A region in the xy-plane is defined by the inequality y lt -x^2 + 4x + 1. Which of the...

1. TRANSLATE the problem information

• Given information:
  • Region defined by: \(\mathrm{y \lt -x^2 + 4x + 1}\)
  • Four points to test: (1,3), (2,4), (3,1), (4,2)
  • Need to find which point is NOT in the region

• What this tells us: A point is in the region if its coordinates satisfy the inequality. Since we want the point NOT in the region, we're looking for the point that makes the inequality false.

2. INFER the approach

• Strategy: Test each point by substituting its coordinates \(\mathrm{(x,y)}\) into the inequality
• For each point, evaluate whether \(\mathrm{y \lt -x^2 + 4x + 1}\) is true or false
• The point that makes the inequality false is our answer

3. SIMPLIFY each test systematically

Testing (1, 3):

• Substitute: Is \(\mathrm{3 \lt -(1)^2 + 4(1) + 1}\)?
• SIMPLIFY: \(\mathrm{3 \lt -1 + 4 + 1}\)
  \(\mathrm{= 3 \lt 4}\) ✓ TRUE
• Point (1, 3) is in the region

Testing (2, 4):

• Substitute: Is \(\mathrm{4 \lt -(2)^2 + 4(2) + 1}\)?
• SIMPLIFY: \(\mathrm{4 \lt -4 + 8 + 1}\)
  \(\mathrm{= 4 \lt 5}\) ✓ TRUE
• Point (2, 4) is in the region

Testing (3, 1):

• Substitute: Is \(\mathrm{1 \lt -(3)^2 + 4(3) + 1}\)?
• SIMPLIFY: \(\mathrm{1 \lt -9 + 12 + 1}\)
  \(\mathrm{= 1 \lt 4}\) ✓ TRUE
• Point (3, 1) is in the region

Testing (4, 2):

• Substitute: Is \(\mathrm{2 \lt -(4)^2 + 4(4) + 1}\)?
• SIMPLIFY: \(\mathrm{2 \lt -16 + 16 + 1}\)
  \(\mathrm{= 2 \lt 1}\) ✗ FALSE
• Point (4, 2) is NOT in the region

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misunderstand what "NOT in the region" means and look for points that satisfy the inequality instead of points that don't satisfy it.

They correctly evaluate all the inequalities but then select one of the points where the inequality is true (like Choice A, B, or C), thinking these are the points "not in the region." This fundamental misreading of the question leads them to select Choice A (1, 3) or another incorrect answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating \(\mathrm{-x^2 + 4x + 1}\), especially with the negative sign in front of \(\mathrm{x^2}\).

For example, with point (4, 2), they might calculate \(\mathrm{-(4)^2}\) as \(\mathrm{-4^2 = -16}\), or forget the negative sign entirely, getting \(\mathrm{16 + 16 + 1 = 33}\) instead of 1. This leads to incorrect comparisons and potentially selecting the wrong answer or getting confused and guessing.

The Bottom Line:

This problem tests whether students can systematically substitute values and correctly interpret what "NOT in the region" means. The key insight is that inequality regions have boundary conditions, and testing points requires both careful arithmetic and clear understanding of the question's logic.