The equation \(\mathrm{x^2 + (y - 1)^2 = 49}\) represents circle A. Circle B is obtained by shifting circle A...

1. TRANSLATE the circle equation into center-radius form

• Given equation: \(\mathrm{x^2 + (y - 1)^2 = 49}\)
• TRANSLATE this into standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\):
  • \(\mathrm{x^2 = (x - 0)^2}\), so \(\mathrm{h = 0}\)
  • \(\mathrm{(y - 1)^2}\), so \(\mathrm{k = 1}\)
  • \(\mathrm{r^2 = 49}\), so \(\mathrm{r = 7}\)
• Circle A has center \(\mathrm{(0, 1)}\) and radius \(\mathrm{7}\)

2. TRANSLATE the transformation instruction

• "Shifting circle A down 2 units" means:
  • Every point \(\mathrm{(x, y)}\) becomes \(\mathrm{(x, y - 2)}\)
  • The center \(\mathrm{(0, 1)}\) becomes \(\mathrm{(0, 1 - 2) = (0, -1)}\)
  • The radius stays the same: \(\mathrm{7}\)

3. INFER the new equation structure

• Circle B has center \(\mathrm{(0, -1)}\) and radius \(\mathrm{7}\)
• Using standard form: \(\mathrm{(x - 0)^2 + (y - (-1))^2 = 7^2}\)
• SIMPLIFY: \(\mathrm{x^2 + (y + 1)^2 = 49}\)

Answer: D. \(\mathrm{x^2 + (y + 1)^2 = 49}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "down 2 units" as adding 2 instead of subtracting 2, or they apply the transformation incorrectly to the equation format.

Some students see "down 2 units" and think the y-term should become \(\mathrm{(y - 1 - 2) = (y - 3)}\), leading to \(\mathrm{x^2 + (y - 3)^2 = 49}\). However, this isn't among the choices, so they get confused and may guess randomly.

Second Most Common Error:

Missing conceptual knowledge about transformations: Students don't realize that shifting affects only the center coordinates, not the radius or overall equation structure.

They might think shifting down means the entire equation changes dramatically, or they might confuse vertical and horizontal shifts. This leads to confusion when trying to match their work to the answer choices and typically results in guessing.

The Bottom Line:

Success on this problem requires clearly understanding that "shifting down 2 units" means subtracting 2 from the y-coordinate of every point, including the center. The key insight is recognizing that transformations are systematic operations on coordinates, not mysterious changes to the entire equation.