The y-intercept of the graph of y = -6x - 32 in the xy-plane is \(\mathrm{(0, y)}\). What is the...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{y = -6x - 32}\)
  • Need to find the y-intercept (0, y)

• What this tells us: The y-intercept occurs when x = 0, so we need to find the y-value at that point.

2. SIMPLIFY by substitution

• Substitute x = 0 into the equation:

\(\mathrm{y = -6(0) - 32}\)

• Perform the arithmetic:

\(\mathrm{y = 0 - 32}\)

\(\mathrm{y = -32}\)

Answer: -32

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "y-intercept" means

Students may think the y-intercept is related to setting y = 0 (finding x-intercept instead) or may not realize they need to substitute x = 0. This leads to confusion and guessing rather than systematic problem-solving.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes during calculation

Students correctly identify they need to substitute x = 0, but make errors like:

• Calculating -6(0) as -6 instead of 0
• Sign errors when computing 0 - 32

This leads to incorrect answers that don't match the correct value of -32.

The Bottom Line:

This problem tests whether students truly understand the definition of y-intercept and can execute basic substitution accurately. The key insight is recognizing that "y-intercept" always means "when x = 0."