The function \(\mathrm{P(x) = -\frac{1}{3}(x - 8)^2 + 75}\) represents a company's weekly profit \(\mathrm{P(x)}\), in hundreds of dollars, when...

1. TRANSLATE the function into vertex form components

• Given: \(\mathrm{P(x) = -\frac{1}{3}(x - 8)^2 + 75}\)
• This matches vertex form: \(\mathrm{P(x) = a(x - h)^2 + k}\)
• TRANSLATE the components:
  • \(\mathrm{a = -\frac{1}{3}}\) (coefficient)
  • \(\mathrm{h = 8}\) (x-coordinate of vertex)
  • \(\mathrm{k = 75}\) (y-coordinate of vertex)
• Therefore: \(\mathrm{vertex = (8, 75)}\)

2. INFER what the vertex represents

• Since \(\mathrm{a = -\frac{1}{3} \lt 0}\), the parabola opens downward
• This means the vertex is the highest point (maximum)
• The vertex \(\mathrm{(8, 75)}\) represents the maximum profit point

3. TRANSLATE the vertex coordinates to real-world meaning

• \(\mathrm{x = 8}\) means 8 thousand units = 8,000 units produced
• \(\mathrm{P(x) = 75}\) means 75 hundreds of dollars = $7,500 profit
• So maximum profit occurs when producing 8,000 units, yielding $7,500

4. APPLY CONSTRAINTS to select the correct interpretation

• Looking at answer choices, we need to identify what the vertex tells us
• Since it's a maximum: "The company's maximum weekly profit was $7,500"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly identify the vertex as \(\mathrm{(8, 75)}\) but fail to properly convert the units. They might think:

• 8 means $8 (ignoring "thousand units")
• 75 means $75 (ignoring "hundreds of dollars")

This confusion about units may lead them to select Choice (B) ($800) if they somehow mix up the coordinate values, or causes them to get stuck and guess.

Second Most Common Error:

Missing INFER reasoning about maximum vs. minimum: Students might identify the vertex but not recognize that \(\mathrm{a \lt 0}\) means it represents a maximum. Without this understanding, they can't distinguish between interpretations about "maximum profit" versus other statements about the vertex.

This conceptual gap leads to confusion when evaluating answer choices and may result in guessing.

The Bottom Line:

This problem requires both technical translation skills (vertex form → coordinates → real units) and conceptual understanding (negative coefficient → maximum). Students often struggle with one or both of these requirements.