For the linear function g, the table shows four values of x and their corresponding values of \(\mathrm{g(x)}\). The function...

1. TRANSLATE the table information

• Given information:
  • Table shows x and g(x) values for a linear function
  • Function form: \(\mathrm{g(x) = mx + b}\)
  • Need to find the value of b (y-intercept)

2. INFER the most efficient approach

• Since we have multiple points, we can find the slope first
• Once we know m, we can use any point to solve for b
• This avoids setting up a complex system of equations

3. SIMPLIFY to find the slope

• Using points (1, 54) and (2, 51):
  \(\mathrm{m = \frac{51 - 54}{2 - 1}}\)
  \(\mathrm{= \frac{-3}{1}}\)
  \(\mathrm{= -3}\)
• Verify with another pair (2, 51) and (3, 48):
  \(\mathrm{m = \frac{48 - 51}{3 - 2}}\)
  \(\mathrm{= \frac{-3}{1}}\)
  \(\mathrm{= -3}\) ✓

4. SIMPLIFY to find b using any point

• Using point (1, 54) in \(\mathrm{g(x) = mx + b}\):
  \(\mathrm{54 = -3(1) + b}\)
  \(\mathrm{54 = -3 + b}\)
  \(\mathrm{b = 57}\)

5. Verify the answer

• Check with point (2, 51): \(\mathrm{g(2) = -3(2) + 57 = 51}\) ✓

Answer: D. 57

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize they need to find the slope first, instead attempting to work directly with the equation \(\mathrm{g(x) = mx + b}\) without a clear strategy.

They might substitute one point randomly, getting \(\mathrm{54 = m + b}\), then feel stuck because they have two unknowns in one equation. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the slope correctly but make sign errors when solving for b.

For example, from \(\mathrm{54 = -3 + b}\), they might subtract 3 instead of adding 3, getting \(\mathrm{b = 51}\). This might lead them to select Choice B (27) if they make additional errors, or cause confusion that leads to guessing.

The Bottom Line:

This problem tests whether students can systematically use multiple data points to determine parameters of a linear function. The key insight is recognizing that the slope can be calculated first, which simplifies finding the y-intercept significantly.