x\(\mathrm{f(x)}\)-110014120For the quadratic function f, the table shows three values of x and their corresponding values of \(\mathrm{f(x)}\). Which...

1. TRANSLATE the table data into mathematical equations

• Given information from table:
  • When \(\mathrm{x = -1}\), \(\mathrm{f(x) = 10}\)
  • When \(\mathrm{x = 0}\), \(\mathrm{f(x) = 14}\)
  • When \(\mathrm{x = 1}\), \(\mathrm{f(x) = 20}\)
• Since f is quadratic, it has form \(\mathrm{f(x) = ax^2 + bx + c}\)
• Each table row gives us one equation when we substitute

2. INFER the most efficient approach

• Notice that \(\mathrm{x = 0}\) gives us the simplest equation
• When \(\mathrm{x = 0}\): \(\mathrm{f(0) = a(0)^2 + b(0) + c = c}\)
• So immediately: \(\mathrm{c = 14}\)

3. TRANSLATE the remaining points into equations

• For \(\mathrm{x = -1}\), \(\mathrm{f(-1) = 10}\):
  \(\mathrm{10 = a(-1)^2 + b(-1) + c}\)
  \(\mathrm{10 = a - b + c}\)
  \(\mathrm{10 = a - b + 14}\)
  \(\mathrm{-4 = a - b}\)
• For \(\mathrm{x = 1}\), \(\mathrm{f(1) = 20}\):
  \(\mathrm{20 = a(1)^2 + b(1) + c}\)
  \(\mathrm{20 = a + b + c}\)
  \(\mathrm{20 = a + b + 14}\)
  \(\mathrm{6 = a + b}\)

4. SIMPLIFY by solving the system of linear equations

• We have:
  • \(\mathrm{a - b = -4}\)
  • \(\mathrm{a + b = 6}\)
• Add the equations: \(\mathrm{(a - b) + (a + b) = -4 + 6}\)
  \(\mathrm{2a = 2}\)
  \(\mathrm{a = 1}\)
• Substitute back: \(\mathrm{1 + b = 6}\), so \(\mathrm{b = 5}\)

5. APPLY CONSTRAINTS to verify our solution

• Our function: \(\mathrm{f(x) = x^2 + 5x + 14}\)
• Check all three points:
  • \(\mathrm{f(-1) = 1 - 5 + 14 = 10}\) ✓
  • \(\mathrm{f(0) = 0 + 0 + 14 = 14}\) ✓
  • \(\mathrm{f(1) = 1 + 5 + 14 = 20}\) ✓

Answer: D. \(\mathrm{f(x) = x^2 + 5x + 14}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not systematically convert each table row into an equation, instead trying to guess-and-check with the answer choices or attempting to find patterns in the y-values without using the quadratic form.

Without systematic equation setup, they might notice that \(\mathrm{f(0) = 14}\) appears in all answer choices as the constant term, but then struggle to determine a and b values. This leads to random checking of answer choices or guessing between options that seem plausible.

This leads to confusion and guessing among the four choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the three equations but make algebraic errors when solving the 2×2 system. Common mistakes include sign errors when manipulating \(\mathrm{a - b = -4}\), or incorrectly adding/subtracting the system equations.

For example, they might solve incorrectly and get \(\mathrm{a = -1}\), \(\mathrm{b = 5}\), leading them to think \(\mathrm{f(x) = -x^2 + 5x + 14}\), which isn't among the choices, causing confusion.

This may lead them to select Choice A (\(\mathrm{f(x) = 3x^2 + 3x + 14}\)) if they abandon systematic solving and pick the choice that "looks different" from their failed attempt.

The Bottom Line:

This problem requires methodical equation setup and careful algebraic manipulation. Students who try shortcuts or make computational errors will struggle to distinguish between the four plausible-looking quadratic expressions.