Question

identifying transformations
match each representation of a transformation to its correct algebraic rule from the bank below.
not all choices will be used.
$(x, y) \to (x, y - 10)$
$(x, y) \to (-x, y)$
$(x, y) \to (x + 10, y - 9)$
$(x, y) \to (y, -x)$
$(x, y) \to (-x, -y)$
$(x, y) \to (x, -y)$
$(x, y) \to (x, y + 10)$
$(x, y) \to (-y, x)$
$(x, y) \to (x - 10, y + 9)$

Explanation:

To solve this, we analyze each transformation rule:

Step 1: Recall Transformation Types

• Translation: Changes position (e.g., \((x,y) \to (x + a, y + b)\)).
• Reflection: Flips over an axis (e.g., \((x,y) \to (-x,y)\) reflects over \(y\)-axis, \((x,y) \to (x,-y)\) reflects over \(x\)-axis, \((x,y) \to (y,x)\) reflects over \(y = x\), \((x,y) \to (-y,-x)\) reflects over \(y=-x\)).

Step 2: Match Each Rule

1. \((x,y) \to (x, y - 10)\): Vertical translation down 10 units.
2. \((x,y) \to (-x, y)\): Reflection over \(y\)-axis.
3. \((x,y) \to (x + 10, y - 9)\): Horizontal translation right 10, vertical down 9.
4. \((x,y) \to (y, -x)\): Rotation or reflection (swap \(x,y\) and negate \(x\), which is a 90° rotation counterclockwise? Wait, standard 90° CCW is \((x,y) \to (-y,x)\), 90° CW is \((x,y) \to (y,-x)\). So this is 90° clockwise rotation.
5. \((x,y) \to (-x, -y)\): Reflection over origin (or 180° rotation).
6. \((x,y) \to (x, -y)\): Reflection over \(x\)-axis.
7. **\((x,y) \to (x + 10)\)? Wait, the last one in the bottom right: \((x,y) \to (x, y + 10)\): Vertical translation up 10.
8. \((x,y) \to (-y, x)\): 90° counterclockwise rotation (since \((x,y) \to (-y,x)\) is 90° CCW).
9. \((x,y) \to (x - 10, y + 9)\): Horizontal translation left 10, vertical up 9.

Since the problem says "Match each representation to its correct algebraic rule", we identify the type for each:

• Translation: \((x,y) \to (x, y - 10)\), \((x,y) \to (x + 10, y - 9)\), \((x,y) \to (x, y + 10)\), \((x,y) \to (x - 10, y + 9)\)
• Reflection: \((x,y) \to (-x, y)\) (over \(y\)-axis), \((x,y) \to (x, -y)\) (over \(x\)-axis), \((x,y) \to (-x, -y)\) (over origin)
• Rotation: \((x,y) \to (y, -x)\) (90° CW), \((x,y) \to (-y, x)\) (90° CCW)

Assuming we need to match each "representation" (the left side) to the rule (right side), but the exact matching depends on the given "representations" (the problem's left column, which we can infer):

For example, if a representation is "Shift down 10", match to \((x,y) \to (x, y - 10)\). If it's "Reflect over \(y\)-axis", match to \((x,y) \to (-x, y)\), etc.

Answer:

To fully solve, we match each transformation description (not shown, but from the rules):

• Vertical down 10: \(\boldsymbol{(x,y) \to (x, y - 10)}\)
• Reflect over \(y\)-axis: \(\boldsymbol{(x,y) \to (-x, y)}\)
• Translate right 10, down 9: \(\boldsymbol{(x,y) \to (x + 10, y - 9)}\)
• 90° CW rotation: \(\boldsymbol{(x,y) \to (y, -x)}\)
• Reflect over origin: \(\boldsymbol{(x,y) \to (-x, -y)}\)
• Reflect over \(x\)-axis: \(\boldsymbol{(x,y) \to (x, -y)}\)
• Vertical up 10: \(\boldsymbol{(x,y) \to (x, y + 10)}\)
• 90° CCW rotation: \(\boldsymbol{(x,y) \to (-y, x)}\)
• Translate left 10, up 9: \(\boldsymbol{(x,y) \to (x - 10, y + 9)}\)