For the linear function f, the graph of \(\mathrm{y = f(x)}\) in the xy-plane has a slope of 39 and...

1. TRANSLATE the problem information

• Given information:
  • Slope of the line = 39
  • Graph passes through point (0, 0)
• What we need: The equation f(x) that defines this linear function

2. INFER the mathematical approach

• Since we have a linear function, we should use slope-intercept form: \(\mathrm{f(x) = mx + b}\)
• We know \(\mathrm{m = 39}\) (the slope)
• We need to find \(\mathrm{b}\) (the y-intercept)

3. INFER the y-intercept value

• The graph passes through \(\mathrm{(0, 0)}\)
• The y-intercept is the y-coordinate when \(\mathrm{x = 0}\)
• Since the point \(\mathrm{(0, 0)}\) is on the line, when \(\mathrm{x = 0}\), \(\mathrm{y = 0}\)
• Therefore, \(\mathrm{b = 0}\)

4. Build the final equation

• Substitute \(\mathrm{m = 39}\) and \(\mathrm{b = 0}\) into \(\mathrm{f(x) = mx + b}\)
• \(\mathrm{f(x) = 39x + 0 = 39x}\)

Answer: D. \(\mathrm{f(x) = 39x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding that "passes through (0, 0)" determines the y-intercept value

Students may think that passing through (0, 0) somehow affects the slope or creates a different form of equation. They might not connect that the y-intercept is simply the y-value when x = 0, leading them to overlook that b = 0. This confusion about the relationship between a point on the line and the equation parameters can cause them to get stuck and guess randomly.

Second Most Common Error:

Conceptual confusion about slope signs: Mixing up positive and negative slopes

Some students may see "slope of 39" but somehow think about negative slopes, especially if they've been working with decreasing functions recently. This misconception would lead them to select Choice A (\(\mathrm{f(x) = -39x}\)).

The Bottom Line:

This problem tests whether students can connect the geometric meaning of "passes through a point" with the algebraic representation in slope-intercept form. The key insight is recognizing that passing through (0, 0) directly gives you the y-intercept.