The function f is defined by \(\mathrm{f(x) = 5(1/4 - x)^2 + 11/4}\). What is the value of \(\mathrm{f(1/4)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 5(1/4 - x)^2 + 11/4}\)
  • Need to find: \(\mathrm{f(1/4)}\)
• This tells us we need to substitute \(\mathrm{x = 1/4}\) into the function

2. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{x = 1/4}\) into \(\mathrm{f(x) = 5(1/4 - x)^2 + 11/4}\):
  \(\mathrm{f(1/4) = 5(1/4 - 1/4)^2 + 11/4}\)

• Evaluate the expression inside parentheses first:
  \(\mathrm{1/4 - 1/4 = 0}\)

• Now we have:
  \(\mathrm{f(1/4) = 5(0)^2 + 11/4}\)

• Square the zero:
  \(\mathrm{0^2 = 0}\)

• Multiply by 5:
  \(\mathrm{5(0) = 0}\)

• Add the remaining term:
  \(\mathrm{f(1/4) = 0 + 11/4 = 11/4}\)

Answer: \(\mathrm{11/4}\) (or 2.75)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what \(\mathrm{f(1/4)}\) means

Students might think they need to manipulate the function algebraically first, or they might be confused about the substitution process. This leads to confusion and guessing rather than the straightforward substitution approach.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during calculation

Students might correctly set up \(\mathrm{f(1/4) = 5(1/4 - 1/4)^2 + 11/4}\) but then make errors like:

• Calculating \(\mathrm{(1/4 - 1/4)}\) incorrectly as something other than 0
• Forgetting that \(\mathrm{0^2 = 0}\)
• Making errors with the final addition

This causes them to arrive at incorrect numerical answers.

The Bottom Line:

This problem tests whether students truly understand function notation and can perform careful arithmetic. The key insight is recognizing that when \(\mathrm{x = 1/4}\), the expression \(\mathrm{(1/4 - x)}\) becomes zero, which dramatically simplifies the calculation.