Math worksheet on geometric reflections across different lines on coordinate grids.

Since the user has set all options (ExplainAnswer, TeachTopic, GeneratePractice) to false, I will proceed directly to solving the problems and providing the final answers in the required format.

Let’s solve each reflection problem step by step:

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Problem 1: Reflection Across Line y = -x

Original shape is in quadrant III. To reflect over y = -x, we swap x and y and change both signs: (x, y) → (-y, -x)

Example point: (-4, -2) → (2, 4)
Another: (-3, -2) → (2, 3)
Another: (-3, -3) → (3, 3)
Another: (-4, -3) → (3, 4)

So reflected shape will be in quadrant I with those points.

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Problem 2: Reflection Across Line y = x

Rule: (x, y) → (y, x)

Original points (approx): (-3, 3), (-2, 4), (-1, 2), (0, 2)

Reflected: (3, -3), (4, -2), (2, -1), (2, 0)

Shape moves to quadrant IV.

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Problem 3: Reflection Across line x = 1

Vertical line at x=1. Distance from point to line is preserved on other side.

For a point (x, y), new x-coordinate = 1 + (1 - x) = 2 - x

Example: (2, 3) → (0, 3)
(4, 3) → (-2, 3)
(4, 1) → (-2, 1)
(2, 1) → (0, 1)

Shape flips left across x=1.

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Problem 4: Reflection Across the y-axis

Rule: (x, y) → (-x, y)

Original triangle: (0,0), (1,4), (5,0)

Reflected: (0,0), (-1,4), (-5,0)

Flips to left side of y-axis.

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Problem 5: Reflection Across line y = 2

Horizontal line. For point (x, y), new y = 2 + (2 - y) = 4 - y

Example: (2, 4) → (2, 0)
(4, 4) → (4, 0)
(4, 2) → (4, 2) [on the line]
(2, 2) → (2, 2)

Shape flips down across y=2.

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Problem 6: Reflection Across the x-axis

Rule: (x, y) → (x, -y)

Original shape in quadrant IV: e.g., (3, -2), (4, -2), (4, -3), (3, -4), (2, -4), (2, -3)

Reflected: (3, 2), (4, 2), (4, 3), (3, 4), (2, 4), (2, 3)

Flips up into quadrant I.

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Final Answer:
The reflected shapes are drawn as follows:

1) Across y = -x: Shape appears in Quadrant I with vertices at approximately (2,4), (2,3), (3,3), (3,4)
2) Across y = x: Shape appears in Quadrant IV with vertices at approximately (3,-3), (4,-2), (2,-1), (2,0)
3) Across x = 1: Shape appears to the left of x=1 with vertices at approximately (0,3), (-2,3), (-2,1), (0,1)
4) Across y-axis: Triangle appears on left with vertices at (0,0), (-1,4), (-5,0)
5) Across y = 2: Shape appears below y=2 with vertices at approximately (2,0), (4,0), (4,2), (2,2)
6) Across x-axis: Shape appears in Quadrant I with vertices at approximately (3,2), (4,2), (4,3), (3,4), (2,4), (2,3)

Note: Since this is a drawing task, the exact plotted points may vary slightly depending on grid interpretation, but the transformation rules above are correct.