Question

choose the algebraic representation (notation) for translating trapezoid abcd to create trapezoid abcd:
$(x,y) \to (y + 3, x - 5)$
$(x,y) \to (x - 5, y + 3)$
$(x,y) \to (-5x, 3y)$
$(x,y) \to (x + 5, y - 3)$

Explanation:

To solve this, we analyze each transformation:

Step 1: Analyze Option 1: \((x,y) \to (y + 3, x - 5)\)

This swaps \(x\) and \(y\) (with adjustments), which is a rotation or reflection, not a translation. Translations do not swap coordinates, so this is incorrect.

Step 2: Analyze Option 2: \((x,y) \to (x - 5, y + 3)\)

A translation in the coordinate plane is a rigid transformation where we add/subtract constants to \(x\) (horizontal shift) and \(y\) (vertical shift). Here, \(x\) is decreased by 5 (shift left 5 units) and \(y\) is increased by 3 (shift up 3 units) – this matches the definition of a translation.

Step 3: Analyze Option 3: \((x,y) \to (-5x, 3y)\)

This scales \(x\) by \(-5\) and \(y\) by \(3\), which is a dilation (non - rigid, changes size), not a translation. So this is incorrect.

Step 4: Analyze Option 4: \((x,y) \to (x + 5, y - 3)\)

This shifts right 5 and down 3, which is the opposite of the direction implied by the trapezoid’s translation (we need left 5 and up 3, not right 5 and down 3). So this is incorrect.

Answer:

\((x,y) \to (x - 5, y + 3)\) (the second option: \((x,y) \to (x - 5, y + 3)\))