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The problem provided involves several geometric transformations applied to a triangle \( PQR \) and a given line equation.

Let's break down each transformation based on the coordinates of the triangle's vertices:
- \( P(-9, -2) \)
- \( Q(7, -4) \)
- \( R(2, 6) \)

### 1. Translation by \( \tau = \left( -4, 6 \right) \)

Translation means shifting each point by the vector \( \tau \).

For each point, apply the translation:
- \( P' = P + (-4, 6) = (-9 - 4, -2 + 6) = (-13, 4) \)
- \( Q' = Q + (-4, 6) = (7 - 4, -4 + 6) = (3, 2) \)
- \( R' = R + (-4, 6) = (2 - 4, 6 + 6) = (-2, 12) \)

### 2. Reflection over the line \( y = x \)

For reflection over the line \( y = x \), switch the coordinates of each point.

New coordinates:
- \( P'' = (-2, -9) \)
- \( Q'' = (-4, 7) \)
- \( R'' = (6, 2) \)

### 3. Dilation with center \( (1, 2) \) and scale factor 3

Use the dilation formula:
\[
P' = \left( x_1 + k(x - x_1), y_1 + k(y - y_1) \right)
\]
where \( k \) is the scale factor and \( (x_1, y_1) \) is the center.

For \( P(-9, -2) \):
- \( P''' = (1 + 3(-9 - 1), 2 + 3(-2 - 2)) = (1 - 30, 2 - 12) = (-29, -10) \)

For \( Q(7, -4) \):
- \( Q''' = (1 + 3(7 - 1), 2 + 3(-4 - 2)) = (1 + 18, 2 - 18) = (19, -16) \)

For \( R(2, 6) \):
- \( R''' = (1 + 3(2 - 1), 2 + 3(6 - 2)) = (1 + 3, 2 + 12) = (4, 14) \)

### 4. Rotation by 90° around the origin \( (0,0) \)

For a 90° counterclockwise rotation, the transformation rule is:
\[
(x', y') = (-y, x)
\]

For each point:
- \( P''' = (2, -9) \)
- \( Q''' = (4, 7) \)
- \( R''' = (-6, 2) \)

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### Next, for the line equation \( y = x^2 - 2x - 8 \), you are tasked to perform four transformations. I will explain the solutions for that separately.

Let me know if you'd like more details or explanations!

**5 Related Questions:**
1. What are the properties of transformations on the coordinate plane?
2. How do you apply a general translation to any geometric shape?
3. How does dilation affect the size of geometric figures?
4. Why does reflection over \( y = x \) result in a coordinate switch?
5. How can you geometrically interpret a 90° rotation?

**Tip**: When applying multiple transformations, always track the coordinates after each step to avoid confusion.

A triangle PQR with vertices P(-9,-2), Q(7,-4), and R(2,6) undergoes transformations: translation, reflection over y=x, dilation, and rotation. A line y = x^2 - 2x - 8 also undergoes similar transformations.

Learn how to apply geometric transformations such as translation, reflection, dilation, and rotation on the vertices of a triangle and a quadratic line equation. This problem includes translation by (-4, 6), reflection over y=x, dilation with scale 3, and rotation by 90°. Suitable for Grade 12 mathematics students.

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