A quadratic function is defined by \(\mathrm{f(x) = a(x - 5)^2 + k}\), where a and k are constants and...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = a(x - 5)^2 + k}\) (vertex form of quadratic)
  • One x-intercept at \(\mathrm{x = 1}\)
  • Second x-intercept at \(\mathrm{x = p}\) (unknown)
• What this tells us: We have a parabola in vertex form and need to find where it crosses the x-axis

2. INFER the key insight about parabola symmetry

• From vertex form \(\mathrm{f(x) = a(x - 5)^2 + k}\), the vertex is at \(\mathrm{x = 5}\)
• This means the axis of symmetry is the vertical line \(\mathrm{x = 5}\)
• Key insight: Parabolas are perfectly symmetric about their axis of symmetry
• Therefore, x-intercepts must be equidistant from the line \(\mathrm{x = 5}\)

3. SIMPLIFY using the symmetry property

• Distance from first x-intercept to axis of symmetry: \(\mathrm{|1 - 5| = 4}\) units
• Since the intercepts are symmetric, the second intercept must also be 4 units from \(\mathrm{x = 5}\)
• The second intercept is on the opposite side: \(\mathrm{p = 5 + 4 = 9}\)

4. Verify the answer

• First intercept: \(\mathrm{x = 1}\) is 4 units left of \(\mathrm{x = 5}\)
• Second intercept: \(\mathrm{x = 9}\) is 4 units right of \(\mathrm{x = 5}\)
• The midpoint is \(\mathrm{(1 + 9)/2 = 5}\), which confirms our axis of symmetry

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the symmetry property of parabolas or fail to connect the vertex form to the axis of symmetry.

They might try to solve algebraically by setting \(\mathrm{f(x) = 0}\) and working with the quadratic formula, but get confused because they don't know the values of a and k. Without recognizing that symmetry provides a much simpler path, they get stuck in complex algebra and end up guessing.

This leads to confusion and guessing among all answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the symmetry concept but make arithmetic errors in applying it.

Common mistakes include calculating the distance as \(\mathrm{|5 - 1| = 4}\) but then finding \(\mathrm{p = 5 - 4 = 1}\) (going the wrong direction) or miscalculating the distance entirely. Some students might confuse which point is the center of symmetry.

This may lead them to select Choice A (7) if they incorrectly calculate \(\mathrm{p = 5 + 2 = 7}\), or other incorrect values.

The Bottom Line:

This problem tests whether students can recognize that geometric properties (symmetry) often provide elegant shortcuts compared to algebraic manipulation. The key insight is seeing the vertex form as immediately revealing the axis of symmetry rather than diving into complex equation-solving.