In the xy-plane, the points \((-1, 4)\) and \((3, -2)\) lie on the graph of which of the following linear...

1. TRANSLATE the problem information

• Given information:
  • Two points: \((-1, 4)\) and \((3, -2)\)
  • Need to find which linear function contains both points
• What this means: Both points must satisfy the equation when we substitute their coordinates

2. INFER the solution strategy

• Since we have specific answer choices, testing each one by substitution will be most efficient
• For each function, substitute both x-values and check if we get the corresponding y-values
• Only one function should work for both points

3. SIMPLIFY by testing each choice systematically

Choice A: \(\mathrm{f(x) = -2x + 1}\)

• Test \((-1, 4)\): \(\mathrm{f(-1) = -2(-1) + 1}\)
• \(\mathrm{= 2 + 1}\)
• \(\mathrm{= 3 \neq 4}\) ❌
• Since the first point fails, Choice A is eliminated

Choice B: \(\mathrm{f(x) = -\frac{3}{2}x + \frac{5}{2}}\)

• Test \((-1, 4)\): \(\mathrm{f(-1) = -\frac{3}{2}(-1) + \frac{5}{2}}\)
• \(\mathrm{= \frac{3}{2} + \frac{5}{2}}\)
• \(\mathrm{= \frac{8}{2} = 4}\) ✓
• Test \((3, -2)\): \(\mathrm{f(3) = -\frac{3}{2}(3) + \frac{5}{2}}\)
• \(\mathrm{= -\frac{9}{2} + \frac{5}{2}}\)
• \(\mathrm{= -\frac{4}{2} = -2}\) ✓
• Both points work! This is likely our answer, but let's verify the others

Choice C: \(\mathrm{f(x) = 2x + 6}\)

• Test \((-1, 4)\): \(\mathrm{f(-1) = 2(-1) + 6}\)
• \(\mathrm{= -2 + 6}\)
• \(\mathrm{= 4}\) ✓
• Test \((3, -2)\): \(\mathrm{f(3) = 2(3) + 6}\)
• \(\mathrm{= 6 + 6}\)
• \(\mathrm{= 12 \neq -2}\) ❌

Choice D: \(\mathrm{f(x) = 3x + 7}\)

• Test \((-1, 4)\): \(\mathrm{f(-1) = 3(-1) + 7}\)
• \(\mathrm{= -3 + 7}\)
• \(\mathrm{= 4}\) ✓
• Test \((3, -2)\): \(\mathrm{f(3) = 3(3) + 7}\)
• \(\mathrm{= 9 + 7}\)
• \(\mathrm{= 16 \neq -2}\) ❌

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make arithmetic errors when working with fractions and negative numbers, particularly with Choice B's calculations like \(\mathrm{-\frac{3}{2}(-1) + \frac{5}{2}}\) or \(\mathrm{-\frac{3}{2}(3) + \frac{5}{2}}\).

Common mistakes include forgetting that \(\mathrm{-\frac{3}{2} \times (-1) = +\frac{3}{2}}\), or incorrectly combining fractions like getting \(\mathrm{\frac{2}{2}}\) instead of \(\mathrm{\frac{8}{2}}\) when adding \(\mathrm{\frac{3}{2} + \frac{5}{2}}\). These calculation errors cause students to incorrectly eliminate the right answer and may lead them to select Choice C (\(\mathrm{f(x) = 2x + 6}\)) or Choice D (\(\mathrm{f(x) = 3x + 7}\)) since these have simpler arithmetic that they're more confident about.

Second Most Common Error:

Incomplete INFER strategy: Students test only the first point for each function rather than verifying both points. Since Choices A, C, and D all satisfy the first point \((-1, 4)\), students who stop after one point end up guessing among the remaining choices instead of systematically checking the second point.

This incomplete approach leads to confusion and random selection among Choice C or Choice D.

The Bottom Line:

This problem rewards systematic verification and careful fraction arithmetic. Students who rush through the calculations or don't check both points will struggle to identify the correct answer confidently.