The graph of the linear function f passes through the points \((-2, 7)\) and \((4, -5)\).Line ell is perpendicular to...

1. TRANSLATE the problem information

• Given information:
  • Function f passes through points \((-2, 7)\) and \((4, -5)\)
  • Line ℓ is perpendicular to f
  • Need to find the slope of line ℓ

2. INFER the approach

• To find the slope of line ℓ, I first need the slope of function f
• Once I have f's slope, I can use the perpendicular relationship
• Strategy: Find slope of f → Apply perpendicular rule → Get slope of ℓ

3. SIMPLIFY to find the slope of function f

• Using slope formula with points \((-2, 7)\) and \((4, -5)\):
• \(\mathrm{m_f} = \frac{\mathrm{y_2 - y_1}}{\mathrm{x_2 - x_1}} = \frac{-5 - 7}{4 - (-2)}\)
• \(\mathrm{m_f} = \frac{-12}{6} = -2\)

4. INFER and SIMPLIFY the perpendicular slope

• Since ℓ is perpendicular to f, their slopes are negative reciprocals
• \(\mathrm{m_ℓ} = \frac{-1}{\mathrm{m_f}} = \frac{-1}{-2} = \frac{1}{2}\)

Answer: \(\frac{1}{2}\)

Other acceptable forms: \(0.5\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the slope of f, particularly with the subtraction in the numerator or denominator.

For example, calculating \((-5 - 7)\) as \(-2\) instead of \(-12\), or \((4 - (-2))\) as \(2\) instead of \(6\). This leads to an incorrect slope for f (like \(-\frac{1}{3}\)), which then gives an incorrect perpendicular slope (like \(-3\)). This causes them to get stuck and guess or select an incorrect numerical answer.

Second Most Common Error:

Conceptual confusion about perpendicular slopes: Students remember that perpendicular lines have "related" slopes but forget the "negative" part of "negative reciprocals."

They might calculate the slope of f correctly as \(-2\), but then think the perpendicular slope is just the reciprocal: \(-\frac{1}{2}\) instead of \(\frac{-1}{-2} = \frac{1}{2}\). This leads to the wrong answer of \(-\frac{1}{2}\).

The Bottom Line:

This problem requires careful arithmetic execution combined with proper application of the perpendicular slope relationship. Many students stumble on the signs—both in the slope calculation and in the negative reciprocal step.