In the xy-plane, the graph of the linear function g passes through the points \((1, 5)\) and \((5, 17)\). Which...

1. TRANSLATE the problem information

• Given information:
  • Linear function g passes through \((1, 5)\) and \((5, 17)\)
  • Need to find equation \(\mathrm{y = g(x)}\)

2. INFER the approach

• To find a linear equation from two points, we need slope first
• Then we can use point-slope form to build the equation

3. Calculate the slope using slope formula

• \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{= \frac{17 - 5}{5 - 1}}\)
• \(\mathrm{= \frac{12}{4}}\)
• \(\mathrm{= 3}\)

4. SIMPLIFY using point-slope form

• Using point \((1, 5)\): \(\mathrm{y - 5 = 3(x - 1)}\)
• Expand: \(\mathrm{y - 5 = 3x - 3}\)
• SIMPLIFY to slope-intercept form:
  • \(\mathrm{y = 3x - 3 + 5}\)
  • \(\mathrm{y = 3x + 2}\)

5. Verify with the second point

• \(\mathrm{g(5) = 3(5) + 2}\)
• \(\mathrm{= 15 + 2}\)
• \(\mathrm{= 17}\) ✓

Answer: A. \(\mathrm{g(x) = 3x + 2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the slope or during algebraic manipulation.

For example, they might calculate \(\mathrm{\frac{12}{4}}\) incorrectly as 4 instead of 3, leading them toward Choice C (\(\mathrm{g(x) = 4x + 1}\)). Or they make sign errors when expanding \(\mathrm{y - 5 = 3(x - 1)}\), getting confused about whether to add or subtract terms.

Second Most Common Error:

Poor INFER reasoning: Students try to substitute points directly into answer choices without understanding they need to find slope first.

Without a systematic approach, they might randomly test answer choices or get overwhelmed by the multiple steps required. This leads to confusion and guessing among the given options.

The Bottom Line:

This problem requires methodical execution of the slope-intercept process. Students who rush through the slope calculation or algebraic steps often select incorrect answers, even when they understand the overall concept.