Question: What is the vertex of the parabola defined by the equation \(\mathrm{y = -2(x - 4)^2 + 3}\)? \(\mathrm{(-4,...

1. TRANSLATE the equation format

• Given equation: \(\mathrm{y = -2(x - 4)^2 + 3}\)
• This matches vertex form: \(\mathrm{y = a(x - h)^2 + k}\)
• We need to identify the values of a, h, and k

2. INFER the parameter values

• Comparing \(\mathrm{y = -2(x - 4)^2 + 3}\) to \(\mathrm{y = a(x - h)^2 + k}\):
  • \(\mathrm{a = -2}\) (coefficient of the squared term)
  • \(\mathrm{h = 4}\) (from x - 4, so h = 4)
  • \(\mathrm{k = 3}\) (constant term added at the end)
• The vertex is at coordinates \(\mathrm{(h, k)}\)

3. TRANSLATE to final coordinates

• \(\mathrm{Vertex = (h, k) = (4, 3)}\)

Answer: D (4, 3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students see \(\mathrm{(x - 4)}\) and incorrectly conclude that \(\mathrm{h = -4}\) instead of \(\mathrm{h = 4}\).

They think: "The expression is \(\mathrm{(x - 4)}\), so the h-value must be -4." This sign confusion happens because they don't carefully compare to the standard form \(\mathrm{y = a(x - h)^2}\), where the h represents the value that makes the parentheses equal zero.

This leads them to select Choice A (-4, 3).

Second Most Common Error:

Poor INFER reasoning: Students recognize the vertex form but mix up which value corresponds to the x-coordinate versus y-coordinate.

They correctly identify \(\mathrm{h = 4}\) and \(\mathrm{k = 3}\), but then write the vertex as (3, 4) instead of (4, 3), confusing the order of coordinates in an ordered pair.

This leads them to confusion since (3, 4) isn't among the choices, causing them to guess.

The Bottom Line:

This problem tests your ability to match patterns carefully and pay attention to signs. The vertex form gives you the answer directly - no complex calculations needed - but you must be precise about extracting the h and k values correctly.