A linear function f is defined by \(\mathrm{f(x) = mx + b}\), where m and b are constants. The table...

1. TRANSLATE the problem information

• Given information:
  • Linear function \(\mathrm{f(x) = mx + b}\)
  • Table with three coordinate pairs: \(\mathrm{(-2, 11), (1, -1), (3, -9)}\)
  • Need to find which answer choice matches this function

• What this tells us: We have three points that lie on the same line and need to determine the slope \(\mathrm{(m)}\) and y-intercept \(\mathrm{(b)}\).

2. INFER your solution approach

• Two viable strategies:
  • Strategy 1: Calculate \(\mathrm{m}\) and \(\mathrm{b}\) directly from the points
  • Strategy 2: Test each answer choice with the given points

• Strategy 2 is often faster with multiple choice questions - let's start there!

3. TRANSLATE one point for testing

• Use the point \(\mathrm{(1, -1)}\) since \(\mathrm{x = 1}\) makes calculations simpler
• This point must satisfy the correct equation: \(\mathrm{f(1) = -1}\)

4. SIMPLIFY by testing each answer choice

Test \(\mathrm{f(1) = -1}\):

• Choice A: \(\mathrm{f(1) = -6(1) - 1 = -7}\) ❌
• Choice B: \(\mathrm{f(1) = 4(1) - 3 = 1}\) ❌
• Choice C: \(\mathrm{f(1) = -4(1) + 3 = -1}\) ✓
• Choice D: \(\mathrm{f(1) = -4(1) + 11 = 7}\) ❌

5. INFER verification step

• Only Choice C worked, but let's verify with another point to be certain
• Test \(\mathrm{(-2, 11)}\) with Choice C: \(\mathrm{f(-2) = -4(-2) + 3 = 8 + 3 = 11}\) ✓

Answer: C) \(\mathrm{f(x) = -4x + 3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when calculating, especially with negative coordinates like \(\mathrm{(-2, 11)}\).

For example, when testing Choice C with point \(\mathrm{(-2, 11)}\):

• Correct: \(\mathrm{f(-2) = -4(-2) + 3 = +8 + 3 = 11}\)
• Common error: \(\mathrm{f(-2) = -4(-2) + 3 = -8 + 3 = -5}\)

This incorrect calculation might lead them to reject the correct choice and select Choice A \(\mathrm{(-6x - 1)}\) or another incorrect option.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which value represents \(\mathrm{x}\) and which represents \(\mathrm{f(x)}\) from the table, or they mix up the order when writing coordinate pairs.

This leads to using wrong points for calculations and getting an entirely different equation, causing them to get stuck and guess randomly.

The Bottom Line:

This problem rewards careful arithmetic and systematic checking. The key insight is that linear functions are completely determined by any two points, so testing one point eliminates most wrong answers immediately.