Question:The functions p and q are defined by the given equations. Which of the following equations displays, as a constant...

1. TRANSLATE the problem requirements

• Given information:
  • Two quadratic functions in standard form
  • Need to determine which displays its minimum value as a constant or coefficient

• What this tells us: We need to find each function's minimum value, then check if that number appears in the original equation.

2. INFER the solution approach

• Both functions are quadratics with positive leading coefficients, so they open upward and have minimum values at their vertices
• Strategy: Use vertex formula to find minimum value, then examine original equation for that number

3. SIMPLIFY to find minimum of function I: \(\mathrm{p(x) = x^2 - 6x + 13}\)

• Vertex x-coordinate: \(\mathrm{x = -b/(2a) = -(-6)/(2 \cdot 1) = 3}\)
• Minimum value:
  \(\mathrm{p(3) = (3)^2 - 6(3) + 13}\)
  \(\mathrm{= 9 - 18 + 13}\)
  \(\mathrm{= 4}\)

4. INFER whether minimum appears in function I

• Constants and coefficients in \(\mathrm{p(x) = x^2 - 6x + 13}\) are: 1, -6, 13
• The minimum value 4 does not match any of these numbers

5. SIMPLIFY to find minimum of function II: \(\mathrm{q(x) = 7x^2 - 14x + 14}\)

• Vertex x-coordinate: \(\mathrm{x = -b/(2a) = -(-14)/(2 \cdot 7) = 1}\)
• Minimum value:
  \(\mathrm{q(1) = 7(1)^2 - 14(1) + 14}\)
  \(\mathrm{= 7 - 14 + 14}\)
  \(\mathrm{= 7}\)

6. INFER whether minimum appears in function II

• Constants and coefficients in \(\mathrm{q(x) = 7x^2 - 14x + 14}\) are: 7, -14, 14
• The minimum value 7 matches the leading coefficient!

Answer: B (II only)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find the minimum values correctly but forget to check whether those values actually appear in the original equations as constants or coefficients.

They calculate the minimum values as 4 and 7, then immediately conclude both functions display their minimum values without examining the original equations. This leads to confusion about why the answer isn't C (both I and II), causing them to get stuck and guess.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students make arithmetic errors when computing the minimum values, particularly when substituting back into the original functions.

For example, they might incorrectly calculate \(\mathrm{q(1) = 7 - 14 + 14 = 21}\) instead of 7, then notice that 21 doesn't appear in the equation and conclude that neither function works. This may lead them to select Choice D (Neither I nor II).

The Bottom Line:

This problem tests whether students understand that finding minimum values is only half the battle - they must also verify that those values actually appear in the original equation structure.