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

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Slope = -2}\)
  • Point on the line: \(\mathrm{(3, 1)}\)
  • Need to find: equation \(\mathrm{g(x)}\)

2. INFER the appropriate approach

• Since we have a slope and one point, point-slope form is the most direct method
• Point-slope form: \(\mathrm{y - y_1 = m(x - x_1)}\)
• We'll convert this to slope-intercept form \(\mathrm{y = mx + b}\) to match the answer choices

3. SIMPLIFY using point-slope form

• Substitute \(\mathrm{m = -2}\) and \(\mathrm{(x_1, y_1) = (3, 1)}\):
  \(\mathrm{y - 1 = -2(x - 3)}\)

4. SIMPLIFY through algebraic expansion

• Distribute the -2:
  \(\mathrm{y - 1 = -2x + 6}\)
• Add 1 to both sides:
  \(\mathrm{y = -2x + 6 + 1}\)
• Combine constants:
  \(\mathrm{y = -2x + 7}\)

5. Express final answer

• Therefore: \(\mathrm{g(x) = -2x + 7}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when expanding \(\mathrm{-2(x - 3)}\)

Many students correctly set up \(\mathrm{y - 1 = -2(x - 3)}\), but then expand incorrectly as \(\mathrm{y - 1 = -2x - 6}\) instead of \(\mathrm{y - 1 = -2x + 6}\). This gives them \(\mathrm{y = -2x - 6 + 1 = -2x - 5}\).

This leads them to select Choice A (\(\mathrm{-2x - 5}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse point coordinates with slope-intercept components

Some students see the point \(\mathrm{(3, 1)}\) and incorrectly think the y-intercept is 1, immediately writing \(\mathrm{g(x) = -2x + 1}\) without using point-slope form.

This leads them to select Choice B (\(\mathrm{-2x + 1}\)).

The Bottom Line:

This problem tests whether students can systematically apply point-slope form and carefully handle negative signs during algebraic manipulation. The key is recognizing that having a slope and point means using point-slope form, not trying to directly identify slope-intercept components.