\(\mathrm{f(x) = 3{,}000(0.75)^x}\) A conservation scientist implemented a program to reduce the population of a certain species in an area....

1. TRANSLATE the function components

• Given function: \(\mathrm{f(x) = 3{,}000(0.75)^x}\)
• \(\mathrm{x}\) = years after 2008
• We need to interpret what 3,000 represents

2. INFER the approach to find meaning

• In exponential functions of the form \(\mathrm{f(x) = a(b)^x}\), the coefficient 'a' represents the initial value
• To find the initial value, we need to determine what happens when \(\mathrm{x = 0}\) (the starting point)

3. SIMPLIFY by evaluating at x = 0

• Substitute \(\mathrm{x = 0}\) into the function:

\(\mathrm{f(0) = 3{,}000(0.75)^0}\)

• Since any number to the power 0 equals 1:

\(\mathrm{f(0) = 3{,}000(1) = 3{,}000}\)

4. TRANSLATE the mathematical result back to context

• \(\mathrm{x = 0}\) means 0 years after 2008, which is the year 2008
• \(\mathrm{f(0) = 3{,}000}\) means the population in 2008 was 3,000
• Therefore, 3,000 represents the initial population in 2008

Answer: D. The estimated initial population for this species and area in 2008

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what the coefficient represents in the exponential function context. They might focus on the decay factor (0.75) and think 3,000 is related to the percent decrease, or they might think it represents the population at the end of the time period rather than the beginning.

This may lead them to select Choice A or B (thinking it's about percent decrease) or Choice C (thinking it's the final population).

Second Most Common Error:

Inadequate INFER execution: Students understand they need to interpret 3,000 but don't recognize that they should evaluate the function at \(\mathrm{x = 0}\) to find the initial value. Instead, they might guess based on the context without doing the mathematical work.

This leads to confusion and random guessing among the answer choices.

The Bottom Line:

This problem tests whether students understand that in exponential functions, the coefficient represents the initial value, and that they can connect the mathematical concept (\(\mathrm{x = 0}\)) to the real-world context (year 2008).