A theater models the number of tickets sold, S, as a quadratic function of the ticket price p (in dollars)....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{S(p)}\) is quadratic (parabolic relationship between price and tickets sold)
  • \(\mathrm{S(8) = 300}\) tickets
  • Maximum sales: \(\mathrm{S(11) = 540}\) tickets
  • Need: Another price where \(\mathrm{S(p) = 300}\)

2. INFER the most efficient approach

• Since we have a quadratic with a known vertex (maximum), we can use the symmetry property
• Key insight: Parabolas are symmetric about their vertex
• Since the maximum occurs at \(\mathrm{p = 11}\), this is our axis of symmetry

3. APPLY symmetry reasoning

• The price \(\mathrm{p = 8}\) is 3 units to the left of the vertex: \(\mathrm{8 = 11 - 3}\)
• By symmetry, the same sales (300 tickets) occur 3 units to the right of the vertex
• Other price: \(\mathrm{p = 11 + 3 = 14}\)

Answer: D (14)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the symmetry property of quadratic functions and attempt to set up complex systems of equations or guess-and-check with the answer choices.

Without the symmetry insight, they may try to find the general quadratic form \(\mathrm{S(p) = ap^2 + bp + c}\) using three conditions, leading to unnecessary algebraic complexity. This often results in calculation errors or abandoning the systematic approach entirely, leading to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the vertex form equation but make errors when solving \(\mathrm{(p - 11)^2 = 9}\), either forgetting the ± when taking the square root or incorrectly calculating \(\mathrm{11 ± 3}\).

These calculation mistakes might lead them to select Choice B (11) by confusing the vertex location with the answer, or Choice A (10) by making arithmetic errors in the final step.

The Bottom Line:

This problem rewards recognizing that quadratic functions have elegant symmetry properties that can dramatically simplify the solution. Students who miss this insight often get bogged down in unnecessary algebraic manipulation.