A linear function f has an average rate of change of 3 between x = 2 and x = 6....

1. TRANSLATE the problem information

• Given information:
  • Average rate of change of 3 between \(\mathrm{x = 2}\) and \(\mathrm{x = 6}\)
  • Function value at \(\mathrm{x = 5}\) is 7 (meaning \(\mathrm{f(5) = 7}\))
  • Need to find the equation \(\mathrm{f(x)}\)

2. INFER the key relationship

• For linear functions, the average rate of change over any interval equals the slope
• Since average rate of change = 3, the slope \(\mathrm{m = 3}\)
• This means our function has the form \(\mathrm{f(x) = 3x + b}\)

3. TRANSLATE the given point condition

• "Function value at \(\mathrm{x = 5}\) is 7" means \(\mathrm{f(5) = 7}\)
• Substitute into our equation: \(\mathrm{f(5) = 3(5) + b = 7}\)

4. SIMPLIFY to find the y-intercept

• \(\mathrm{3(5) + b = 7}\)
• \(\mathrm{15 + b = 7}\)
• \(\mathrm{b = 7 - 15 = -8}\)

5. Write the final equation

• \(\mathrm{f(x) = 3x + (-8) = 3x - 8}\)

Answer: B. \(\mathrm{f(x) = 3x - 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that for linear functions, average rate of change equals slope everywhere. They might try to use the average rate of change formula directly with the given interval, getting confused about what information they actually need.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "function value at \(\mathrm{x = 5}\) is 7" and don't realize this gives them the point \(\mathrm{(5, 7)}\) to work with. Without this crucial point, they can't determine the y-intercept even if they correctly identify the slope as 3.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students understand the fundamental property that distinguishes linear functions - their constant rate of change. The key insight is connecting "average rate of change" to "slope" for linear functions.