In the xy-plane, a parabola has vertex \((-2, 12)\) and intersects the y-axis at \((0, 8)\). If the equation of...

1. TRANSLATE the problem information

• Given information:
  • Vertex: \((-2, 12)\)
  • Y-intercept: \((0, 8)\)
  • Need: \(\mathrm{a + b + c}\) where \(\mathrm{y = ax^2 + bx + c}\)

2. INFER the best approach

• Since we have vertex information, vertex form is most efficient
• Vertex form: \(\mathrm{y = a(x - h)^2 + k}\) where \((\mathrm{h}, \mathrm{k})\) is the vertex
• With vertex \((-2, 12)\): \(\mathrm{y = a(x - (-2))^2 + 12 = a(x + 2)^2 + 12}\)

3. TRANSLATE the y-intercept condition

• Y-intercept \((0, 8)\) means when \(\mathrm{x = 0}\), \(\mathrm{y = 8}\)
• Substitute into vertex form: \(\mathrm{8 = a(0 + 2)^2 + 12}\)

4. SIMPLIFY to find parameter a

• \(\mathrm{8 = a(2)^2 + 12}\)
• \(\mathrm{8 = 4a + 12}\)
• \(\mathrm{-4 = 4a}\)
• \(\mathrm{a = -1}\)

5. SIMPLIFY to expand to standard form

• \(\mathrm{y = -1(x + 2)^2 + 12}\)
• \(\mathrm{y = -(x^2 + 4x + 4) + 12}\)
• \(\mathrm{y = -x^2 - 4x - 4 + 12}\)
• \(\mathrm{y = -x^2 - 4x + 8}\)

6. INFER the final answer

• From \(\mathrm{y = -x^2 - 4x + 8}\): \(\mathrm{a = -1}\), \(\mathrm{b = -4}\), \(\mathrm{c = 8}\)
• Therefore: \(\mathrm{a + b + c = -1 + (-4) + 8 = 3}\)

Answer: B. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to set up a system of equations using multiple points instead of recognizing that vertex form is the direct path when given vertex information.

They might attempt to find three points and create a system:

• Use vertex \((-2, 12)\)
• Use y-intercept \((0, 8)\)
• Try to find a third point

This leads to unnecessary complexity and potential arithmetic errors, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up vertex form but make algebraic errors when expanding \(\mathrm{(x + 2)^2}\) or when distributing the negative sign.

Common mistakes:

• \(\mathrm{(x + 2)^2 = x^2 + 4x + 2}\) (forgetting the constant term is 4, not 2)
• Incorrect distribution: \(\mathrm{-(x^2 + 4x + 4) = -x^2 - 4x - 4}\) (missing the negative on the constant)

These errors lead to wrong coefficients and thus wrong values for \(\mathrm{a + b + c}\), potentially selecting Choice A (-1) or Choice D (12).

The Bottom Line:

This problem tests whether students can efficiently connect vertex form with standard form. The key insight is recognizing that vertex form provides a direct path to finding the parameter 'a', making the expansion to standard form straightforward.