Question:y = 4x - 60The equation above represents the profit y, in dollars, from a small business when x items...

1. TRANSLATE the question requirements

• Given: \(\mathrm{y = 4x - 60}\) (profit equation)
• Need: Interpretation of y-coordinate of y-intercept in business context
• This means we need to find where the line crosses the y-axis and explain what that means for the business

2. INFER how to find the y-intercept

• The y-intercept occurs when \(\mathrm{x = 0}\)
• We substitute \(\mathrm{x = 0}\) into the equation to find the y-coordinate
• In business terms: \(\mathrm{x = 0}\) means no items are sold

3. SIMPLIFY by substituting x = 0

\(\mathrm{y = 4(0) - 60}\)

\(\mathrm{y = 0 - 60}\)

\(\mathrm{y = -60}\)

4. TRANSLATE the mathematical result into business context

• When \(\mathrm{x = 0}\) (no items sold), \(\mathrm{y = -60}\) (profit is -$60)
• Negative profit means the business loses money
• This $60 loss represents costs the business must pay even when selling nothing
• These are fixed daily costs (rent, utilities, wages, etc.)

5. INFER which answer choice matches this interpretation

• (A) Profit per item sold → This is the slope (4), not y-intercept
• (B) Break-even point → This would be when \(\mathrm{y = 0}\), not the y-intercept
• (C) Initial daily costs → This matches our interpretation ✓
• (D) Maximum profit → Line has positive slope, so no maximum exists

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students find the y-intercept correctly (-60) but misinterpret its business meaning. They might think "negative profit" simply means "bad for business" without understanding it specifically represents fixed costs that occur regardless of sales volume.

This leads to confusion when evaluating answer choices, often resulting in guessing or selecting Choice (A) because they recognize the slope (4) represents "profit per item" and assume that's what the question is asking about.

Second Most Common Error:

Poor INFER reasoning: Students confuse the y-intercept with the break-even point. They know break-even is important in business contexts and might calculate where profit equals zero (\(\mathrm{x = 15}\)) instead of finding where \(\mathrm{x = 0}\).

This may lead them to select Choice (B) or causes confusion about what exactly the question is asking for.

The Bottom Line:

This problem tests whether students can connect algebraic procedures (finding intercepts) with real-world interpretation. The mathematical work is straightforward, but translating mathematical results into meaningful business context requires careful attention to what each variable represents.