\(\mathrm{f(x) = (x - 2)(x + 15)}\) The function f is defined by the given equation. For what value of...

1. INFER the problem strategy

• Given: \(\mathrm{f(x) = (x - 2)(x + 15)}\) in factored form
• Need to find: value of \(\mathrm{x}\) where \(\mathrm{f(x)}\) reaches its minimum
• Key insight: Since this expands to a quadratic with positive leading coefficient, the parabola opens upward and has a minimum at its vertex

2. INFER the approach to find the vertex

• The vertex (minimum point) of a parabola occurs at the midpoint between its x-intercepts
• From the factored form, we can directly read the x-intercepts

3. TRANSLATE the x-intercepts from factored form

• From \(\mathrm{f(x) = (x - 2)(x + 15)}\), the function equals zero when:
  • \(\mathrm{x - 2 = 0}\) → \(\mathrm{x = 2}\)
  • \(\mathrm{x + 15 = 0}\) → \(\mathrm{x = -15}\)
• So the x-intercepts are at \(\mathrm{x = 2}\) and \(\mathrm{x = -15}\)

4. SIMPLIFY to find the midpoint

• Midpoint between \(\mathrm{x = 2}\) and \(\mathrm{x = -15}\):
• \(\mathrm{x = \frac{2 + (-15)}{2}}\)
• \(\mathrm{= -\frac{13}{2}}\)
• \(\mathrm{= -6.5}\) (use calculator for decimal conversion)

Answer: \(\mathrm{-\frac{13}{2}}\) or \(\mathrm{-6.5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the vertex of the parabola to locate the minimum. They might try random substitution of values or attempt to solve \(\mathrm{f(x) = 0}\) instead of finding where the minimum occurs.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating the midpoint, particularly getting confused with the negative numbers. They might calculate \(\mathrm{\frac{2 + (-15)}{2}}\) as \(\mathrm{\frac{2 - 15}{2} = -\frac{13}{2}}\) but then make a sign error, getting \(\mathrm{+\frac{13}{2} = +6.5}\) instead of \(\mathrm{-6.5}\).

This leads them to select a positive answer if one is available among choices, or causes confusion.

The Bottom Line:

This problem tests whether students understand the connection between factored form and vertex location. The key insight is recognizing that the vertex lies exactly halfway between the x-intercepts—a geometric property that many students miss.