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

Let’s solve each reflection problem step by step. We’ll find the new coordinates of each shape after reflecting it over the given line.

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Problem 1: Reflection Across Line y = -x

Original points (from graph):
Looking at the green L-shape in quadrant III:
- Bottom-left corner: (-3, -4)
- Top-left corner: (-3, -2)
- Bottom-right corner: (-1, -4)

Rule for reflection over y = -x:
Swap x and y, then change both signs → (x, y) → (-y, -x)

Apply to each point:

- (-3, -4) → (4, 3)
- (-3, -2) → (2, 3)
- (-1, -4) → (4, 1)

So reflected points are: (4,3), (2,3), (4,1)

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Problem 2: Reflection Across Line y = x

Original triangle points (in quadrant II):
From graph:
- (-4, 2)
- (-2, 3)
- (0, 1)

Rule for reflection over y = x: Swap x and y → (x, y) → (y, x)

Apply:

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

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

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Problem 3: Reflection Across the line x = 1

Original shape (green polygon in quadrant I):
Points from graph:
- (1, 2)
- (1, 4)
- (4, 4)
- (4, 3)
- (2, 3)
- (2, 2)

To reflect over vertical line x = 1, use formula:
New x = 1 - (original x - 1) = 2 - original x
Keep y same.

So: (x, y) → (2 - x, y)

Apply:

- (1,2) → (2-1, 2) = (1,2) ← stays on line
- (1,4) → (1,4) ← stays
- (4,4) → (2-4,4) = (-2,4)
- (4,3) → (-2,3)
- (2,3) → (0,3)
- (2,2) → (0,2)

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

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Problem 4: Reflection Across the y-axis

Original triangle (quadrant I):
Points:
- (1,1)
- (1,4)
- (5,1)

Rule for reflection over y-axis: Change sign of x → (x,y) → (-x, y)

Apply:

- (1,1) → (-1,1)
- (1,4) → (-1,4)
- (5,1) → (-5,1)

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

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Problem 5: Reflection Across the line y = 2

Original parallelogram (quadrant I):
Points:
- (1,3)
- (2,5)
- (5,5)
- (4,3)

To reflect over horizontal line y = 2:
New y = 2 - (original y - 2) = 4 - original y
Keep x same.

So: (x, y) → (x, 4 - y)

Apply:

- (1,3) → (1, 4-3) = (1,1)
- (2,5) → (2, 4-5) = (2,-1)
- (5,5) → (5,-1)
- (4,3) → (4,1)

Reflected points: (1,1), (2,-1), (5,-1), (4,1)

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Problem 6: Reflection Across the x-axis

Original L-shape (quadrant IV):
Points:
- (2,-5)
- (2,-3)
- (4,-3)
- (4,-2)
- (5,-2)
- (5,-4)

Rule for reflection over x-axis: Change sign of y → (x,y) → (x, -y)

Apply:

- (2,-5) → (2,5)
- (2,-3) → (2,3)
- (4,-3) → (4,3)
- (4,-2) → (4,2)
- (5,-2) → (5,2)
- (5,-4) → (5,4)

Reflected points: (2,5), (2,3), (4,3), (4,2), (5,2), (5,4)

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Final Answer:

1) Reflected points across y = -x: (4,3), (2,3), (4,1)
2) Reflected points across y = x: (2,-4), (3,-2), (1,0)
3) Reflected points across x = 1: (1,2), (1,4), (-2,4), (-2,3), (0,3), (0,2)
4) Reflected points across y-axis: (-1,1), (-1,4), (-5,1)
5) Reflected points across y = 2: (1,1), (2,-1), (5,-1), (4,1)
6) Reflected points across x-axis: (2,5), (2,3), (4,3), (4,2), (5,2), (5,4)