In the xy-plane, a parabola has vertex \((-5, 16)\) and intersects the x-axis at two distinct points. If the equation...

1. TRANSLATE the problem information

• Given information:
  • Vertex: \((-5, 16)\)
  • Parabola intersects x-axis at two distinct points
  • Equation form: \(\mathrm{y = ax^2 + bx + c}\)
  • Need to find possible value of \(\mathrm{a - b + c}\)

2. INFER what the expression \(\mathrm{a - b + c}\) represents

• The key insight: \(\mathrm{a - b + c}\) is actually the value of the function when \(\mathrm{x = -1}\)
• Here's why: \(\mathrm{y(-1) = a(-1)^2 + b(-1) + c}\)
  \(\mathrm{= a(1) - b + c}\)
  \(\mathrm{= a - b + c}\)
• This transforms our problem from abstract algebra to concrete function evaluation

3. TRANSLATE to vertex form and evaluate

• Using vertex form: \(\mathrm{y = a(x - h)^2 + k}\) with vertex \((-5, 16)\)
• This gives us: \(\mathrm{y = a(x + 5)^2 + 16}\)
• Now evaluate at \(\mathrm{x = -1}\): \(\mathrm{y(-1) = a(-1 + 5)^2 + 16}\)
  \(\mathrm{= a(4)^2 + 16}\)
  \(\mathrm{= 16a + 16}\)

4. INFER the constraint on parameter a

• The vertex \((-5, 16)\) is above the x-axis (positive y-coordinate)
• The parabola crosses the x-axis at two distinct points
• For this to happen, the parabola must open downward
• Therefore: \(\mathrm{a \lt 0}\)

5. APPLY CONSTRAINTS to eliminate answer choices

• Since \(\mathrm{a \lt 0}\), we have: \(\mathrm{16a + 16 \lt 0 + 16 = 16}\)
• So \(\mathrm{a - b + c \lt 16}\)
• Checking choices:
  • (A) \(\mathrm{7 \lt 16}\) ✓ Possible
  • (B) \(\mathrm{16 = 16}\) ✗ Not less than 16
  • (C) \(\mathrm{19 \gt 16}\) ✗ Too large
  • (D) \(\mathrm{24 \gt 16}\) ✗ Too large

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(\mathrm{a - b + c}\) represents \(\mathrm{y(-1)}\)

Students often see "\(\mathrm{a - b + c}\)" as just an abstract algebraic expression and try to work directly with the standard form \(\mathrm{y = ax^2 + bx + c}\). Without the key insight that this expression equals the function value at \(\mathrm{x = -1}\), they get stuck trying to manipulate the general form or attempt to find specific values of a, b, and c. This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about parabola orientation: Thinking that a positive vertex y-coordinate means \(\mathrm{a \gt 0}\)

Students might reason: "The vertex has \(\mathrm{y = 16 \gt 0}\), so the parabola opens upward, meaning \(\mathrm{a \gt 0}\)." This leads them to conclude that \(\mathrm{16a + 16 \gt 16}\), making choices (C) or (D) seem reasonable. This may lead them to select Choice C (19) or Choice D (24).

The Bottom Line:

This problem tests whether students can connect abstract algebraic expressions to concrete function evaluations, and whether they truly understand how parabola orientation relates to x-axis intersections rather than just vertex position.