The table shows three values of x and their corresponding values of \(\mathrm{g(x)}\), where \(\mathrm{g(x) = \frac{f(x)}{x+3}}\) and f is...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = \frac{f(x)}{x+3}}\) where f is linear
  • Table showing x and g(x) values: (-27, 3), (-9, 0), (21, 5)
  • Need to find y-intercept of f(x)

• This tells us we need to work backwards from \(\mathrm{g(x)}\) to find \(\mathrm{f(x)}\)

2. INFER the approach

• Since \(\mathrm{g(x) = \frac{f(x)}{x+3}}\), we can solve for \(\mathrm{f(x)}\): \(\mathrm{f(x) = g(x) \times (x+3)}\)
• We'll use the table to find specific points on \(\mathrm{f(x)}\)
• Since f is linear, we can then find its equation and y-intercept

3. SIMPLIFY to find points on f(x)

• Calculate \(\mathrm{f(x) = g(x) \times (x+3)}\) for each table entry:
  • \(\mathrm{f(-27) = 3 \times (-27+3)}\)
    \(\mathrm{= 3 \times (-24)}\)
    \(\mathrm{= -72}\)
  • \(\mathrm{f(-9) = 0 \times (-9+3)}\)
    \(\mathrm{= 0 \times (-6)}\)
    \(\mathrm{= 0}\)
  • \(\mathrm{f(21) = 5 \times (21+3)}\)
    \(\mathrm{= 5 \times 24}\)
    \(\mathrm{= 120}\)

• Points on \(\mathrm{f(x)}\): (-27, -72), (-9, 0), (21, 120)

4. INFER linear function properties

• Since f is linear: \(\mathrm{f(x) = mx + b}\) where b is the y-intercept
• We need to find slope m and y-intercept b

5. SIMPLIFY to find slope and y-intercept

• Using points (-27, -72) and (-9, 0):
  \(\mathrm{m = \frac{0-(-72)}{-9-(-27)}}\)
  \(\mathrm{= \frac{72}{18}}\)
  \(\mathrm{= 4}\)

• Using \(\mathrm{f(x) = 4x + b}\) and point (-9, 0):
  \(\mathrm{0 = 4(-9) + b}\)
  \(\mathrm{0 = -36 + b}\)
  \(\mathrm{b = 36}\)

Answer: A. (0, 36)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse \(\mathrm{g(x)}\) values with \(\mathrm{f(x)}\) values and try to use the table points directly as points on \(\mathrm{f(x)}\).

They see the table and think (-27, 3), (-9, 0), and (21, 5) are points on \(\mathrm{f(x)}\). Using these to find slope gives \(\mathrm{m = \frac{0-3}{-9-(-27)} = \frac{-3}{18} = -\frac{1}{6}}\), and using point (-9, 0) gives \(\mathrm{f(x) = -\frac{1}{6}x + b}\), so \(\mathrm{0 = -\frac{1}{6}(-9) + b}\), meaning \(\mathrm{b = -\frac{3}{2}}\). This incorrect approach leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that \(\mathrm{f(x) = g(x) \times (x+3)}\) but make arithmetic errors when calculating the products or finding the slope.

For example, calculating \(\mathrm{f(-27) = 3 \times (-24) = -72}\) incorrectly as positive 72, or making errors in the slope calculation. These calculation mistakes lead to incorrect values for the y-intercept. This may lead them to select Choice B (0, 12) or Choice C (0, 4).

The Bottom Line:

This problem requires students to understand the relationship between transformed and original functions, then systematically work backwards to reconstruct the original linear function. Success depends on careful translation of the given relationship and accurate algebraic execution.