The function C, which gives the cost in dollars to produce n items, is defined by the equation \(\mathrm{C(n) =...

1. TRANSLATE the problem information

• Given information:
  • Original function: \(\mathrm{C(n) = n^2 - 40n + 600}\)
  • New function: \(\mathrm{P(n) = 2C(n) - 100}\)
  • \(\mathrm{m}\) represents the minimum value of \(\mathrm{C(n)}\)

• We need to find the minimum value of \(\mathrm{P(n)}\) in terms of \(\mathrm{m}\)

2. INFER the key relationship

• Notice that \(\mathrm{P(n) = 2C(n) - 100}\) represents transformations of \(\mathrm{C(n)}\)
• This means \(\mathrm{P(n)}\) is created by taking \(\mathrm{C(n)}\), multiplying by 2, then subtracting 100
• Key insight: When we transform a function, we transform its extreme values the same way

3. INFER the transformation effects

• The transformation has two parts:
  • Multiply by 2 (vertical stretch)
  • Subtract 100 (vertical shift down)

• Since \(\mathrm{m}\) is the minimum value of \(\mathrm{C(n)}\), applying these same transformations:
  • First: \(\mathrm{2m}\) (stretch the minimum by factor of 2)
  • Then: \(\mathrm{2m - 100}\) (shift down by 100)

4. Verify the logic

• Both \(\mathrm{C(n)}\) and \(\mathrm{P(n)}\) are quadratic functions that open upward
• They achieve their minimum values at the same n-value
• Therefore, \(\mathrm{P(n)_{min} = 2C(n)_{min} - 100 = 2m - 100}\)

Answer: C (\(\mathrm{2m - 100}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that function transformations apply to extreme values in the same way.

Instead of applying the transformations 2×(stuff) - 100 to the minimum value \(\mathrm{m}\), they might think the minimum of \(\mathrm{P(n)}\) is somehow different or requires finding it from scratch. This leads to confusion about how to connect \(\mathrm{P(n)}\) back to the given information about \(\mathrm{m}\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what \(\mathrm{m}\) represents or how \(\mathrm{P(n)}\) relates to \(\mathrm{C(n)}\).

They might think \(\mathrm{m}\) is just some number rather than specifically the minimum value of \(\mathrm{C(n)}\), or they might not see that \(\mathrm{P(n) = 2C(n) - 100}\) means "take \(\mathrm{C(n)}\), double it, subtract 100." This disconnect prevents them from seeing how transformations work.

This may lead them to select Choice A (\(\mathrm{m - 100}\)) by incorrectly applying only the "-100" part of the transformation.

The Bottom Line:

This problem tests whether students understand that transforming a function transforms its extreme values in exactly the same way. The key insight is recognizing that mathematical operations applied to functions are also applied to their special values like minimums and maximums.