The function f is defined by \(\mathrm{f(x) = -9x + 9}\). What is the y-coordinate of the y-intercept of the...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = -9x + 9}\)
• Need to find: y-coordinate of the y-intercept of \(\mathrm{y = f(x)}\)
• What this means: Find the y-value where the graph crosses the y-axis

2. INFER the approach

• The y-intercept occurs when \(\mathrm{x = 0}\) (by definition)
• To find the y-coordinate, substitute \(\mathrm{x = 0}\) into the function
• This gives us \(\mathrm{f(0)}\), which equals the y-coordinate we want

3. SIMPLIFY by substituting x = 0

• \(\mathrm{f(0) = -9(0) + 9}\)
• \(\mathrm{f(0) = 0 + 9}\)
• \(\mathrm{f(0) = 9}\)

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about y-intercept: Students may confuse y-intercept with x-intercept, thinking they need to find where \(\mathrm{y = 0}\) instead of where \(\mathrm{x = 0}\). This fundamental misunderstanding about what 'y-intercept' means leads them to set \(\mathrm{f(x) = 0}\) and solve \(\mathrm{-9x + 9 = 0}\), getting \(\mathrm{x = 1}\). This leads to confusion since the question asks for a y-coordinate, not an x-coordinate, causing them to abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students truly understand the definition of y-intercept and can translate verbal descriptions into mathematical operations. The key insight is recognizing that 'y-intercept' automatically means 'substitute \(\mathrm{x = 0}\).'