Geometry Transformation Composition Key - Free Printable

We are given a composition of transformations applied to a set of points:
Pre-image points:
- D(9, -3)
- E(6, -7)
- F(3, -3)
- G(5, -1)

Transformations (in order):
$$
R_{x=2} \circ T_{180^\circ} \circ T_{\langle -10, -1 \rangle}
$$

This means we apply the transformations from right to left:

1. $ T_{\langle -10, -1 \rangle} $: Translate by vector $ \langle -10, -1 \rangle $
2. $ T_{180^\circ} $: Rotate 180° about the origin
3. $ R_{x=2} $: Reflect over the vertical line $ x = 2 $

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Let’s go step-by-step for each point.

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Step 1: Apply $ T_{\langle -10, -1 \rangle} $

This transformation moves each point by:
$$
(x, y) \to (x - 10, y - 1)
$$

Apply this to all points:

- D(9, -3) → D': (9 - 10, -3 - 1) = (-1, -4)
- E(6, -7) → E': (6 - 10, -7 - 1) = (-4, -8)
- F(3, -3) → F': (3 - 10, -3 - 1) = (-7, -4)
- G(5, -1) → G': (5 - 10, -1 - 1) = (-5, -2)

So after first transformation, we have:

| Point | Coordinates |
|-------|-------------|
| D' | (-1, -4) |
| E' | (-4, -8) |
| F' | (-7, -4) |
| G' | (-5, -2) |

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Step 2: Apply $ T_{180^\circ} $ — Rotation 180° about the origin

Rotation 180° about the origin maps:
$$
(x, y) \to (-x, -y)
$$

Apply this to D', E', F', G':

- D'(-1, -4) → D'': (1, 4)
- E'(-4, -8) → E'': (4, 8)
- F'(-7, -4) → F'': (7, 4)
- G'(-5, -2) → G'': (5, 2)

So after second transformation:

| Point | Coordinates |
|-------|-------------|
| D'' | (1, 4) |
| E'' | (4, 8) |
| F'' | (7, 4) |
| G'' | (5, 2) |

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Step 3: Apply $ R_{x=2} $ — Reflection over the line $ x = 2 $

To reflect a point $ (x, y) $ over the vertical line $ x = 2 $, use the formula:
$$
x' = 2 \cdot 2 - x = 4 - x
$$
So the new point is $ (4 - x, y) $

Apply this to D'', E'', F'', G'':

- D''(1, 4) → D''': (4 - 1, 4) = (3, 4)
- E''(4, 8) → E''': (4 - 4, 8) = (0, 8)
- F''(7, 4) → F''': (4 - 7, 4) = (-3, 4)
- G''(5, 2) → G''': (4 - 5, 2) = (-1, 2)

So final image points:

| Point | Coordinates |
|-------|-------------|
| D''' | (3, 4) |
| E''' | (0, 8) |
| F''' | (-3, 4) |
| G''' | (-1, 2) |

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Final Answer Table:

| Order of Transformations | Coordinates: |
|--------------------------|--------------|
| $ T_{\langle -10, -1 \rangle} $ | D'(-1, -4), E'(-4, -8), F'(-7, -4), G'(-5, -2) |
| $ T_{180^\circ} $ | D''(1, 4), E''(4, 8), F''(7, 4), G''(5, 2) |
| $ R_{x=2} $ | D'''(3, 4), E'''(0, 8), F'''(-3, 4), G'''(-1, 2) |

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Summary:

The final image of the figure after all three transformations is:

- D'''(3, 4)
- E'''(0, 8)
- F'''(-3, 4)
- G'''(-1, 2)

You can plot these on the graph to see the final position.

Let me know if you'd like a sketch or explanation of how the shape changes!