A parabola in the xy-plane has equation y = x^2 - 4x + 1. Which of the following points does...

1. TRANSLATE the problem information

• Given information:
  • Parabola equation: \(\mathrm{y = x^2 - 4x + 1}\)
  • Four points to test: (0,0), (2,-4), (3,-3), (4,3)
  • Need to find which point does NOT lie below the parabola

• What this tells us: A point is below the parabola when its y-coordinate is less than the parabola's y-value at that same x-coordinate.

2. INFER the approach

• For each point (x, y), we need to:
  • Calculate the parabola value at that x-coordinate
  • Compare the point's y-coordinate to this parabola value
  • The point that has \(\mathrm{y \geq parabola\ value}\) is NOT below the curve

3. SIMPLIFY each evaluation systematically

Point A: (0, 0)

• Parabola value: \(\mathrm{y = 0^2 - 4(0) + 1}\)
• \(\mathrm{= 0 - 0 + 1}\)
• \(\mathrm{= 1}\)
• Comparison: Is \(\mathrm{0 \lt 1}\)? Yes → Point is below parabola

Point B: (2, -4)

• Parabola value: \(\mathrm{y = 2^2 - 4(2) + 1}\)
• \(\mathrm{= 4 - 8 + 1}\)
• \(\mathrm{= -3}\)
• Comparison: Is \(\mathrm{-4 \lt -3}\)? Yes → Point is below parabola

Point C: (3, -3)

• Parabola value: \(\mathrm{y = 3^2 - 4(3) + 1}\)
• \(\mathrm{= 9 - 12 + 1}\)
• \(\mathrm{= -2}\)
• Comparison: Is \(\mathrm{-3 \lt -2}\)? Yes → Point is below parabola

Point D: (4, 3)

• Parabola value: \(\mathrm{y = 4^2 - 4(4) + 1}\)
• \(\mathrm{= 16 - 16 + 1}\)
• \(\mathrm{= 1}\)
• Comparison: Is \(\mathrm{3 \lt 1}\)? No, \(\mathrm{3 \gt 1}\) → Point is NOT below parabola

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what "below the parabola" means and think they need to find points that ARE below the parabola instead of the one that is NOT below.

They correctly calculate all the parabola values and comparisons but then select one of the points that actually IS below the parabola (like point B or C) because they forgot the question asks for the exception.

This may lead them to select Choice B (-4) or Choice C (-3).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating the quadratic expression, particularly with the negative terms and order of operations.

For example, when calculating \(\mathrm{3^2 - 4(3) + 1}\), they might compute \(\mathrm{9 - 12 + 1}\) incorrectly as \(\mathrm{9 - 13 = -4}\) instead of \(\mathrm{9 - 12 + 1 = -2}\). This leads to wrong comparisons and incorrect conclusions about which points lie below the parabola.

This causes them to get stuck and guess among the choices.

The Bottom Line:

This problem tests both careful translation of geometric language into algebraic conditions and systematic algebraic evaluation, but the trickiest part is remembering that you're looking for the point that does NOT satisfy the condition.