In the xy-plane, the vertex of the parabola \(\mathrm{y = f(x)}\) is at the point \(\mathrm{(h, k)}\). Which of the...

1. TRANSLATE the problem information

• Given information:
  • Parabola equation: \(\mathrm{y = f(x)}\)
  • Vertex coordinates: \(\mathrm{(h, k)}\)
• What this tells us: The point \(\mathrm{(h, k)}\) lies on the parabola

2. INFER the fundamental relationship

• Since the vertex \(\mathrm{(h, k)}\) is ON the parabola \(\mathrm{y = f(x)}\), this point must satisfy the equation
• Key insight: Any point \(\mathrm{(x, y)}\) on the curve \(\mathrm{y = f(x)}\) means \(\mathrm{y = f(x)}\)
• For our vertex: when \(\mathrm{x = h}\), then \(\mathrm{y = k}\)

3. Apply function notation directly

• By definition of function notation: \(\mathrm{f(h)}\) = the y-value when \(\mathrm{x = h}\)
• Since the y-coordinate of the vertex is k: \(\mathrm{k = f(h)}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students see the letter k in both the vertex coordinates and the answer choices, leading them to think k relates to \(\mathrm{f(k)}\) rather than understanding that k represents the y-coordinate at \(\mathrm{x = h}\).

They might reason: "Since k is in the vertex, the answer should involve k" and incorrectly select Choice B (\(\mathrm{f(k)}\)).

Second Most Common Error:

Conceptual confusion about function notation: Students understand that \(\mathrm{(h, k)}\) is the vertex but don't make the connection between coordinate pairs and function notation. They might think the problem is asking about special function values and guess Choice C (\(\mathrm{f(0)}\)) thinking it relates to the y-intercept.

The Bottom Line:

This problem tests whether students truly understand that function notation \(\mathrm{f(x)}\) simply represents the y-coordinate when the x-coordinate is the input. The vertex relationship is just an application of this fundamental concept.