For the quadratic function f, the table shows three values of x and their corresponding values of \(\mathrm{f(x)}\). Function f...

1. TRANSLATE the table information into equations

• Given information:
  • \(\mathrm{f(x) = ax^2 + bx + c}\) (general quadratic form)
  • \(\mathrm{f(1) = -32}\), \(\mathrm{f(2) = 8}\), \(\mathrm{f(3) = 88}\)
• What this gives us: Three equations by substituting each x-value
  • f(1): \(\mathrm{a(1)^2 + b(1) + c = -32}\) → \(\mathrm{a + b + c = -32}\)
  • f(2): \(\mathrm{a(2)^2 + b(2) + c = 8}\) → \(\mathrm{4a + 2b + c = 8}\)
  • f(3): \(\mathrm{a(3)^2 + b(3) + c = 88}\) → \(\mathrm{9a + 3b + c = 88}\)

2. INFER the most efficient solution strategy

• We have 3 equations with 3 unknowns, but we only need to find \(\mathrm{a - b}\)
• Strategy: Use elimination to get rid of c and work with just a and b
• This avoids finding c entirely, making the solution faster

3. SIMPLIFY using elimination method

• Eliminate c by subtracting equations:
  • Equation (2) - Equation (1): \(\mathrm{(4a + 2b + c) - (a + b + c) = 8 - (-32)}\)
  • This gives us: \(\mathrm{3a + b = 40}\)
• Similarly: Equation (3) - Equation (2): \(\mathrm{(9a + 3b + c) - (4a + 2b + c) = 88 - 8}\)
  • This gives us: \(\mathrm{5a + b = 80}\)

4. SIMPLIFY the two-equation system

• Now we have:
  • \(\mathrm{3a + b = 40}\)
  • \(\mathrm{5a + b = 80}\)
• Subtract the first from the second: \(\mathrm{(5a + b) - (3a + b) = 80 - 40}\)
• This gives us: \(\mathrm{2a = 40}\), so \(\mathrm{a = 20}\)
• Substitute back: \(\mathrm{3(20) + b = 40}\) → \(\mathrm{b = -20}\)

5. Calculate the final answer

• \(\mathrm{a - b = 20 - (-20) = 20 + 20 = 40}\)

Answer: D (40)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often struggle to correctly set up the three equations from the function notation. They might confuse \(\mathrm{f(1) = -32}\) with "f times 1 equals negative 32" instead of understanding it means "when x = 1, the function value is -32."

This conceptual confusion leads them to set up incorrect equations like "\(\mathrm{f \times 1 = a + b + c = -32}\)" or miss the proper substitution entirely. Without correct initial equations, their entire solution becomes wrong, leading them to guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the initial system but make algebraic mistakes during the elimination process. Common errors include:

• Sign errors when subtracting equations (getting \(\mathrm{-3a - b = 40}\) instead of \(\mathrm{3a + b = 40}\))
• Arithmetic mistakes when calculating \(\mathrm{8 - (-32) = 40}\)
• Forgetting the negative sign when computing \(\mathrm{a - b = 20 - (-20)}\)

These calculation errors often lead them to select Choice A (-40) or Choice B (-20) instead of the correct positive result.

The Bottom Line:

This problem tests whether students can bridge the gap between function notation and algebraic manipulation. The key insight is that each table entry gives you one linear equation, and solving the resulting system efficiently requires strategic elimination rather than finding all three coefficients.