The function h is defined by \(\mathrm{h(x) = px^2 + qx + r}\), where p, q, and r are constants....

1. TRANSLATE the axis of symmetry information

• Given information:
  • \(\mathrm{h(x) = px^2 + qx + r}\) (quadratic function)
  • Axis of symmetry at \(\mathrm{x = -2}\)
  • \(\mathrm{p \gt 1}\) and p is an integer
  • Need to find possible value of \(\mathrm{q - p}\)
• What this tells us: We can use the axis of symmetry formula to create an equation

2. INFER the relationship between p and q

• The axis of symmetry formula for \(\mathrm{y = ax^2 + bx + c}\) is \(\mathrm{x = -b/(2a)}\)
• For our function \(\mathrm{h(x) = px^2 + qx + r}\), this becomes \(\mathrm{x = -q/(2p)}\)
• Since the axis is at \(\mathrm{x = -2}\), we have: \(\mathrm{-2 = -q/(2p)}\)

3. SIMPLIFY to find the relationship

• Starting with: \(\mathrm{-2 = -q/(2p)}\)
• Multiply both sides by \(\mathrm{-2p}\):
  \(\mathrm{(-2)(-2p) = (-q/(2p))(-2p)}\)
• This gives us:
  \(\mathrm{4p = q}\)

4. INFER what q - p equals

• We found \(\mathrm{q = 4p}\)
• Therefore:
  \(\mathrm{q - p = 4p - p = 3p}\)

5. APPLY CONSTRAINTS to find possible values

• Since \(\mathrm{p \gt 1}\) and p is an integer, p can be 2, 3, 4, 5...
• This means \(\mathrm{q - p = 3p}\) can be:
  • If \(\mathrm{p = 2}\):
    \(\mathrm{q - p = 3(2) = 6}\)
  • If \(\mathrm{p = 3}\):
    \(\mathrm{q - p = 3(3) = 9}\)
  • If \(\mathrm{p = 4}\):
    \(\mathrm{q - p = 3(4) = 12}\)
• Checking answer choices, only 9 is listed

Answer: D (9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the axis of symmetry formula, thinking it's \(\mathrm{x = -b/a}\) instead of \(\mathrm{x = -b/(2a)}\).

This leads them to set up \(\mathrm{-2 = -q/p}\) instead of \(\mathrm{-2 = -q/(2p)}\), giving them \(\mathrm{q = 2p}\) rather than \(\mathrm{q = 4p}\). Then \(\mathrm{q - p = 2p - p = p}\). With \(\mathrm{p \gt 1}\), they might calculate \(\mathrm{p = 2}\) gives \(\mathrm{q - p = 2}\), or \(\mathrm{p = 3}\) gives \(\mathrm{q - p = 3}\), leading them to select Choice (B) (3).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{-2 = -q/(2p)}\) but make an algebraic error when solving for q.

They might multiply by \(\mathrm{-2}\) instead of \(\mathrm{-2p}\), or make a sign error, potentially getting \(\mathrm{q = -4p}\) or other incorrect relationships. This creates confusion about what \(\mathrm{q - p}\) should equal, causing them to get stuck and guess randomly.

The Bottom Line:

This problem requires solid knowledge of the axis of symmetry formula and careful algebraic manipulation. The key insight is recognizing that the constraint \(\mathrm{p \gt 1}\) (integer) creates multiple possible values for \(\mathrm{q - p = 3p}\), but only one matches the given choices.