The linear function r satisfies \(\mathrm{r(4) = 12}\) and has the property that \(\mathrm{r(a + 3) = r(a) - 6}\)...

1. TRANSLATE the problem information

• Given information:
  • r is a linear function
  • \(\mathrm{r(4) = 12}\)
  • \(\mathrm{r(a + 3) = r(a) - 6}\) for all values of a

2. INFER what the functional property tells us

• The property \(\mathrm{r(a + 3) = r(a) - 6}\) means:
  • When the input increases by 3, the output decreases by 6
  • This gives us the rate of change (slope) = \(\mathrm{-6/3 = -2}\)

3. INFER the approach using slope-intercept form

• Since \(\mathrm{r(x)}\) is linear, we can write: \(\mathrm{r(x) = mx + b}\)
• We know \(\mathrm{m = -2}\), so: \(\mathrm{r(x) = -2x + b}\)
• We need to find b using the given point

4. SIMPLIFY to find the y-intercept

• Since \(\mathrm{r(4) = 12}\):
  \(\mathrm{12 = -2(4) + b}\)
  \(\mathrm{12 = -8 + b}\)
  \(\mathrm{b = 20}\)

5. Write the final equation

• \(\mathrm{r(x) = -2x + 20}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret \(\mathrm{r(a + 3) = r(a) - 6}\) and think the slope is just -6 instead of recognizing it represents -6 change over +3 change in input.

They might calculate \(\mathrm{r(x) = -6x + b}\), then use \(\mathrm{r(4) = 12}\) to get:
\(\mathrm{12 = -6(4) + b}\)
\(\mathrm{12 = -24 + b}\)
\(\mathrm{b = 36}\)

This leads them to select Choice B (\(\mathrm{r(x) = -6x + 36}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the slope as -2 but make arithmetic errors when solving for b.

From \(\mathrm{12 = -2(4) + b}\), they might incorrectly calculate:
\(\mathrm{12 = -8 + b}\)
\(\mathrm{b = 12 + 8 = 20}\)... wait, that's wrong
\(\mathrm{b = 12 - 8 = 4}\)

This causes confusion and may lead to guessing or selecting Choice A (\(\mathrm{r(x) = 2x + 4}\)) if they also mix up the slope sign.

The Bottom Line:

The key insight is recognizing that a functional relationship like \(\mathrm{r(a + 3) = r(a) - 6}\) describes the slope through the ratio of output change to input change, not just the output change alone.