A projectile's height above ground is modeled by the function \(\mathrm{q(x) = (x - 2)(x - 18) + 7}\), where...

1. INFER the problem type and strategy

• Given: \(\mathrm{q(x) = (x - 2)(x - 18) + 7}\) represents projectile height
• Need: horizontal distance where minimum height occurs
• Key insight: This is a quadratic function, and we need to find its vertex (minimum point)

2. INFER the most efficient approach

• The expression \(\mathrm{(x - 2)(x - 18)}\) tells us the zeros are at \(\mathrm{x = 2}\) and \(\mathrm{x = 18}\)
• Since this quadratic has positive leading coefficient, it opens upward
• The minimum occurs at the axis of symmetry, which is the midpoint between zeros

3. SIMPLIFY to find the minimum location

• Midpoint between zeros: \(\mathrm{x = (2 + 18)/2 = 20/2 = 10}\)

Alternative approach:

• SIMPLIFY by expanding: \(\mathrm{(x - 2)(x - 18) = x^2 - 20x + 36}\)
• So \(\mathrm{q(x) = x^2 - 20x + 43}\)
• APPLY vertex formula: \(\mathrm{x = -b/(2a) = -(-20)/(2 \times 1) = 10}\)

Answer: B) 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they're looking for the vertex of a parabola. Instead, they might try to set the function equal to zero or look for where the height equals 7, missing that 'minimum height' means finding the vertex.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize they need the vertex but make calculation errors, such as finding the midpoint incorrectly as \(\mathrm{(18 - 2)/2 = 8}\) instead of \(\mathrm{(18 + 2)/2 = 10}\), or making sign errors when applying the vertex formula.

This may lead them to select Choice A (8).

The Bottom Line:

This problem requires recognizing the connection between factored form and vertex location. The key insight is that for any quadratic \(\mathrm{(x - a)(x - b)}\), the vertex occurs at \(\mathrm{x = (a + b)/2}\), regardless of any additional constants.