Question: What is the maximum value of the function g defined by \(\mathrm{g(x) = -(x - 3)^2 + 5}\)?-5-335

1. TRANSLATE the function into vertex form components

• Given function: \(\mathrm{g(x) = -(x - 3)^2 + 5}\)
• This matches vertex form: \(\mathrm{g(x) = a(x - h)^2 + k}\)
• Identifying components:
  • \(\mathrm{a = -1}\)
  • \(\mathrm{h = 3}\)
  • \(\mathrm{k = 5}\)

2. INFER what the coefficient tells us about the parabola

• Since \(\mathrm{a = -1 \lt 0}\), the parabola opens downward
• Downward-opening parabolas have their highest point (maximum) at the vertex
• Upward-opening parabolas (\(\mathrm{a \gt 0}\)) have their lowest point (minimum) at the vertex

3. INFER the vertex location and maximum value

• The vertex is located at \(\mathrm{(h, k) = (3, 5)}\)
• Since this parabola opens downward, the vertex represents the maximum point
• Therefore, the maximum value of the function is \(\mathrm{k = 5}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see the negative coefficient \(\mathrm{a = -1}\) and incorrectly assume this means the maximum value must be negative, leading them to select Choice A (-5) or Choice B (-3).

The reasoning error: "Since there's a negative sign in front, the maximum must be negative." This completely misses that the negative coefficient affects the parabola's orientation, not the sign of the maximum value.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that \(\mathrm{h = 3}\) from the vertex form but confuse the x-coordinate of the vertex with the maximum value itself, leading them to select Choice C (3).

The reasoning error: "The vertex is at x = 3, so the maximum value is 3." This confuses the input value (x-coordinate) with the output value (function value).

The Bottom Line:

This problem tests whether students truly understand vertex form beyond just memorizing the formula. The key insight is connecting the sign of the coefficient 'a' to whether you're looking for a maximum or minimum, then correctly identifying that 'k' gives the actual maximum/minimum value, not 'h'.