A quadratic function q satisfies \(\mathrm{q(-4) = 12}\), \(\mathrm{q(0) = 12}\), and \(\mathrm{q(1) = 10}\). Which equation defines q?

1. TRANSLATE the problem information

• Given conditions:
  • \(\mathrm{q(-4) = 12}\)
  • \(\mathrm{q(0) = 12}\)
  • \(\mathrm{q(1) = 10}\)
  • q is a quadratic function
• What this tells us: We need to find a quadratic function that passes through these three points.

2. INFER the approach

• Since q is quadratic, use general form: \(\mathrm{q(x) = ax^2 + bx + c}\)
• Three unknown parameters (a, b, c) require three equations
• Each given condition will provide one equation

3. TRANSLATE each condition into equations

• From \(\mathrm{q(0) = 12}\): Substitute \(\mathrm{x = 0}\)
  • \(\mathrm{a(0)^2 + b(0) + c = 12}\)
  • \(\mathrm{c = 12}\)
• From \(\mathrm{q(-4) = 12}\): Substitute \(\mathrm{x = -4}\)
  • \(\mathrm{a(-4)^2 + b(-4) + c = 12}\)
  • \(\mathrm{16a - 4b + 12 = 12}\)
  • \(\mathrm{16a - 4b = 0}\)
• From \(\mathrm{q(1) = 10}\): Substitute \(\mathrm{x = 1}\)
  • \(\mathrm{a(1)^2 + b(1) + c = 10}\)
  • \(\mathrm{a + b + 12 = 10}\)
  • \(\mathrm{a + b = -2}\)

4. SIMPLIFY the system of equations

• From \(\mathrm{16a - 4b = 0}\): Divide by 4
  • \(\mathrm{4a - b = 0}\)
  • \(\mathrm{b = 4a}\)
• Substitute \(\mathrm{b = 4a}\) into \(\mathrm{a + b = -2}\):
  • \(\mathrm{a + 4a = -2}\)
  • \(\mathrm{5a = -2}\)
  • \(\mathrm{a = -\frac{2}{5}}\)
• Find b: \(\mathrm{b = 4(-\frac{2}{5}) = -\frac{8}{5}}\)

5. INFER the final answer

• We have: \(\mathrm{a = -\frac{2}{5}, b = -\frac{8}{5}, c = 12}\)
• Therefore: \(\mathrm{q(x) = -\frac{2}{5}x^2 - \frac{8}{5}x + 12}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when solving the system of equations, particularly with fraction arithmetic or sign errors.

For example, they might incorrectly solve \(\mathrm{5a = -2}\) as \(\mathrm{a = \frac{2}{5}}\) instead of \(\mathrm{a = -\frac{2}{5}}\), or calculate \(\mathrm{b = 4a}\) incorrectly with the wrong sign. These computational errors lead to selecting one of the incorrect answer choices.

This may lead them to select Choice B (\(\mathrm{q(x) = \frac{2}{5}x^2 - \frac{8}{5}x + 12}\)) or Choice C (\(\mathrm{q(x) = -\frac{2}{5}x^2 + \frac{8}{5}x + 12}\)).

Second Most Common Error:

Inadequate INFER reasoning: Students don't recognize that they need to set up a system of equations using the general quadratic form.

Instead, they might try to work backwards from the answer choices or attempt pattern recognition without systematic algebraic setup. This leads to confusion about which coefficients correspond to which terms, causing them to guess randomly among the choices.

The Bottom Line:

This problem tests whether students can systematically translate function conditions into algebraic equations and solve the resulting system accurately. The key insight is recognizing that three points uniquely determine a quadratic function.