Question:The function f is linear.It satisfies \(\mathrm{f(0) = 8}\) and \(\mathrm{f(1) = 18}\).What is \(\mathrm{f(8)}\)?18808898

1. TRANSLATE the problem information

• Given information:
  • f is a linear function
  • \(\mathrm{f(0) = 8}\)
  • \(\mathrm{f(1) = 18}\)
  • Need to find \(\mathrm{f(8)}\)

2. INFER the approach

• Since f is linear, it has the standard form \(\mathrm{f(x) = mx + b}\)
• We need to find the slope m and y-intercept b
• We can use the two given points to determine these values

3. TRANSLATE the first condition to find b

• From \(\mathrm{f(0) = 8}\):
  • Substitute: \(\mathrm{f(0) = m(0) + b = b = 8}\)
  • Therefore: \(\mathrm{b = 8}\)
  • So our function becomes: \(\mathrm{f(x) = mx + 8}\)

4. TRANSLATE the second condition and SIMPLIFY to find m

• From \(\mathrm{f(1) = 18}\):
  • Substitute: \(\mathrm{f(1) = m(1) + 8 = m + 8 = 18}\)
  • Solve for m: \(\mathrm{m = 18 - 8 = 10}\)
  • Complete function: \(\mathrm{f(x) = 10x + 8}\)

5. SIMPLIFY to find f(8)

• Substitute \(\mathrm{x = 8}\):
  • \(\mathrm{f(8) = 10(8) + 8}\)
  • \(\mathrm{f(8) = 80 + 8}\)
  • \(\mathrm{f(8) = 88}\)

Answer: C. 88

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly set up \(\mathrm{f(x) = 10x + 8}\) but make a calculation error when finding \(\mathrm{f(8)}\). They might calculate \(\mathrm{10(8) = 80}\) but forget to add the constant term \(\mathrm{(+8)}\).

This may lead them to select Choice B (80)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what they're looking for and simply recall that \(\mathrm{f(1) = 18}\), thinking this might be the answer without recognizing they need \(\mathrm{f(8)}\).

This may lead them to select Choice A (18)

The Bottom Line:

This problem tests whether students can systematically work with the linear function form \(\mathrm{f(x) = mx + b}\). Success depends on carefully translating each given condition into an equation and methodically solving for the unknown parameters.