In the xy-plane, point P is translated 4 units to the right and 3 units downward to create point P'....

1. TRANSLATE the problem information

• Given information:
  • Point P is translated 4 units right and 3 units down
  • The resulting point P' has coordinates \(\mathrm{(1, -5)}\)
  • Need to find original coordinates of P

• What this tells us: If P starts at \(\mathrm{(x, y)}\), then P' is at \(\mathrm{(x + 4, y - 3)}\)

2. INFER the approach

• Since we know where P ended up after translation, we need to work backwards
• Set up equations using the translation relationship: \(\mathrm{P' = (x + 4, y - 3) = (1, -5)}\)

3. SIMPLIFY to find each coordinate

• For x-coordinate: \(\mathrm{x + 4 = 1}\)
  Subtract 4 from both sides:
  \(\mathrm{x = 1 - 4 = -3}\)

• For y-coordinate: \(\mathrm{y - 3 = -5}\)
  Add 3 to both sides:
  \(\mathrm{y = -5 + 3 = -2}\)

4. State the final coordinates

Answer: B \(\mathrm{(-3, -2)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse the direction of translation when working backwards. They might think "if P' moved 4 units right from P, then P is 4 units right of P'" and incorrectly calculate P as \(\mathrm{(1 + 4, -5 + 3) = (5, -2)}\).

This leads them to select Choice C \(\mathrm{(5, -2)}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x + 4 = 1}\) and \(\mathrm{y - 3 = -5}\), but make sign errors when isolating variables. For example, they might calculate \(\mathrm{y = -5 - 3 = -8}\) instead of \(\mathrm{y = -5 + 3 = -2}\).

This may lead them to select Choice A \(\mathrm{(-3, -8)}\) or Choice D \(\mathrm{(5, -8)}\) depending on which coordinate they mess up.

The Bottom Line:

Translation problems require careful attention to direction - the key insight is that to find where something came FROM, you reverse the translation direction. Moving right means the original was to the left, and moving down means the original was above.