For the linear function g, the graph of \(\mathrm{y = g(x)}\) in the xy-plane has a slope of 2 and...

1. TRANSLATE the problem information

• Given information:
  • Slope = 2
  • Graph passes through point (1, 14)
  • Need to find equation g(x)

2. INFER the approach

• Since this is a linear function with known slope, use slope-intercept form
• Set up: \(\mathrm{g(x) = mx + b}\) where \(\mathrm{m = 2}\)
• This gives us: \(\mathrm{g(x) = 2x + b}\)
• Now need to find the y-intercept b using the given point

3. TRANSLATE the point condition into an equation

• "Passes through (1, 14)" means when \(\mathrm{x = 1, g(x) = 14}\)
• Substitute into our equation: \(\mathrm{14 = 2(1) + b}\)

4. SIMPLIFY to find b

• \(\mathrm{14 = 2(1) + b}\)
• \(\mathrm{14 = 2 + b}\)
• \(\mathrm{b = 14 - 2 = 12}\)

5. Write the final equation

• \(\mathrm{g(x) = 2x + 12}\)

Answer: C. g(x) = 2x + 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students see the point (1, 14) and think the y-intercept b must equal 14, without recognizing they need to substitute the point into the equation to solve for b.

They incorrectly reason: "The function passes through (1, 14), so the equation must have 14 in it somewhere." This leads them to select Choice D (g(x) = 2x + 14).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{14 = 2 + b}\) but make arithmetic errors, such as adding instead of subtracting: \(\mathrm{b = 14 + 2 = 16}\).

Since 16 isn't among the y-intercept values in the choices, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students understand that "passing through a point" creates a constraint equation that must be solved, rather than directly incorporating the point's coordinates into the final answer.