Question:\(\mathrm{g(x) = -(x - 8)(x + 12)}\)The function g is defined by the given equation. For what value of x...

1. TRANSLATE the function information

• Given: \(\mathrm{g(x) = -(x - 8)(x + 12)}\)
• Need to find: value of x where \(\mathrm{g(x)}\) reaches its maximum

2. INFER the parabola's behavior

• The negative sign in front tells us the leading coefficient is negative
• This means the parabola opens downward and has a maximum (not a minimum)
• For any parabola, the maximum or minimum occurs at the vertex

3. TRANSLATE to find the roots

• Set \(\mathrm{g(x) = 0}\): \(\mathrm{-(x - 8)(x + 12) = 0}\)
• This gives us: \(\mathrm{(x - 8) = 0}\) or \(\mathrm{(x + 12) = 0}\)
• So the roots are: \(\mathrm{x = 8}\) and \(\mathrm{x = -12}\)

4. INFER the vertex location using symmetry

• Parabolas are symmetric about their vertex
• The vertex occurs at the midpoint between the two roots
• Midpoint formula: \(\mathrm{x = \frac{root_1 + root_2}{2}}\)

5. SIMPLIFY to find the exact location

• \(\mathrm{x = \frac{8 + (-12)}{2}}\)
• \(\mathrm{x = \frac{-4}{2}}\)
• \(\mathrm{x = -2}\)

Answer: C (-2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the factored form to finding roots, or don't realize that the vertex lies at the midpoint of the roots. Instead, they might try to expand the function or use complicated methods, getting lost in algebra and selecting a wrong answer through guessing.

Second Most Common Error:

Poor TRANSLATE execution: Students correctly identify that they need roots but make sign errors when solving \(\mathrm{(x - 8) = 0}\) and \(\mathrm{(x + 12) = 0}\), thinking the roots are \(\mathrm{x = -8}\) and \(\mathrm{x = 12}\) instead of \(\mathrm{x = 8}\) and \(\mathrm{x = -12}\). This leads to calculating the wrong midpoint: \(\mathrm{\frac{-8 + 12}{2} = 2}\), causing them to select Choice D (2).

The Bottom Line:

This problem rewards students who recognize the elegant connection between factored form and vertex location through the symmetry of parabolas, rather than those who attempt more complex algebraic manipulations.