Question:The functions f and g are defined by the given equations, where x geq 0. Which of the following equations...

1. TRANSLATE the problem requirements

• Given information:
  • Function I: \(\mathrm{f(x) = -2(x - 3)^2 + 8}\) (vertex form)
  • Function II: \(\mathrm{g(x) = -2x^2 + 12x + 10}\) (standard form)
  • Domain: \(\mathrm{x \geq 0}\) for both functions
• What we need to find: Which equation displays its maximum value as a constant or coefficient

2. INFER the approach strategy

• Key insight: "Displays" means the maximum value appears directly in the equation - we shouldn't have to calculate or derive it
• Both functions have negative leading coefficients (-2), so both open downward and have maximum values
• The question isn't asking which HAS a maximum, but which SHOWS the maximum directly

3. INFER analysis of Function I

• \(\mathrm{f(x) = -2(x - 3)^2 + 8}\) is in vertex form: \(\mathrm{a(x - h)^2 + k}\)
• The vertex is at \(\mathrm{(3, 8)}\), making 8 the maximum value
• Since \(\mathrm{x = 3}\) is in our domain (\(\mathrm{x \geq 0}\)), this maximum is achievable
• The maximum value 8 appears directly as the constant term

4. SIMPLIFY Function II to find its maximum

• \(\mathrm{g(x) = -2x^2 + 12x + 10}\) is in standard form - maximum isn't obvious
• Complete the square:
  • \(\mathrm{g(x) = -2(x^2 - 6x) + 10}\)
  • \(\mathrm{g(x) = -2(x^2 - 6x + 9 - 9) + 10}\)
  • \(\mathrm{g(x) = -2(x - 3)^2 + 18 + 10}\)
  • \(\mathrm{g(x) = -2(x - 3)^2 + 28}\)

5. INFER the final comparison

• Function II has maximum value 28 at \(\mathrm{x = 3}\) (within our domain)
• However, this maximum value 28 does NOT appear in the original equation \(\mathrm{g(x) = -2x^2 + 12x + 10}\)
• We had to derive it through completing the square

Answer: A (I only)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on whether functions HAVE maximum values rather than whether they DISPLAY them directly in the equation.

They correctly identify that both functions open downward and have maximum values, but miss the crucial distinction that the question asks which equations show the maximum as a visible constant or coefficient. They might think "both have maximums, so the answer should be C" without recognizing that Function II requires calculation to reveal its maximum.

This leads them to select Choice C (I and II).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt to find maximums but make algebraic errors while completing the square for Function II.

Common mistakes include sign errors or arithmetic mistakes during the completing-the-square process, leading them to calculate an incorrect maximum value. When their derived maximum doesn't match any constants in the original equation, they might incorrectly conclude that neither function displays its maximum.

This may lead them to select Choice D (Neither I nor II).

The Bottom Line:

The key challenge is distinguishing between mathematical existence and explicit display - just because a function has a maximum doesn't mean the equation shows it directly. Students must recognize that vertex form immediately reveals the maximum while standard form conceals it.