In the xy-plane, a parabola with the equation \(\mathrm{y = 2(x - 5)^2 + k}\), where k is a constant,...

1. TRANSLATE the problem information

• Given information:
  • Parabola equation: \(\mathrm{y = 2(x - 5)^2 + k}\)
  • The parabola passes through point \(\mathrm{(3, 11)}\)
• What "passes through" means: The coordinates \(\mathrm{(3, 11)}\) must satisfy the equation

2. TRANSLATE this condition into mathematics

• Since point \(\mathrm{(3, 11)}\) lies on the parabola, substitute \(\mathrm{x = 3}\) and \(\mathrm{y = 11}\) into the equation:
  \(\mathrm{11 = 2(3 - 5)^2 + k}\)

3. SIMPLIFY to solve for k

• Work inside the parentheses first: \(\mathrm{3 - 5 = -2}\)
  \(\mathrm{11 = 2(-2)^2 + k}\)
• Calculate the exponent: \(\mathrm{(-2)^2 = 4}\)
  \(\mathrm{11 = 2(4) + k}\)
• Multiply: \(\mathrm{2(4) = 8}\)
  \(\mathrm{11 = 8 + k}\)
• Solve for k: \(\mathrm{k = 11 - 8 = 3}\)

Answer: B) 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding that "passes through the point" means the coordinates must satisfy the equation.

Some students might try to work with the vertex form conceptually instead of using the specific point given. They might attempt to identify the vertex as \(\mathrm{(5, k)}\) and try to use that information, missing that they need to use the point \(\mathrm{(3, 11)}\) directly. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors or arithmetic mistakes in the calculation steps.

The most common mistake is calculating \(\mathrm{(-2)^2}\) incorrectly as -4 instead of +4. This would give:
\(\mathrm{11 = 2(-4) + k}\)
\(\mathrm{11 = -8 + k}\)
\(\mathrm{k = 19}\)

This may lead them to select Choice E (19).

The Bottom Line:

This problem tests whether students understand the fundamental relationship between points and equations - that coordinates of points on a curve must satisfy the curve's equation. Success depends on proper translation of this geometric concept into algebraic substitution.