Question: A researcher studying growth patterns defines a linear function g by \(\mathrm{g(x) = mx + b}\), where m and...

1. TRANSLATE the problem information

• Given information:
  • Linear function: \(\mathrm{g(x) = mx + b}\)
  • Rate of change = \(\mathrm{\frac{2}{3}}\)
  • \(\mathrm{g(9) = 8}\)
  • Need to find: \(\mathrm{g(0)}\)

2. INFER what the given information means

• Rate of change of a linear function = slope, so \(\mathrm{m = \frac{2}{3}}\)
• \(\mathrm{g(9) = 8}\) gives us the point \(\mathrm{(9, 8)}\) on the line
• \(\mathrm{g(0)}\) represents the y-intercept, which equals \(\mathrm{b}\) in the equation

3. INFER the solution strategy

• Since \(\mathrm{g(0) = b}\), we need to find the value of \(\mathrm{b}\)
• We can use the point \(\mathrm{(9, 8)}\) and slope \(\mathrm{m = \frac{2}{3}}\) to solve for \(\mathrm{b}\)

4. SIMPLIFY by substituting into the equation

• Substitute the point \(\mathrm{(9, 8)}\) into \(\mathrm{g(x) = mx + b}\):
  \(\mathrm{g(9) = m(9) + b = 8}\)
• Substitute \(\mathrm{m = \frac{2}{3}}\):
  \(\mathrm{\frac{2}{3}(9) + b = 8}\)
  \(\mathrm{6 + b = 8}\)
  \(\mathrm{b = 2}\)

5. INFER the final answer

• Since \(\mathrm{g(0) = b}\) and we found \(\mathrm{b = 2}\):
  \(\mathrm{g(0) = 2}\)

Answer: B) 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misunderstand what "rate of change" means in the context of linear functions. They might think it refers to something other than the slope, or they might confuse the concepts of slope and y-intercept.

This conceptual confusion leads them to set up incorrect equations, potentially leading them to select Choice A (-2) if they somehow use negative values, or to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{m = \frac{2}{3}}\) and need to solve for \(\mathrm{b}\), but make arithmetic errors when calculating \(\mathrm{\frac{2}{3}(9)}\). They might calculate this as 3 instead of 6, leading to:
\(\mathrm{3 + b = 8}\), so \(\mathrm{b = 5}\)

Since none of the choices show 5, this leads to confusion and guessing, or they might misremember their calculation and select Choice C (6).

The Bottom Line:

This problem tests whether students truly understand the connection between the verbal description "rate of change" and the mathematical concept of slope, combined with their ability to use point-slope relationships to find the y-intercept.