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

1. TRANSLATE the problem information

• Given information:
  • Table shows three points: \((1, 13), (3, 17), (5, 21)\)
  • \(\mathrm{g(x)}\) is a linear function
  • Need to find which equation defines \(\mathrm{g(x)}\)

• What this tells us: We need to find the equation \(\mathrm{g(x) = mx + b}\)

2. INFER the approach

• Since \(\mathrm{g(x)}\) is linear, it has the form \(\mathrm{g(x) = mx + b}\)
• Strategy: Find the slope (m) first, then use one point to find the y-intercept (b)
• Any two points can give us the slope

3. SIMPLIFY to find the slope

• Using points \((1, 13)\) and \((3, 17)\):

\(\mathrm{m = \frac{17 - 13}{3 - 1}}\)
\(\mathrm{m = \frac{4}{2}}\)
\(\mathrm{m = 2}\)

4. SIMPLIFY to find the y-intercept

• Substitute point \((1, 13)\) into \(\mathrm{g(x) = 2x + b}\):

\(\mathrm{13 = 2(1) + b}\)
\(\mathrm{13 = 2 + b}\)
\(\mathrm{b = 11}\)

5. TRANSLATE back to complete equation

• Therefore: \(\mathrm{g(x) = 2x + 11}\)
• Check against answer choices: This matches choice (A)

6. Verify with remaining point

• \(\mathrm{g(5) = 2(5) + 11 = 21}\) ✓

Answer: A. \(\mathrm{g(x) = 2x + 11}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find slope first, or they try to use the table values directly as coefficients without understanding the slope-intercept relationship.

They might see the pattern in the table (13, 17, 21) and think one of these numbers should be a coefficient, leading them to select Choice (B) (\(\mathrm{g(x) = 13x + 17}\)) or another incorrect option.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for slope and y-intercept but make arithmetic errors in the calculation process.

For example, calculating slope incorrectly as \(\mathrm{\frac{17-13}{1-3} = \frac{4}{-2} = -2}\), or making errors when solving \(\mathrm{13 = 2(1) + b}\). This leads to confusion and potentially guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand that linear functions follow \(\mathrm{y = mx + b}\) form and can systematically find both parameters, rather than just pattern-matching from the given values.