The equations x = -4 and \(\mathrm{x = (y - 3)^2 + 2}\) are graphed in the xy-plane.How many points...

1. TRANSLATE the problem information

• Given equations:
  • \(\mathrm{x = -4}\) (vertical line)
  • \(\mathrm{x = (y - 3)^2 + 2}\) (horizontal parabola opening right)
• Find: Number of intersection points

2. INFER the domain constraint for the parabola

• Since \(\mathrm{(y - 3)^2 \geq 0}\) for any real value of y
• Therefore: \(\mathrm{x = (y - 3)^2 + 2 \geq 0 + 2 = 2}\)
• The parabola only exists for x-values \(\mathrm{\geq 2}\)

3. INFER whether intersection is possible

• The vertical line is at \(\mathrm{x = -4}\)
• The parabola exists only for \(\mathrm{x \geq 2}\)
• Since \(\mathrm{-4 \lt 2}\), there's no overlap between the graphs

4. Verify algebraically (optional but recommended)

• For intersection: \(\mathrm{-4 = (y - 3)^2 + 2}\)

SIMPLIFY: \(\mathrm{(y - 3)^2 = -6}\)

• Since squares cannot be negative, no real solutions exist

Answer: (A) Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve \(\mathrm{-4 = (y - 3)^2 + 2}\) algebraically without first recognizing the domain constraint. They get \(\mathrm{(y - 3)^2 = -6}\) but don't immediately recognize this has no real solutions, or they make computational errors trying to find complex solutions they think might count.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Insufficient TRANSLATE understanding: Students don't properly visualize what \(\mathrm{x = (y - 3)^2 + 2}\) represents as a horizontal parabola, instead thinking of it as a vertical parabola. This prevents them from using the domain analysis approach and forces them to rely solely on algebraic methods they might execute incorrectly.

This may lead them to select Choice (B) (Exactly one) thinking the parabola and line must intersect somewhere.

The Bottom Line:

This problem rewards students who recognize that understanding the domain of the parabola makes the solution immediate, without needing complex algebra. The key insight is that \(\mathrm{(y - 3)^2 \geq 0}\) creates a natural boundary that eliminates any possibility of intersection.