. Apply rule: $(-4,1) \rightarrow (1,-(-4)) = (1,4)$.

16. Identify the image of $(8,-4)$ after dilation about the origin with scale factor $\frac{1}{2}$.

    $(4,-2)$. Multiply by $\frac{1}{2}$: $(8,-4) → (4,-2)$.

17. Identify the transformation that maps $(x,y)$ to $(x,-y)$.

    Reflection across the $x$-axis. Negating only $y$ reflects across the $x$-axis.

18. Identify the image of $P(3,-7)$ after reflection across the line $y=x$.

    $(-7,3)$. Swap coordinates: $(3,-7) \rightarrow (-7,3)$.

19. What is the coordinate rule for dilating a point $(x,y)$ by scale factor $k$ about the origin?

    $(x,y)\rightarrow(kx,ky)$. Multiply both coordinates by scale factor $k$.

20. What is the coordinate rule for reflecting a point $(x,y)$ across the $y$-axis?

    $(x,y)\rightarrow(-x,y)$. Negate the $x$-coordinate to flip across the vertical axis.

21. What is the image of $(x,y)$ after a $90^\circ$ counterclockwise rotation about the origin?

    $(-y,x)$. Rotate $90°$ CCW: $(x,y) → (-y,x)$ by switching and negating.

22. What is the image of the point $(4,1)$ after a $90^\circ$ counterclockwise rotation about the origin?

    $(-1,4)$. Apply $90°$ CCW rule: $(4,1) → (-1,4)$.

23. What happens to lengths and angle measures under a dilation with scale factor $k$?

    Lengths multiply by $|k|$; angles stay the same. Dilation scales distances but preserves angle measures.

24. Identify the image of the point $(3,-2)$ after the translation $(x,y)\rightarrow(x-5,y+4)$.

    $(-2,2)$. Apply the rule: $3-5=-2$ for $x$, $-2+4=2$ for $y$.

25. What is the image of $(x,y)$ after a reflection across the $x$-axis?

    $(x,-y)$. Flip across $x$-axis by negating the $y$-coordinate.

26. Identify the transformation that maps $(x,y)$ to $(x-6,y+2)$.

    Translation by $(-6,2)$. Subtracting 6 from $x$ and adding 2 to $y$ translates by $(-6,2)$.

27. What is the coordinate rule for reflecting a point $(x,y)$ across the $x$-axis?

    $(x,y)\rightarrow(x,-y)$. Negate the $y$-coordinate to flip across the horizontal axis.

28. What is the coordinate rule for reflecting a point $(x,y)$ across the line $y=-x$?

    $(x,y)\rightarrow(-y,-x)$. Swap and negate both coordinates for reflection across $y=-x$.

29. What is the coordinate rule for reflecting a point across the $x$-axis?

    $(x,y)\rightarrow(x,-y)$. Negate the $y$-coordinate while keeping $x$ unchanged.

30. Identify the image of $(2,7)$ after a $90^\circ$ counterclockwise rotation about the origin.

    $(-7,2)$. Apply rule: $(2,7) → (-7,2)$ by swapping and negating new $x$.

31. Identify the transformation that maps $(x,y)$ to $(-x,-y)$.

    Rotation $180^\circ$ about the origin. Negating both coordinates rotates $180°$.

32. What is the coordinate rule for translating a point by $(a,b)$?

    $(x,y)\rightarrow(x+a,y+b)$. Add $a$ to $x$-coordinate and $b$ to $y$-coordinate to shift the point.

33. What is the coordinate rule for reflecting a point $(x,y)$ across the line $y=x$?

    $(x,y)\rightarrow(y,x)$. Swap coordinates to reflect across the diagonal line $y=x$.

34. Identify the scale factor $k$ if $P(2,3)$ dilates about the origin to $P'(6,9)$.

    $k=3$. Since $(2 cdot 3, 3 cdot 3) = (6,9)$, the scale factor is $3$.

