The function \(\mathrm{P(d) = -5(d - 40)^2 + 18000}\) models the estimated monthly profit, \(\mathrm{P(d)}\), in dollars, for a company...

1. TRANSLATE the function form

• Given: \(\mathrm{P(d) = -5(d - 40)^2 + 18000}\)
• This matches vertex form: \(\mathrm{y = a(x - h)^2 + k}\)
• What this tells us: We can directly identify the vertex coordinates

2. INFER the vertex location and type

• Comparing forms: \(\mathrm{a = -5}\), \(\mathrm{h = 40}\), \(\mathrm{k = 18000}\)
• Vertex is at \(\mathrm{(40, 18000)}\)
• Since \(\mathrm{a = -5}\) is negative, parabola opens downward
• Downward opening means vertex is a maximum point

3. TRANSLATE coordinates into context

• \(\mathrm{d\text{-coordinate } (40)}\) represents the price: $40
• \(\mathrm{P\text{-coordinate } (18000)}\) represents the profit: $18,000
• Combined meaning: maximum monthly profit of $18,000 occurs at price $40

4. INFER the correct answer choice

• We need the interpretation of the vertex
• Vertex represents maximum profit of $18,000
• This matches choice A exactly

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly identify the vertex as \(\mathrm{(40, 18000)}\) but confuse which coordinate represents which variable. They might think the maximum profit is $40 instead of recognizing that $40 is the optimal price, and $18,000 is the maximum profit.

This may lead them to select Choice B ($40 maximum profit).

Second Most Common Error:

Poor INFER reasoning: Students identify the vertex correctly but fail to determine whether it represents a maximum or minimum. They might assume all vertices represent minimums or simply guess, not realizing the negative leading coefficient indicates a maximum.

This may lead them to select Choice C (minimum profit of $18,000).

The Bottom Line:

This problem tests whether students can connect the abstract mathematical concept of a vertex with real-world meaning, requiring careful attention to which variable represents price versus profit.