Question:The function g is defined by \(\mathrm{g(y) = y^2 - 12y + 11}\). What is the value of \(\mathrm{g(-2)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(y) = y^2 - 12y + 11}\)
  • Need to find: \(\mathrm{g(-2)}\)
• This means substitute \(\mathrm{y = -2}\) into the function definition

2. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{y = -2}\): \(\mathrm{g(-2) = (-2)^2 - 12(-2) + 11}\)
• Evaluate each part using order of operations:
  • \(\mathrm{(-2)^2 = 4}\) (negative squared gives positive)
  • \(\mathrm{-12(-2) = 24}\) (negative times negative gives positive)
  • Keep the constant: \(\mathrm{+11}\)
• Rewrite: \(\mathrm{g(-2) = 4 + 24 + 11}\)

3. SIMPLIFY the final arithmetic

• Add from left to right: \(\mathrm{4 + 24 = 28}\)
• Final addition: \(\mathrm{28 + 11 = 39}\)

Answer: 39

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative numbers: Students incorrectly evaluate \(\mathrm{(-2)^2 = -4}\) instead of \(\mathrm{+4}\), forgetting that squaring a negative number gives a positive result.

This leads to \(\mathrm{g(-2) = -4 + 24 + 11 = 31}\), causing them to select an incorrect answer or become confused when 31 doesn't match any expected result.

Second Most Common Error:

Poor SIMPLIFY reasoning with sign operations: Students incorrectly calculate \(\mathrm{-12(-2) = -24}\) instead of \(\mathrm{+24}\), missing that multiplying two negatives gives a positive.

This leads to \(\mathrm{g(-2) = 4 + (-24) + 11 = -9}\), resulting in confusion since function values this negative seem unexpected for this context.

The Bottom Line:

This problem tests careful handling of negative number operations within function evaluation. Success requires systematic application of order of operations while maintaining attention to positive/negative sign rules at each step.