A linear function f has a constant rate of change of 4. The function satisfies \(\mathrm{f(5) = 26}\). What is...

1. TRANSLATE the problem information

• Given information:
  • f is a linear function
  • Constant rate of change = 4
  • \(\mathrm{f(5) = 26}\)
  • Find \(\mathrm{f(0)}\)

• TRANSLATE "constant rate of change of 4" → \(\mathrm{slope = 4}\)

2. INFER the solution approach

• Since we need \(\mathrm{f(0)}\) and we know the slope, this suggests using slope-intercept form
• \(\mathrm{f(x) = mx + b}\) where \(\mathrm{m = 4}\) (our slope) and b is what we need to find
• Recognizing that \(\mathrm{f(0) = b}\), so we're actually looking for the y-intercept

3. Set up the linear function

• Write \(\mathrm{f(x) = 4x + b}\)
• We need to find b using the given point

4. SIMPLIFY using point substitution

• Substitute the known point \(\mathrm{f(5) = 26}\):
  \(\mathrm{4(5) + b = 26}\)
  \(\mathrm{20 + b = 26}\)
  \(\mathrm{b = 26 - 20 = 6}\)

5. Find f(0)

• Now that we know \(\mathrm{b = 6}\):
  \(\mathrm{f(0) = 4(0) + 6 = 6}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not recognize that "constant rate of change" means slope in a linear function context. They might try to work with the rate of change as a separate concept rather than understanding it equals the slope coefficient.

This confusion leads them to set up incorrect equations or approach the problem without a clear strategy, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{f(x) = 4x + b}\) and substitute \(\mathrm{f(5) = 26}\), but make arithmetic errors when solving \(\mathrm{20 + b = 26}\). They might incorrectly calculate \(\mathrm{b = 20 + 26 = 46}\) instead of \(\mathrm{b = 26 - 20 = 6}\).

This leads to \(\mathrm{f(0) = 46}\), which isn't among the answer choices, causing confusion and potentially selecting Choice C (20) by using just the coefficient they calculated.

The Bottom Line:

This problem tests whether students can bridge the language of "rate of change" with the mathematical concept of slope, then execute a straightforward point-substitution strategy. The key insight is recognizing that finding \(\mathrm{f(0)}\) means finding the y-intercept of the linear function.