In the xy-plane, a parabola has vertex \((-8, 6)\) and intersects the y-axis at exactly two distinct points. If the...

1. TRANSLATE the problem information

• Given information:
  • Parabola has vertex \(\mathrm{(-8, 6)}\)
  • Equation form: \(\mathrm{x = ay^2 + by + c}\)
  • Intersects y-axis at exactly two distinct points
• What this tells us: This is a horizontal parabola (x as a function of y)

2. INFER the approach

• Since we have a horizontal parabola with known vertex, use vertex form first
• The phrase "intersects y-axis at exactly two distinct points" is key - this will create a constraint on our coefficients
• We'll need to expand to standard form, then use the y-axis intersection condition

3. TRANSLATE vertex form to standard form

• Vertex form for horizontal parabola: \(\mathrm{x = a(y - k)^2 + h}\) where \(\mathrm{(h, k)}\) is vertex
• With vertex \(\mathrm{(-8, 6)}\): \(\mathrm{x = a(y - 6)^2 + (-8) = a(y - 6)^2 - 8}\)

4. SIMPLIFY by expanding

• \(\mathrm{x = a(y^2 - 12y + 36) - 8}\)
• \(\mathrm{x = ay^2 - 12ay + 36a - 8}\)
• Comparing with \(\mathrm{x = ay^2 + by + c}\):
  • Coefficient of \(\mathrm{y^2}\): \(\mathrm{a = a}\)
  • Coefficient of y: \(\mathrm{b = -12a}\)
  • Constant term: \(\mathrm{c = 36a - 8}\)
• Therefore: \(\mathrm{a + b + c = a + (-12a) + (36a - 8) = 25a - 8}\)

5. INFER the constraint from y-axis intersection

• "Intersects y-axis at exactly two distinct points" means when \(\mathrm{x = 0}\), we get exactly two different y-values
• Setting \(\mathrm{x = 0}\): \(\mathrm{ay^2 + by + c = 0}\)
• For exactly two distinct real solutions, discriminant must be positive: \(\mathrm{\Delta \gt 0}\)

6. SIMPLIFY the discriminant condition

• \(\mathrm{\Delta = b^2 - 4ac = (-12a)^2 - 4a(36a - 8)}\)
• \(\mathrm{\Delta = 144a^2 - 144a^2 + 32a = 32a}\)
• For \(\mathrm{\Delta \gt 0}\): \(\mathrm{32a \gt 0}\), so \(\mathrm{a \gt 0}\)

7. APPLY CONSTRAINTS to eliminate answer choices

• Since \(\mathrm{a \gt 0}\): \(\mathrm{a + b + c = 25a - 8 \gt -8}\)
• Checking choices:
  • (A) \(\mathrm{-17 \lt -8}\) ✗
  • (B) \(\mathrm{-13 \lt -8}\) ✗
  • (C) \(\mathrm{-8 = -8}\) ✗ (we need \(\mathrm{\gt -8}\))
  • (D) \(\mathrm{-5 \gt -8}\) ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "intersects y-axis at exactly two distinct points" to the discriminant condition. They may correctly expand the vertex form and find \(\mathrm{a + b + c = 25a - 8}\), but then get stuck because they don't realize they need additional constraints to eliminate answer choices. Without the discriminant analysis, all they know is that \(\mathrm{a + b + c = 25a - 8}\) for some value of a, leaving them unable to distinguish between the choices. This leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students may not remember that for a horizontal parabola, the vertex form is \(\mathrm{x = a(y - k)^2 + h}\). Instead, they might try to use the vertical parabola form \(\mathrm{y = a(x - h)^2 + k}\), leading them down a completely wrong path. This conceptual confusion causes them to abandon systematic solution and guess.

The Bottom Line:

This problem requires connecting geometric language ("intersects y-axis at exactly two distinct points") to algebraic conditions (discriminant \(\mathrm{\gt 0}\)). Students who can't make this connection will struggle to find the constraint that makes the problem solvable.