For the linear function g, the graph of \(\mathrm{y = g(x)}\) in the xy-plane passes through the points \(\mathrm{(1, 4)}\)...

1. TRANSLATE the problem information

• Given information:
  • Linear function g passes through points \((1, 4)\) and \((3, 10)\)
  • Need to find which equation defines g

2. INFER the solution strategy

• With two points on a line, I can find the slope first
• Then use either point with the slope to write the equation
• This is more direct than trying to test each answer choice

3. Calculate the slope using the slope formula

• Using points \((1, 4)\) and \((3, 10)\):
• \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3}\)

4. SIMPLIFY using point-slope form

• Take point \(\mathrm{(1, 4)}\) and slope \(\mathrm{m = 3}\):
• \(\mathrm{y - 4 = 3(x - 1)}\)
• \(\mathrm{y - 4 = 3x - 3}\)
• \(\mathrm{y = 3x + 1}\)
• Therefore: \(\mathrm{g(x) = 3x + 1}\)

5. Verify the solution

• Check with point \((3, 10)\): \(\mathrm{g(3) = 3(3) + 1 = 10}\) ✓
• This matches answer choice C

Answer: C. \(\mathrm{g(x) = 3x + 1}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make algebraic mistakes when expanding the point-slope form. They might incorrectly expand \(\mathrm{y - 4 = 3(x - 1)}\) as \(\mathrm{y - 4 = 3x - 1}\) instead of \(\mathrm{y - 4 = 3x - 3}\), leading to \(\mathrm{y = 3x + 3}\). Looking for something close to this incorrect result, they might select Choice D (\(\mathrm{g(x) = 6x - 2}\)) or get confused and guess.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the systematic approach of finding slope first. Instead, they try to plug the given points into each answer choice, but make calculation errors or get overwhelmed by the process. This leads to confusion and random answer selection.

The Bottom Line:

This problem rewards students who recognize the standard two-step strategy (find slope, then use point-slope form) and can execute the algebraic manipulation accurately. The key insight is that working systematically is much more reliable than trying to test answer choices.