A marine biologist implements a containment measure to reduce a jellyfish bloom beginning in 2019. The function \(\mathrm{J(y) = 3,200(0.82)^{(y...

1. TRANSLATE the function structure

• Given function: \(\mathrm{J(y) = 3,200(0.82)^{(y - 2019)}}\)
• This is an exponential function in the form \(\mathrm{A \cdot b^{(time)}}\)
• We need to determine what the coefficient 3,200 represents

2. INFER the strategy to find meaning

• In exponential functions, the coefficient A represents the value when the exponent equals zero
• To find when the exponent equals zero: \(\mathrm{(y - 2019) = 0}\), so \(\mathrm{y = 2019}\)
• This means we should evaluate \(\mathrm{J(2019)}\)

3. SIMPLIFY by substituting the reference year

• \(\mathrm{J(2019) = 3,200(0.82)^{(2019 - 2019)}}\)
• \(\mathrm{J(2019) = 3,200(0.82)^0}\)
• \(\mathrm{J(2019) = 3,200 \times 1 = 3,200}\)

4. TRANSLATE the mathematical result back to context

• Since \(\mathrm{J(2019) = 3,200}\), this means there were 3,200 jellyfish in 2019
• Therefore, 3,200 represents the estimated number of jellyfish in the region in 2019

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand the structure of exponential functions and confuse which parameter represents what. They might think 3,200 represents the final amount or a percentage rather than the initial value.

This confusion often stems from not recognizing that the coefficient in \(\mathrm{A \cdot b^t}\) always represents the value when \(\mathrm{t = 0}\) (the reference point). Students may randomly guess or select Choice B (the 2026 value) thinking the coefficient somehow represents a later time period.

Second Most Common Error:

Missing conceptual knowledge about exponential decay: Students might confuse the coefficient 3,200 with the decay rate information. They see 0.82 and might incorrectly associate 3,200 with percentage decrease rather than understanding that 0.82 (which represents 82% remaining, or 18% decrease) is the decay factor.

This may lead them to select Choice C or Choice D, incorrectly interpreting 3,200 as representing some form of percentage change.

The Bottom Line:

This problem requires students to understand that in exponential models, you must substitute the reference value to find what the coefficient represents. The key insight is recognizing that when the exponent equals zero, the function equals just the coefficient, revealing its meaning as the initial value.