Question:\(\mathrm{g(x) = -2x^2 + 16x - 14}\)The given equation defines the function g. For what value of x does \(\mathrm{g(x)}\)...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = -2x^2 + 16x - 14}\)
  • Need to find: x-value where g(x) reaches its maximum

• What this tells us: This is a quadratic function, and we need to find where it peaks

2. INFER the approach

• Since this is a quadratic function, its graph is a parabola
• The maximum or minimum occurs at the vertex of the parabola
• We need to determine if this parabola has a maximum or minimum, then find the vertex

3. INFER whether vertex is maximum or minimum

• Looking at the coefficient of \(\mathrm{x^2}\): \(\mathrm{a = -2}\)
• Since \(\mathrm{a = -2 \lt 0}\), the parabola opens downward
• Therefore, the vertex represents a maximum point

4. SIMPLIFY using the vertex formula

• For \(\mathrm{g(x) = ax^2 + bx + c}\), the x-coordinate of vertex is: \(\mathrm{x = \frac{-b}{2a}}\)
• Here: \(\mathrm{a = -2, b = 16, c = -14}\)
• Substitute: \(\mathrm{x = \frac{-b}{2a} = \frac{-16}{2(-2)}}\)
• SIMPLIFY: \(\mathrm{x = \frac{-16}{-4} = 4}\)

Answer: B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a vertex-finding problem and instead try to set the function equal to zero or use other inappropriate methods.

They might attempt to solve \(\mathrm{-2x^2 + 16x - 14 = 0}\), thinking they need to find where the function equals zero rather than where it reaches its maximum value. This leads to confusion since they're solving the wrong equation entirely.

This causes them to get stuck and randomly select an answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students use the correct vertex formula but make arithmetic errors in the calculation.

Common mistakes include: \(\mathrm{x = \frac{-16}{2(-2)} = \frac{-16}{4} = -4}\) (forgetting the negative sign) or \(\mathrm{x = \frac{16}{2(-2)} = \frac{16}{-4} = -4}\) (dropping the negative from the numerator).

This may lead them to select Choice A (2) if they get confused about signs, or they might pick a different incorrect answer choice.

The Bottom Line:

Success on this problem requires recognizing that "maximum value" connects to the vertex of a quadratic function, then executing the vertex formula accurately. The key insight is strategic - knowing what mathematical tool applies to this type of question.