\(\mathrm{f(x) = 9,000(0.66)^x}\) The given function f models the number of advertisements a company sent to its clients each year,...

1. INFER what we need to find

• The question asks for the interpretation of the y-intercept
• Key insight: The y-intercept occurs where the graph crosses the y-axis, which happens when \(\mathrm{x = 0}\)

2. SIMPLIFY to find the y-intercept value

• Substitute \(\mathrm{x = 0}\) into \(\mathrm{f(x) = 9,000(0.66)^x}\)
• \(\mathrm{f(0) = 9,000(0.66)^0}\)
• Since any number to the power of 0 equals 1: \(\mathrm{(0.66)^0 = 1}\)
• \(\mathrm{f(0) = 9,000(1) = 9,000}\)
• The y-intercept is the point \(\mathrm{(0, 9,000)}\)

3. TRANSLATE the mathematical result into context

• \(\mathrm{x}\) represents the number of years since 1997
• When \(\mathrm{x = 0}\), this means 0 years since 1997
• Therefore, \(\mathrm{x = 0}\) corresponds to the year 1997
• \(\mathrm{f(0) = 9,000}\) means 9,000 advertisements were sent in 1997

4. APPLY CONSTRAINTS to select the correct interpretation

• Looking at the answer choices, we need the one that states:
  • The year 1997 (not about minimum values)
  • The value 9,000 (not 1,708)

Answer: D. The estimated number of advertisements the company sent to its clients in 1997 was 9,000.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly calculate \(\mathrm{f(0) = 9,000}\) but misinterpret what \(\mathrm{x = 0}\) represents in the context. They might think \(\mathrm{x = 0}\) means "the starting point" without connecting it to the specific year 1997, or they get confused about what "years since 1997" actually means.

This leads to confusion when trying to match their mathematical result with the contextual answer choices, causing them to guess or select an answer based on the numerical value alone rather than the complete contextual meaning.

Second Most Common Error:

Conceptual confusion about y-intercept: Students might confuse the y-intercept with other characteristics of the function, such as minimum or maximum values. Since this is a decreasing exponential function, some students might incorrectly think the y-intercept represents the minimum value over the given domain.

This may lead them to select Choice A or B (which mention minimum values) instead of focusing on what the y-intercept specifically represents.

The Bottom Line:

This problem tests whether students can bridge the gap between mathematical procedures (finding where \(\mathrm{x = 0}\)) and real-world interpretation (understanding what "0 years since 1997" means). Success requires both technical accuracy and careful contextual translation.