y = x^2 - 2Which table gives three values of x and their corresponding values of y for the given...

1. TRANSLATE the problem requirements

• Given information:
  • Equation: \(\mathrm{y = x^2 - 2}\)
  • Need to find which table has correct x and y pairs
• What this means: I must substitute each x-value from the tables into the equation and check if the resulting y-value matches what's shown in that table

2. INFER the systematic approach

• Strategy: Test each table by substituting all three x-values
• Key insight: All three pairs in a table must be correct for that table to be the answer
• Start with any table and work methodically

3. SIMPLIFY by testing Option B first

• For \(\mathrm{x = 1}\):
  \(\mathrm{y = (1)^2 - 2}\)
  \(\mathrm{= 1 - 2}\)
  \(\mathrm{= -1}\) ✓ (matches table)
• For \(\mathrm{x = 2}\):
  \(\mathrm{y = (2)^2 - 2}\)
  \(\mathrm{= 4 - 2}\)
  \(\mathrm{= 2}\) ✓ (matches table)
• For \(\mathrm{x = 3}\):
  \(\mathrm{y = (3)^2 - 2}\)
  \(\mathrm{= 9 - 2}\)
  \(\mathrm{= 7}\) ✓ (matches table)

4. INFER that Option B is likely correct

• All three pairs work perfectly
• But let me verify one other option to be thorough

5. SIMPLIFY by testing Option A as verification

• For \(\mathrm{x = 1}\):
  \(\mathrm{y = 1 - 2}\)
  \(\mathrm{= -1}\), but table shows 0 ✗
• Since the first pair doesn't work, this table is incorrect

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors, especially with negative numbers. They might calculate \(\mathrm{1 - 2 = 1}\) instead of \(\mathrm{1 - 2 = -1}\), or forget that squaring comes before subtraction in the order of operations.

This type of calculation error would make the correct table (B) appear wrong and might lead them to select Choice A or cause confusion that leads to guessing.

Second Most Common Error:

Incomplete INFER reasoning: Students check only one or two x-values instead of verifying all three pairs in each table. They might find that \(\mathrm{x = 2}\) gives \(\mathrm{y = 2}\) in both Options A and B, assume either could be right, and guess randomly.

This incomplete verification process may lead them to select Choice A (since it matches the \(\mathrm{x = 2}\) pair) or causes them to get stuck and guess.

The Bottom Line:

This problem tests careful arithmetic with negative results and systematic verification. Students who rush through calculations or don't check all three pairs in each table are most likely to make errors.