An entomologist recommended a program to reduce a certain invasive beetle population in an area. The given function \(\mathrm{f(x) =...

1. TRANSLATE the problem information

• Given function: \(\mathrm{f(x) = 4{,}000(0.75)^x}\)
• Context: \(\mathrm{x}\) = years after 2012, beetle population estimate
• Question: What does 4,000 represent?

2. INFER the exponential function structure

• This follows the form \(\mathrm{f(x) = a(b)^x}\) where:
  • \(\mathrm{a = 4{,}000}\) (the coefficient)
  • \(\mathrm{b = 0.75}\) (the base)
• In exponential functions, the coefficient represents the initial value

3. TRANSLATE what "initial value" means in context

• Initial value occurs when \(\mathrm{x = 0}\)
• \(\mathrm{x = 0}\) means "0 years after 2012" = the year 2012
• \(\mathrm{f(0) = 4{,}000(0.75)^0}\)
  \(\mathrm{= 4{,}000(1)}\)
  \(\mathrm{= 4{,}000}\)

4. INFER the correct interpretation

• 4,000 is the estimated beetle population in 2012
• This matches Choice A exactly

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not clearly understand what \(\mathrm{x = 0}\) represents in the context, or they confuse the roles of the coefficient (4,000) and the base (0.75) in the exponential function.

Some students think the coefficient might represent the final population or a rate of change rather than recognizing it as the starting value. This conceptual confusion about exponential function parameters leads to uncertainty and potentially selecting any of the incorrect choices.

Second Most Common Error:

Inadequate INFER reasoning: Students may recognize that 0.75 relates to the 25% decrease each year (since \(\mathrm{1 - 0.75 = 0.25 = 25\%}\)) but then incorrectly think that 4,000 must also represent some kind of percentage or rate.

This flawed reasoning might lead them to select Choice C or get confused about what 4,000 could represent as a percentage, leading to guessing.

The Bottom Line:

This problem tests whether students understand the standard form of exponential functions and can correctly interpret parameters within a real-world context. Success requires clearly connecting mathematical structure (\(\mathrm{f(x) = a(b)^x}\)) with contextual meaning (population over time).