Question:The functions f and p are defined by the given equations, where x is a real number.I. \(\mathrm{f(x) = 2(x-3)(x-9)}\)II....

1. TRANSLATE the problem requirements

• Given: Two quadratic functions in different forms
• Need to find: Which equation displays its minimum value as a constant or coefficient
• Key insight: We must find the actual minimum value of each function, then check if that value appears in the original equation

2. INFER the solution strategy

• Since both are quadratic functions with positive leading coefficients, they have minimum values
• Strategy: Find minimum value for each function → Check if it appears in original equation
• This requires finding vertex coordinates for each parabola

3. SIMPLIFY to find minimum of function I: \(\mathrm{f(x) = 2(x-3)(x-9)}\)

• In factored form, vertex x-coordinate is midpoint between roots
• x-coordinate: \(\mathrm{\frac{3 + 9}{2} = 6}\)
• Minimum value: \(\mathrm{f(6) = 2(6-3)(6-9) = 2(3)(-3) = -18}\)
• Constants/coefficients in original equation: 2, 3, 9
• Since \(\mathrm{-18 \neq}\) any of these values, equation I does NOT display its minimum

4. SIMPLIFY to find minimum of function II: \(\mathrm{p(x) = 3(x-5)^2 + 2x}\)

• First expand: \(\mathrm{p(x) = 3x^2 - 30x + 75 + 2x = 3x^2 - 28x + 75}\)
• Use vertex formula: \(\mathrm{x = \frac{-(-28)}{2 \cdot 3} = \frac{14}{3}}\)
• Minimum value: \(\mathrm{p(\frac{14}{3}) = 3(\frac{14}{3}-5)^2 + 2(\frac{14}{3}) = 3(-\frac{1}{3})^2 + \frac{28}{3} = \frac{29}{3}}\)
• Constants/coefficients in original equation: 3, 5, 2
• Since \(\mathrm{\frac{29}{3} \neq}\) any of these values, equation II does NOT display its minimum

5. INFER the final conclusion

• Neither function displays its minimum value as a constant or coefficient
• Answer: D

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misunderstand what "displays as a constant or coefficient" means and think it refers to any characteristic of the function rather than specifically the minimum value appearing in the equation.

They might reason: "Function I has roots at 3 and 9, and these appear in the equation, so it displays important values" or "Function II has its vertex form showing x = 5, so it displays the vertex information." This leads them to select Choice C (I and II) without actually computing the minimum values.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly identify that they need to find minimum values but make calculation errors, particularly with function II where the +2x term complicates the vertex form.

They might expand incorrectly or substitute wrong values, leading to incorrect minimum values that might coincidentally match constants in the equations. This causes confusion and may lead them to select Choice A (I only) or Choice B (II only).

The Bottom Line:

This problem tests whether students can distinguish between values that appear in a function's equation versus values that represent key characteristics (like minimum values) of the function. The key insight is that displaying a minimum value means that exact numerical value appears as a constant or coefficient in the given equation.