\(\mathrm{h(x) = 2x^2 - 24x + 80}\)The function h is defined by the given equation. Let n_sub denote the minimum...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(x) = 2x^2 - 24x + 80}\)
  • \(\mathrm{n_{sub}}\) = minimum value of \(\mathrm{h(x)}\)
  • \(\mathrm{k(x) = h(x) - 8}\)
• Find: Expression representing the minimum value of \(\mathrm{k(x)}\)

2. INFER the approach

• Since \(\mathrm{h(x)}\) is quadratic with positive leading coefficient, it has a minimum
• Need to find this minimum first, then apply the transformation to \(\mathrm{k(x)}\)
• The key insight: when we subtract 8 from every output of \(\mathrm{h(x)}\), we subtract 8 from the minimum too

3. SIMPLIFY to find the minimum of h(x)

• Use vertex formula: \(\mathrm{x = -b/(2a)}\) = \(\mathrm{-(-24)/(2 \cdot 2)}\) = \(\mathrm{24/4 = 6}\)
• Substitute \(\mathrm{x = 6}\) into \(\mathrm{h(x)}\):
  \(\mathrm{h(6) = 2(36) - 24(6) + 80}\)
  \(\mathrm{= 72 - 144 + 80}\)
  \(\mathrm{= 8}\)
• Therefore: \(\mathrm{n_{sub} = 8}\)

4. INFER the minimum of k(x)

• Since \(\mathrm{k(x) = h(x) - 8}\), we subtract 8 from every output of \(\mathrm{h(x)}\)
• The minimum of \(\mathrm{k(x)}\) occurs at the same x-value as \(\mathrm{h(x)}\) (\(\mathrm{x = 6}\))
• Minimum value of \(\mathrm{k(x)}\) = minimum value of \(\mathrm{h(x)}\) - 8 = \(\mathrm{n_{sub} - 8}\)

Answer: A (\(\mathrm{n_{sub} - 8}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students may not recognize the relationship between transformations and extreme values. They might think that subtracting 8 from the function changes where the minimum occurs, or they may try to work backwards from the answer choices without understanding the transformation concept.

This leads to confusion and guessing among the available expressions.

Second Most Common Error:

Poor TRANSLATE skills: Students might misinterpret what \(\mathrm{n_{sub}}\) represents, thinking it's a variable rather than the specific minimum value of \(\mathrm{h(x)}\). This confusion about notation can lead them to manipulate \(\mathrm{n_{sub}}\) algebraically in incorrect ways.

This may lead them to select Choice B (\(\mathrm{n_{sub} + 8}\)) by incorrectly thinking they need to add rather than subtract.

The Bottom Line:

This problem tests your understanding of how vertical transformations affect the range of a function. The key insight is that subtracting a constant from every output shifts all function values down by that amount, including the minimum value.