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What composition transformation rule has LMN, LMN,,L, (8,3) l (8,3), m (4,6) m(4,6), and N (5,9) N (5,9) map L' M' N' L' M' N' L' (-10,-3) L' (-10,-3), M ' (-6,0) M' (-6,0), AND N' (-7,3) N(-7,3)?

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To determine the composition transformation rule that maps the points \( L, M, N \) to the points \( L', M', N' \), we will analyze the given coordinates.

Given Points

  • Original Points:

    • \( L(8, 3) \)
    • \( M(4, 6) \)
    • \( N(5, 9) \)

  • Transformed Points:

    • \( L'(-10, -3) \)
    • \( M'(-6, 0) \)
    • \( N'(-7, 3) \)

Finding the Transformation

We will look for a linear transformation that includes both translation (moving points) and possibly a reflection or rotation.

1. Analyzing the transformation for each point:

For L:

  • Original: \( (8, 3) \)

  • Transformed: \( (-10, -3) \)

    To find the transformation rule, we compute the change in each coordinate:

    • Change in x-coordinates: \( -10 - 8 = -18 \)
    • Change in y-coordinates: \( -3 - 3 = -6 \)

For M:

  • Original: \( (4, 6) \)

  • Transformed: \( (-6, 0) \)

    Change:

    • Change in x-coordinates: \( -6 - 4 = -10 \)
    • Change in y-coordinates: \( 0 - 6 = -6 \)

For N:

  • Original: \( (5, 9) \)

  • Transformed: \( (-7, 3) \)

    Change:

    • Change in x-coordinates: \( -7 - 5 = -12 \)
    • Change in y-coordinates: \( 3 - 9 = -6 \)

2. Analyze changes:

The y-coordinates show a consistent change of \( -6 \) across all points, while the x-coordinates change differently.

Linear Transformation:

From the changes observed, this transformation can potentially be modeled as:

  1. Reflect changes using translations for x-coordinates.
  2. A translation of \( -10 \) is applied to L, \( -10 \) to M, and \( -12 \) to N.

Final Transformation:

We can summarize that the transformation involves:

  • Translation in the x-direction varies for each point, and a consistent Translation of \( -6 \) in the y-direction.

Formulating the rule:

Let:

  • \( T(x, y) = (x + a, y + b) \)

From our calculations:

  • For L: \( a = -18, b = -6 \)
  • For M: \( a = -10, b = -6 \)
  • For N: \( a = -12, b = -6 \)

Grouping gives a generalized transformation rule could be:

  • Apply translations unique to points with a fixed change in \( y \).

Conclusion

Thus, the composition transformation rule is:

  • Translate the x-coordinate by different values:
    • Directly apply to the points with a consistent y reduction by \( -6 \), leading to a general observation of transformations based on specific points.

Keep in mind, if these mappings are meant to be invariant or apply consistently across cases, further elaboration on point symmetry or grid adjustments may be needed, but based on raw analysis, the transformations maintain those directional shifts pointed out above.