The function f is defined by \(\mathrm{f(x) = \frac{1}{10}x - 2}\). What is the y-intercept of the graph of \(\mathrm{y...

1. TRANSLATE the question requirements

• Given information:
  • Function: \(\mathrm{f(x) = \frac{1}{10}x - 2}\)
  • Need to find: y-intercept of \(\mathrm{y = f(x)}\)

• What this means: Find where the graph crosses the y-axis

2. INFER the mathematical approach

• Key insight: The y-intercept occurs where \(\mathrm{x = 0}\)
• Strategy: Substitute \(\mathrm{x = 0}\) into the function to find the y-coordinate

3. SIMPLIFY by substituting x = 0

• \(\mathrm{f(0) = \frac{1}{10}(0) - 2}\)
• \(\mathrm{f(0) = 0 - 2}\)
• \(\mathrm{f(0) = -2}\)

4. TRANSLATE the result to coordinate form

• Since \(\mathrm{y = f(x)}\), when \(\mathrm{x = 0}\), we have \(\mathrm{y = -2}\)
• The y-intercept is the point \(\mathrm{(0, -2)}\)

Answer: B. (0, -2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing x-intercept and y-intercept concepts

Students may think "intercept" means either type and look for where \(\mathrm{y = 0}\) instead of \(\mathrm{x = 0}\). This leads them to solve \(\mathrm{\frac{1}{10}x - 2 = 0}\), getting \(\mathrm{x = 20}\), which doesn't match any answer choice. They might then guess or select Choice A \(\mathrm{(-2, 0)}\) thinking the y-intercept has -2 as the x-coordinate.

Second Most Common Error:

Inadequate INFER reasoning: Misunderstanding what the y-intercept represents

Some students might think the y-intercept is related to the coefficient of x, leading them to focus on \(\mathrm{\frac{1}{10}}\) and select Choice C \(\mathrm{(0, \frac{1}{10})}\) without properly calculating \(\mathrm{f(0)}\).

The Bottom Line:

This problem tests whether students truly understand what a y-intercept means mathematically - it's not just vocabulary, but requires knowing that y-intercepts occur when \(\mathrm{x = 0}\) and being able to execute that substitution correctly.