A parabola in the xy-plane has the equation \(\mathrm{y - 6 = (x - n)^2}\). Which of the following gives...

1. TRANSLATE the equation into standard vertex form

• Given equation: \(\mathrm{y - 6 = (x - n)²}\)
• Add 6 to both sides: \(\mathrm{y = (x - n)² + 6}\)
• This is now in vertex form: \(\mathrm{y = a(x - h)² + k}\)

2. INFER the vertex form parameters

• Comparing \(\mathrm{y = (x - n)² + 6}\) to \(\mathrm{y = a(x - h)² + k}\):
  • \(\mathrm{a = 1}\) (coefficient of the squared term)
  • \(\mathrm{h = n}\) (the value subtracted from x)
  • \(\mathrm{k = 6}\) (the constant term)

3. INFER the vertex and axis of symmetry

• From vertex form \(\mathrm{y = a(x - h)² + k}\):
  • Vertex is always at \(\mathrm{(h, k)}\)
  • Axis of symmetry is always \(\mathrm{x = h}\) (for vertical parabolas)
• Therefore:
  • Vertex: \(\mathrm{(n, 6)}\)
  • Axis of symmetry: \(\mathrm{x = n}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the order of coordinates in the vertex.

They correctly identify that the vertex involves n and 6, but mix up which coordinate is which. Since they see "6" prominently in the equation and "n" as the variable, they might think the vertex is \(\mathrm{(6, n)}\) rather than \(\mathrm{(n, 6)}\). They forget that in vertex form \(\mathrm{y = a(x - h)² + k}\), the vertex is specifically \(\mathrm{(h, k)}\) where h comes from the x-term and k is the y-shift.

This may lead them to select Choice B (\(\mathrm{(6, n)}\) and axis \(\mathrm{x = n}\)).

Second Most Common Error:

Conceptual confusion about axis of symmetry: Students confuse horizontal and vertical axes of symmetry.

They might correctly find the vertex as \(\mathrm{(n, 6)}\) but then think the axis of symmetry is \(\mathrm{y = 6}\) because they see the 6 coordinate and assume the axis passes through that value. They don't recognize that for upward/downward opening parabolas, the axis of symmetry is always vertical (x = something), not horizontal.

This may lead them to select Choice C (\(\mathrm{(n, 6)}\) and axis \(\mathrm{y = 6}\)).

The Bottom Line:

The key challenge is keeping track of which parameter controls which aspect of the parabola. Students need to remember that in \(\mathrm{y = a(x - h)² + k}\), the vertex is \(\mathrm{(h, k)}\) in that specific order, and the axis is always \(\mathrm{x = h}\) for vertical parabolas.