A taxi fare is modeled by the linear function \(\mathrm{F(m) = 1.75m + b}\), where m is the number of...

1. TRANSLATE the problem information

• Given information:
  • Taxi fare function: \(\mathrm{F(m) = 1.75m + b}\)
  • A 5-mile trip costs $12.25
• What this tells us: \(\mathrm{F(5) = 12.25}\)

2. INFER the solution approach

• To find the unknown parameter \(\mathrm{b}\), we need to use the given point
• Strategy: Substitute the known values into the function and solve for \(\mathrm{b}\)

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{m = 5}\) into \(\mathrm{F(m) = 1.75m + b}\):
  \(\mathrm{F(5) = 1.75(5) + b}\)
• Since \(\mathrm{F(5) = 12.25}\):
  \(\mathrm{12.25 = 1.75(5) + b}\)
  \(\mathrm{12.25 = 8.75 + b}\)
• Solve for \(\mathrm{b}\):
  \(\mathrm{b = 12.25 - 8.75 = 3.5}\)

Answer: 3.5 or $3.50

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students may not recognize that "a 5-mile trip costs $12.25" means \(\mathrm{F(5) = 12.25}\). Instead, they might try to set up equations like "\(\mathrm{5 \times 1.75 = 12.25}\)" or get confused about which variable represents what.

This leads to confusion and inability to set up the correct equation, causing them to guess randomly.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{F(5) = 12.25 = 1.75(5) + b}\) but make arithmetic errors, such as calculating \(\mathrm{1.75 \times 5}\) incorrectly as 9.25 instead of 8.75, or subtracting wrong: \(\mathrm{12.25 - 9.25 = 3.00}\).

This leads them to calculate \(\mathrm{b = 3.00}\) instead of \(\mathrm{b = 3.5}\).

The Bottom Line:

This problem tests whether students can connect real-world function applications with algebraic manipulation. The key challenge is recognizing that a specific input-output pair gives you enough information to find the unknown parameter.