35. Identify the image of $P(7,-3)$ after a $180^\circ$ rotation about the origin.

    $(-7,3)$. Negate both: $(7,-3) \rightarrow (-7,3)$

36. Identify the image of $P(2,-5)$ after a $90^\circ$ counterclockwise rotation about the origin.

    $(5,2)$. Apply rule: $(2,-5) \rightarrow (-(-5),2) = (5,2)$.

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

    $(x,y)\rightarrow(-x,-y)$. Negate both coordinates for half-turn around origin.

38. What is the coordinate rule for reflecting a point across the line $y=x$?

    $(x,y)\rightarrow(y,x)$. Swap the $x$ and $y$ coordinates.

39. What is the image of $(x,y)$ after a $90^\circ$ clockwise rotation about the origin?

    $(y,-x)$. Rotate $90°$ CW: $(x,y) → (y,-x)$ by switching and negating.

40. What is the image of $(x,y)$ after a reflection across the $y$-axis?

    $(-x,y)$. Flip across $y$-axis by negating the $x$-coordinate.

41. What is the image of the point $(-2,6)$ after a reflection across the $y$-axis?

    $(2,6)$. Reflecting $(-2,6)$ across $y$-axis negates $x$: $(2,6)$.

42. What is the image of $(x,y)$ after a dilation with scale factor $k$ centered at the origin?

    $(kx,ky)$. Multiply both coordinates by scale factor $k$.

43. Identify the image of $(4,-1)$ after reflection across the $x$-axis.

    $(4,1)$. Negate $y$-coordinate: $-1$ becomes $1$, $x$ stays $4$.

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

    $(x,y)\rightarrow(-y,x)$. Swap coordinates and negate new $x$ for $90°$ CCW rotation.

45. What is the image of $(x,y)$ after a reflection across the line $y=-x$?

    $(-y,-x)$. Swap and negate both coordinates for $y=-x$ reflection.

46. What happens to angle measures under translations, rotations, reflections, and dilations?

    Angles stay equal for all four transformations. All four transformations preserve angle measures.

47. Identify the image of $P(6,-1)$ after reflection across the $x$-axis.

    $(6,1)$. Reflecting across $x$-axis negates $y$: $(6,-1) \rightarrow (6,1)$.

48. Identify the transformation that maps $(x,y)$ to $(-x,y)$.

    Reflection across the $y$-axis. Negating only $x$ reflects across the $y$-axis.

49. Identify the transformation that maps $(x,y)$ to $(y,-x)$.

    Rotation $90^\circ$ clockwise about the origin. Pattern $(y,-x)$ indicates $90°$ CW rotation.

50. Identify the image of $P(-3,4)$ after translation by $(5,-2)$.

    $(2,2)$. Apply translation: $(-3+5, 4+(-2)) = (2,2)$.

51. What is always true about side lengths after a dilation with scale factor $k$?

    Each length is multiplied by $|k|$. Dilation scales all distances by the absolute value of $k$.

52. Identify the transformation that maps $(x,y)$ to $(-y,x)$.

    Rotation $90^\circ$ counterclockwise about the origin. Pattern $(-y,x)$ indicates $90°$ CCW rotation.

53. What is the image of the point $(-3,2)$ after a dilation with scale factor $\frac{1}{2}$ centered at the origin?

    $\left(-\frac{3}{2},1\right)$. Scale by $\frac{1}{2}$: $(-3,2) → (-\frac{3}{2},1)$.

54. What is the coordinate rule for translating a point $(x,y)$ by $(a,b)$?

    $(x,y)\rightarrow(x+a,y+b)$. Add $a$ to $x$-coordinate and $b$ to $y$-coordinate to shift the point.

55. What is the image of the point $(3,-5)$ after a translation by $(4,-2)$?

    $(7,-7)$. $(3,-5) + (4,-2) = (3+4, -5+(-2)) = (7,-7)$.

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