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
  • Need \(\mathrm{a + b + c}\) where \(\mathrm{y = ax^2 + bx + c}\)

2. INFER the key constraint

• 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 \(\mathrm{(a \gt 0)}\)

3. SIMPLIFY by converting 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. SIMPLIFY the target expression

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

5. APPLY CONSTRAINTS to eliminate impossible choices

• Since \(\mathrm{a \gt 0}\), we need \(\mathrm{64a - 14}\) where a is positive
• Check each choice by solving \(\mathrm{64a - 14 = }\)[choice value]:
  • Choice A: \(\mathrm{a = -9/64}\) (negative) ❌
  • Choice B: \(\mathrm{a = -5/64}\) (negative) ❌
  • Choice C: \(\mathrm{a = 0}\) (not a parabola) ❌
  • Choice D: \(\mathrm{a = 1/32}\) (positive) ✓

Answer: D. -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that the vertex position and x-intercept information constrains the parabola orientation. They might correctly derive \(\mathrm{a + b + c = 64a - 14}\) but then accept any answer choice without checking whether the resulting value of a makes geometric sense. Since choices A and B both give negative values of a, the parabola would open downward, meaning it couldn't intersect the x-axis at two points when the vertex is already below the x-axis.

This may lead them to select Choice A (-23) or Choice B (-19) without checking the constraint.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding the vertex form or collecting like terms. Common mistakes include sign errors when expanding \(\mathrm{(x - 9)^2}\) or arithmetic errors when computing coefficients. These calculation errors lead to an incorrect expression for \(\mathrm{a + b + c}\), making it impossible to identify the correct constraint relationship.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires both algebraic fluency and geometric reasoning. Students must not only convert between parabola forms accurately but also understand how the vertex position and x-intercept behavior constrain the parabola's orientation.