When the quadratic function f is graphed in the xy-plane, where \(\mathrm{x = f(y)}\), its vertex is \(\mathrm{(4, 2)}\). One...

1. TRANSLATE the problem information

• Given information:
  • Quadratic function where \(\mathrm{x = f(y)}\) (this is key!)
  • Vertex at \(\mathrm{(4, 2)}\)
  • One y-intercept at \(\mathrm{(0, 9/2)}\)
  • Need to find the other y-intercept

2. INFER the parabola's orientation and symmetry

• Since \(\mathrm{x = f(y)}\) (not \(\mathrm{y = f(x)}\)), this parabola opens horizontally
• The vertex \(\mathrm{(4, 2)}\) tells us the axis of symmetry is the horizontal line \(\mathrm{y = 2}\)
• Y-intercepts occur where \(\mathrm{x = 0}\), and they should be symmetric about \(\mathrm{y = 2}\)

3. SIMPLIFY to find the distance from the axis of symmetry

• Given y-intercept: \(\mathrm{(0, 9/2)}\)
• Distance from axis: \(\mathrm{|9/2 - 2|}\) \(\mathrm{= |9/2 - 4/2|}\)
  \(\mathrm{= |5/2|}\)
  \(\mathrm{= 5/2}\)
• This point is \(\mathrm{5/2}\) units above the axis of symmetry

4. APPLY CONSTRAINTS using symmetry property

• By symmetry, the other y-intercept must be \(\mathrm{5/2}\) units below the axis
• Other y-coordinate: \(\mathrm{y = 2 - 5/2}\)
  \(\mathrm{= 4/2 - 5/2}\)
  \(\mathrm{= -1/2}\)
• Other y-intercept: \(\mathrm{(0, -1/2)}\)

Answer: B \(\mathrm{(0, -1/2)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students don't recognize that \(\mathrm{x = f(y)}\) represents a horizontal parabola. They treat it like a standard vertical parabola with \(\mathrm{y = f(x)}\), leading them to think about x-intercepts instead of y-intercepts or to misidentify the axis of symmetry as vertical rather than horizontal.

This confusion about the parabola's orientation leads to incorrect analysis and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly understand the horizontal orientation but make calculation errors when finding the distance from the axis of symmetry \(\mathrm{(9/2 - 2 = 5/2)}\) or when applying symmetry \(\mathrm{(2 - 5/2 = -1/2)}\).

This may lead them to select Choice A \(\mathrm{(-5/2)}\) if they incorrectly calculate \(\mathrm{2 - 9/2}\) instead of using the distance properly.

The Bottom Line:

The unusual notation \(\mathrm{x = f(y)}\) requires students to mentally rotate their typical understanding of parabolas, and then apply symmetry properties correctly - a challenging combination of conceptual flexibility and precise calculation.