In the xy-plane, the line y = -3 and the parabola \(\mathrm{y = (x + 4)^2 - 3}\) intersect at...

1. TRANSLATE the problem setup

• Given information:
  • Line: \(\mathrm{y = -3}\) (horizontal line)
  • Parabola: \(\mathrm{y = (x + 4)^2 - 3}\)
  • They intersect at exactly one point

• What this tells us: At the intersection point, both curves have the same y-value, so we set the equations equal

2. TRANSLATE into mathematical equation

Set the y-values equal:

\(\mathrm{-3 = (x + 4)^2 - 3}\)

3. SIMPLIFY through algebraic manipulation

• Add 3 to both sides:
  \(\mathrm{0 = (x + 4)^2}\)

• INFER the next step: Since \(\mathrm{(x + 4)^2 = 0}\), we take the square root
  \(\mathrm{0 = x + 4}\)

• Solve for x:
  \(\mathrm{x = -4}\)

4. Verify the solution

Substitute \(\mathrm{x = -4}\) back into the parabola:

\(\mathrm{y = (-4 + 4)^2 - 3}\)

\(\mathrm{= 0^2 - 3}\)

\(\mathrm{= -3}\) ✓

This matches the line \(\mathrm{y = -3}\).

Answer: C. -4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not realize that "intersect" means the y-values must be equal. Instead, they might try to graph both equations or use more complex methods, leading to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make sign errors when manipulating the equation, such as forgetting to add 3 to both sides correctly, or making errors when taking the square root. This could lead them to select Choice A (-7) or Choice B (-5) based on calculation mistakes.

The Bottom Line:

This problem tests whether students understand the fundamental concept that intersections occur where curves have equal outputs, combined with solid algebraic manipulation skills. The key insight is recognizing that the setup leads to a perfect square equal to zero.