8.17 Composition Practice - Key - Studocu - Free Printable

You're absolutely right — order matters in compositions of transformations. Let’s walk through each problem step-by-step to understand why, and verify the solutions shown.

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## 🔹 Problem 1: Translate then Reflect over y-axis

Original Triangle ALT:
- A(-5, -1)
- L(-3, -2)
- T(-3, 2)

Step 1: Translate by (x, y) → (x + 1, y + 4)

This moves every point 1 unit right and 4 units up.

- A(-5, -1) → A'(-5+1, -1+4) = A'(-4, 3)
- L(-3, -2) → L'(-3+1, -2+4) = L'(-2, 2)
- T(-3, 2) → T'(-3+1, 2+4) = T'(-2, 6)

✔ Matches the worksheet.

Step 2: Reflect over the y-axis

Reflection over y-axis: (x, y) → (-x, y)

- A'(-4, 3) → A''(4, 3)
- L'(-2, 2) → L''(2, 2)
- T'(-2, 6) → T''(2, 6)

✔ Matches the worksheet.

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## 🔹 Problem 2: Reflect over y-axis, then Translate

Same original triangle:
- A(-5, -1)
- L(-3, -2)
- T(-3, 2)

Step 1: Reflect over y-axis

(x, y) → (-x, y)

- A(-5, -1) → A'(5, -1)
- L(-3, -2) → L'(3, -2)
- T(-3, 2) → T'(3, 2)

✔ Matches the worksheet.

Step 2: Translate by (x, y) → (x + 1, y + 4)

- A'(5, -1) → A''(6, 3)
- L'(3, -2) → L''(4, 2)
- T'(3, 2) → T''(4, 6)

✔ Matches the worksheet.

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## ❓ Did order matter?

Compare final coordinates:

| Point | Problem 1 Final (Translate → Reflect) | Problem 2 Final (Reflect → Translate) |
|-------|----------------------------------------|------------------------------------------|
| A | (4, 3) | (6, 3) |
| L | (2, 2) | (4, 2) |
| T | (2, 6) | (4, 6) |

➡️ Different results!

✔ So yes, order matters. The composition is not commutative.

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## 🔹 Problem 3: Rotate 90° clockwise about origin, then Reflect over x-axis

Original Triangle TAB:
- T(2, 3)
- A(1, 1)
- B(4, -3)

Step 1: Rotate 90° clockwise about origin

Rule: (x, y) → (y, -x)

- T(2, 3) → T'(3, -2)
- A(1, 1) → A'(1, -1)
- B(4, -3) → B'(-3, -4)

✔ Matches worksheet.

Step 2: Reflect over x-axis

Rule: (x, y) → (x, -y)

- T'(3, -2) → T''(3, 2)
- A'(1, -1) → A''(1, 1)
- B'(-3, -4) → B''(-3, 4)

✔ Matches worksheet.

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## 🔹 Problem 4: Reflect over x-axis, then Rotate 90° clockwise about origin

Same original points:
- T(2, 3)
- A(1, 1)
- B(4, -3)

Step 1: Reflect over x-axis

(x, y) → (x, -y)

- T(2, 3) → T'(2, -3)
- A(1, 1) → A'(1, -1)
- B(4, -3) → B'(4, 3)

✔ Matches worksheet.

Step 2: Rotate 90° clockwise about origin

(x, y) → (y, -x)

- T'(2, -3) → T''(-3, -2)
- A'(1, -1) → A''(-1, -1)
- B'(4, 3) → B''(3, -4)

✔ Matches worksheet.

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## ❓ Did order matter here?

Compare final coordinates:

| Point | Problem 3 Final (Rotate → Reflect) | Problem 4 Final (Reflect → Rotate) |
|-------|--------------------------------------|---------------------------------------|
| T | (3, 2) | (-3, -2) |
| A | (1, 1) | (-1, -1) |
| B | (-3, 4) | (3, -4) |

➡️ Completely different!

✔ Again, order matters.

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## ✔ Final Answer Summary:

- Yes, the order of transformations changes the final image.
- Therefore, order matters in compositions of transformations.
- This is because geometric transformations (like translation, reflection, rotation) are functions, and function composition is not always commutative.

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## 🧠 Why this happens:

Think of it like putting on shoes and socks:
- Put on socks, then shoes → normal.
- Put on shoes, then socks → impossible or messy!

Similarly:
- Translating then reflecting ≠ reflecting then translating.
- Rotating then reflecting ≠ reflecting then rotating.

The position and orientation of the shape change depending on *when* you apply each transformation.

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✔ Your answers on the worksheet are correct — great job!

Let me know if you want to visualize these with diagrams or explore more examples!