In the xy-plane, a parabola has vertex \(\mathrm{(9, -14)}\) and intersects the x-axis at two points. If the equation of...

1. TRANSLATE the problem information

• Given information:
  • Vertex: \(\mathrm{(9, -14)}\)
  • Parabola intersects x-axis at two points
  • Equation form: \(\mathrm{y = ax^2 + bx + c}\)

• What this tells us: We need to find \(\mathrm{a + b + c}\), and the intersection condition gives us constraints on the parabola's orientation.

2. INFER the parabola's orientation

• Since the vertex \(\mathrm{(9, -14)}\) is below the x-axis AND the parabola intersects the x-axis at two points, the parabola must open upward
• This means the coefficient a must be positive \(\mathrm{(a \gt 0)}\)
• Key insight: This constraint will help us eliminate wrong answer choices

3. SIMPLIFY from vertex form to standard form

• Start with vertex form: \(\mathrm{y = a(x - 9)^2 - 14}\)
• Expand the squared term: \(\mathrm{y = a(x^2 - 18x + 81) - 14}\)
• Distribute: \(\mathrm{y = ax^2 - 18ax + 81a - 14}\)
• Comparing to \(\mathrm{y = ax^2 + bx + c}\):
  • Coefficient of \(\mathrm{x^2}\): \(\mathrm{a = a}\)
  • Coefficient of x: \(\mathrm{b = -18a}\)
  • Constant term: \(\mathrm{c = 81a - 14}\)

4. Calculate \(\mathrm{a + b + c}\)

• \(\mathrm{a + b + c = a + (-18a) + (81a - 14)}\)
• \(\mathrm{a + b + c = 64a - 14}\)

5. APPLY CONSTRAINTS to test answer choices

• Since \(\mathrm{a \gt 0}\), test each choice:
• Choice A: \(\mathrm{64a - 14 = -23}\) → \(\mathrm{a = -9/64 \lt 0}\) ❌
• Choice B: \(\mathrm{64a - 14 = -19}\) → \(\mathrm{a = -5/64 \lt 0}\) ❌
• Choice C: \(\mathrm{64a - 14 = -14}\) → \(\mathrm{a = 0}\) ❌ (not a parabola)
• Choice D: \(\mathrm{64a - 14 = -12}\) → \(\mathrm{a = 1/32 \gt 0}\) ✓

Answer: D. -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that the parabola must open upward given the vertex location and intersection conditions. They might convert the forms correctly but then accept any answer choice without considering the constraint that \(\mathrm{a \gt 0}\). This leads them to potentially select Choice A (-23) or Choice B (-19) since the algebra seems to work out, not realizing these require negative values of a.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x - 9)^2}\) or when collecting like terms to find \(\mathrm{a + b + c}\). Common mistakes include forgetting to distribute the 'a' properly or sign errors when expanding. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires both solid algebraic manipulation and conceptual understanding of parabola behavior. The key insight is recognizing the physical constraint \(\mathrm{(a \gt 0)}\) that eliminates most answer choices, making it as much a reasoning problem as an algebra problem.