Question:\(\mathrm{g(t) = (14 - t)(t + 6)}\)The function g is defined by the given equation. For what value of t...

1. TRANSLATE the factored form to find roots

• Given: \(\mathrm{g(t) = (14 - t)(t + 6)}\)
• To find roots, set each factor equal to zero:
  • \(\mathrm{14 - t = 0}\) → \(\mathrm{t = 14}\)
  • \(\mathrm{t + 6 = 0}\) → \(\mathrm{t = -6}\)
• The roots are \(\mathrm{t = -6}\) and \(\mathrm{t = 14}\)

2. INFER the location of the maximum

• For any quadratic function, the vertex (maximum or minimum) occurs exactly halfway between the two roots
• This happens because quadratic functions are symmetric about their axis of symmetry
• The maximum will be at the midpoint of the roots

3. SIMPLIFY to find the midpoint

• Midpoint = (first root + second root)/2
• \(\mathrm{Midpoint = \frac{-6 + 14}{2}}\)
  \(\mathrm{= \frac{8}{2}}\)
  \(\mathrm{= 4}\)
• Therefore, the maximum occurs at \(\mathrm{t = 4}\)

Answer: B (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect that the vertex lies at the midpoint between roots. Instead, they might try to expand the function and complete the square or use the vertex formula \(\mathrm{-\frac{b}{2a}}\), making the problem much more complicated than necessary.

This leads to confusion and potentially incorrect calculations, causing them to select the wrong answer or abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the midpoint but make arithmetic errors. For example, calculating \(\mathrm{\frac{-6 + 14}{2}}\) as \(\mathrm{\frac{-6 + 14}{2} = \frac{8}{2} = 2}\), or forgetting the division by 2 entirely.

This may lead them to select Choice (C) (10) if they add without dividing, or other incorrect choices based on their arithmetic mistake.

The Bottom Line:

The key insight is recognizing that you don't need to expand or use complex formulas - the factored form directly gives you the roots, and the maximum is simply their midpoint. Students who miss this connection make the problem much harder than it needs to be.