A quadratic function models a company's daily revenue \(\mathrm{R(p)}\), in dollars, as a function of the price p, in dollars,...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{R(p)}\) models daily revenue as a quadratic function of price \(\mathrm{p}\)
  • \(\mathrm{R(0) = 275}\) (revenue is $275 when price is $0)
  • Maximum revenue is $500 when price is $5
  • Need to find revenue when price is $7
• What this tells us: We have a parabola with vertex at (5, 500)

2. INFER the mathematical approach

• Since we know the vertex (maximum point), use vertex form: \(\mathrm{R(p) = a(p - 5)^2 + 500}\)
• We need to find the value of parameter 'a' using the additional point \(\mathrm{R(0) = 275}\)
• Then we can calculate \(\mathrm{R(7)}\)

3. SIMPLIFY to find parameter 'a'

• Substitute the known point \(\mathrm{R(0) = 275}\):
  \(\mathrm{275 = a(0 - 5)^2 + 500}\)
  \(\mathrm{275 = a(-5)^2 + 500}\)
  \(\mathrm{275 = 25a + 500}\)
  \(\mathrm{25a = 275 - 500}\)
  \(\mathrm{25a = -225}\)
  \(\mathrm{a = -9}\)

4. SIMPLIFY to find R(7)

• Now that we know \(\mathrm{a = -9}\), our function is: \(\mathrm{R(p) = -9(p - 5)^2 + 500}\)
• Calculate \(\mathrm{R(7)}\):
  \(\mathrm{R(7) = -9(7 - 5)^2 + 500}\)
  \(\mathrm{R(7) = -9(2)^2 + 500}\)
  \(\mathrm{R(7) = -9(4) + 500}\)
  \(\mathrm{R(7) = -36 + 500}\)
  \(\mathrm{R(7) = 464}\)

Answer: 464

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to connect the business context "maximum revenue occurs at price $5" with the mathematical concept of vertex coordinates (5, 500). They may try to use standard form or not recognize that they should use vertex form at all.

This leads to complicated algebra or getting stuck completely, causing them to guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the vertex form but make arithmetic errors when solving \(\mathrm{25a = -225}\) (getting \(\mathrm{a = 9}\) instead of \(\mathrm{a = -9}\)) or when calculating \(\mathrm{(-9)(4) = -36}\).

With \(\mathrm{a = 9}\), they would get \(\mathrm{R(7) = 9(4) + 500 = 536}\), which isn't among the choices, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students can bridge business language with quadratic function properties. The key insight is recognizing that "maximum revenue of 500 at price 5" directly gives you the vertex (5, 500) for vertex form.