A quadratic function \(\mathrm{h(x)}\) can be written in the form \(\mathrm{h(x) = a(x - 2)(x - 8)}\) where a is...

1. TRANSLATE the factored form to find zeros

• Given: \(\mathrm{h(x) = a(x - 2)(x - 8)}\)
• To find zeros, set each factor equal to zero:
  • \(\mathrm{x - 2 = 0}\) → \(\mathrm{x = 2}\)
  • \(\mathrm{x - 8 = 0}\) → \(\mathrm{x = 8}\)
• The function has zeros at \(\mathrm{x = 2}\) and \(\mathrm{x = 8}\)

2. INFER the location of the vertex

• Key insight: For any quadratic function, the vertex lies exactly halfway between the two zeros
• This is because parabolas are symmetric about their vertex
• The x-coordinate of the vertex equals the midpoint of the zeros

3. SIMPLIFY to find the midpoint

• Use the midpoint formula: \(\mathrm{(x₁ + x₂)/2}\)
• x-coordinate of vertex = \(\mathrm{(2 + 8)/2 = 10/2 = 5}\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Incorrectly determining the zeros from the factored form

Students see \(\mathrm{h(x) = a(x - 2)(x - 8)}\) and think the zeros are at \(\mathrm{x = -2}\) and \(\mathrm{x = -8}\), essentially ignoring the subtraction signs within the factors. They reason: "If \(\mathrm{x - 2}\) is a factor, then \(\mathrm{x = -2}\) makes it zero."

This leads them to calculate the midpoint as \(\mathrm{(-2 + -8)/2 = -10/2 = -5}\), giving an incorrect x-coordinate for the vertex.

Second Most Common Error:

Poor INFER reasoning: Not connecting the vertex location to the zeros

Some students recognize the zeros correctly but don't realize that the vertex x-coordinate is simply the midpoint between them. Instead, they might try to use the given y-coordinate of 18 to somehow calculate the x-coordinate, or they might attempt to expand the factored form and use the vertex formula \(\mathrm{x = -b/(2a)}\), leading to unnecessary complexity and potential algebraic errors.

This causes them to get stuck and abandon a systematic approach.

The Bottom Line:

This problem tests whether students truly understand what the factored form tells them about a quadratic function. The key insight is recognizing that once you have the zeros, finding the vertex x-coordinate becomes a simple midpoint calculation - no need for complex algebra or the vertex formula.