For the linear function \(\mathrm{f(x)}\), the table shows three values of x and their corresponding values of \(\mathrm{f(x)}\). Function f...

1. TRANSLATE the problem information

• Given information:
  • Linear function: \(\mathrm{f(x) = ax + b}\)
  • Table values: \(\mathrm{f(1) = -64}\), \(\mathrm{f(2) = 0}\), \(\mathrm{f(3) = 64}\)
  • Need to find: \(\mathrm{a - b}\)

2. INFER the approach

• Since we have a linear function with two unknown constants (a and b), we need two equations
• Any two points from the table will give us these equations
• I'll use \(\mathrm{f(1) = -64}\) and \(\mathrm{f(2) = 0}\) for cleaner calculations

3. TRANSLATE table values into equations

From \(\mathrm{f(1) = -64}\):

\(\mathrm{a(1) + b = -64}\)

\(\mathrm{a + b = -64}\)

From \(\mathrm{f(2) = 0}\):

\(\mathrm{a(2) + b = 0}\)

\(\mathrm{2a + b = 0}\)

4. SIMPLIFY the system of equations

Subtract the first equation from the second:

\(\mathrm{(2a + b) - (a + b) = 0 - (-64)}\)

\(\mathrm{2a + b - a - b = 64}\)

\(\mathrm{a = 64}\)

5. SIMPLIFY to find b

Substitute \(\mathrm{a = 64}\) into the first equation:

\(\mathrm{64 + b = -64}\)

\(\mathrm{b = -128}\)

6. SIMPLIFY to find the final answer

\(\mathrm{a - b = 64 - (-128)}\)

\(\mathrm{= 64 + 128}\)

\(\mathrm{= 192}\)

Answer: D. 192

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when computing \(\mathrm{a - b}\), especially with the negative value of b.

They correctly find \(\mathrm{a = 64}\) and \(\mathrm{b = -128}\), but then calculate \(\mathrm{a - b = 64 - 128 = -64}\) instead of \(\mathrm{a - b = 64 - (-128) = 192}\).

This leads them to select Choice A (-64).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to solve a system of equations and instead try to work backwards from the answer choices or use only one data point.

This leads to confusion about the relationship between a and b, causing them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem tests whether students can systematically convert function notation into equations and carefully handle negative numbers in algebraic operations. The key insight is recognizing that subtracting a negative number is the same as adding a positive number.