Question:The equation above defines a parabola in the xy-plane. What are the coordinates of the vertex of the parabola?y =...

1. INFER the solution strategy

• Given: \(\mathrm{y = x^2 + 12x + 11}\) in standard form
• Need: vertex coordinates \(\mathrm{(h, k)}\)
• Strategy: Complete the square to get vertex form \(\mathrm{y = a(x - h)^2 + k}\)

2. SIMPLIFY by completing the square

• Take half the coefficient of x: \(\mathrm{12 \div 2 = 6}\)
• Square this result: \(\mathrm{6^2 = 36}\)
• Add and subtract 36 to maintain equality:
  \(\mathrm{y = x^2 + 12x + 36 - 36 + 11}\)

3. SIMPLIFY by factoring and combining

• Factor the perfect square: \(\mathrm{y = (x + 6)^2 - 36 + 11}\)
• Combine the constants: \(\mathrm{y = (x + 6)^2 - 25}\)

4. INFER the vertex coordinates from vertex form

• The equation \(\mathrm{y = (x + 6)^2 - 25}\) is in vertex form
• This means \(\mathrm{y = (x - (-6))^2 + (-25)}\)
• Therefore, the vertex is \(\mathrm{(-6, -25)}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when interpreting the vertex form. After correctly getting \(\mathrm{y = (x + 6)^2 - 25}\), they might think the vertex is \(\mathrm{(6, -25)}\) instead of \(\mathrm{(-6, -25)}\) because they forget that vertex form shows \(\mathrm{(x - h)}\), so \(\mathrm{(x + 6)}\) means \(\mathrm{h = -6}\).

This leads them to select Choice (D) \(\mathrm{(6, 25)}\) or get confused about signs.

Second Most Common Error:

Poor SIMPLIFY skills: Students make arithmetic errors while completing the square, such as incorrectly calculating \(\mathrm{6^2 = 36}\) or making errors when combining \(\mathrm{-36 + 11 = -25}\). These calculation mistakes produce wrong vertex coordinates.

This causes them to select incorrect answer choices or abandon the systematic approach and guess.

The Bottom Line:

Completing the square requires careful attention to signs and arithmetic. The key insight is recognizing that \(\mathrm{(x + 6)}\) in vertex form means the x-coordinate of the vertex is -6, not +6.