The table shows three values of x and their corresponding values of y for a quadratic function. Which equation represents...

1. TRANSLATE the problem information

• Given information:
  • Table showing three coordinate points: \((1,1), (2,2), (3,5)\)
  • Four possible quadratic equations as answer choices
  • Need to find which equation produces these exact y-values for the given x-values

2. INFER the most efficient approach

• Since we have multiple choice answers, testing each option by substitution will be faster than solving a system of equations
• For each equation, substitute \(\mathrm{x = 1, 2, and\ 3}\), then check if we get \(\mathrm{y = 1, 2, and\ 5}\) respectively
• The correct equation must work for ALL three points

3. SIMPLIFY by testing Option A: \(\mathrm{y = x^2 - 2x + 2}\)

• When \(\mathrm{x = 1}\):
  \(\mathrm{y = (1)^2 - 2(1) + 2}\)
  \(\mathrm{= 1 - 2 + 2}\)
  \(\mathrm{= 1}\) ✓
• When \(\mathrm{x = 2}\):
  \(\mathrm{y = (2)^2 - 2(2) + 2}\)
  \(\mathrm{= 4 - 4 + 2}\)
  \(\mathrm{= 2}\) ✓
• When \(\mathrm{x = 3}\):
  \(\mathrm{y = (3)^2 - 2(3) + 2}\)
  \(\mathrm{= 9 - 6 + 2}\)
  \(\mathrm{= 5}\) ✓

All three points work! But let's verify by checking that the other options don't work.

4. SIMPLIFY by testing the remaining options

• Option B: \(\mathrm{y = x^2 + 2x + 2}\)
  • When \(\mathrm{x = 1}\): \(\mathrm{y = 1 + 2 + 2 = 5}\) ✗ (should be 1)
• Option C: \(\mathrm{y = x^2 - 2x - 2}\)
  • When \(\mathrm{x = 1}\): \(\mathrm{y = 1 - 2 - 2 = -3}\) ✗ (should be 1)
• Option D: \(\mathrm{y = 2x^2 - 2x + 2}\)
  • When \(\mathrm{x = 1}\): \(\mathrm{y = 2 - 2 + 2 = 2}\) ✗ (should be 1)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when substituting values, especially with negative signs and order of operations. For example, when evaluating \(\mathrm{x^2 - 2x + 2}\) for \(\mathrm{x = 3}\), they might calculate \(\mathrm{9 - 6 + 2}\) as \(\mathrm{9 - (6 + 2) = 1}\) instead of \(\mathrm{(9 - 6) + 2 = 5}\).

This computational error makes them think Option A doesn't work, leading them to select an incorrect choice or abandon the systematic checking process and guess.

Second Most Common Error:

Incomplete INFER reasoning: Students test only the first point \((1,1)\) with each equation and select the first option that works, without verifying all three points. Since multiple equations might work for just one point, they may incorrectly select Choice B, C, or D without completing the full verification process.

The Bottom Line:

This problem rewards systematic checking and careful arithmetic. The key insight is that ALL three points must satisfy the correct equation, so thorough verification of each option is essential for confidence in the answer.