A certain college had 3,000 students enrolled in 2015. The college predicts that after 2015, the number of students enrolled...

1. TRANSLATE the problem information

• Given information:
  • Initial enrollment in 2015: 3,000 students
  • Each year after 2015: enrollment decreases by 2%
  • Need function \(\mathrm{f(x)}\) where \(\mathrm{x}\) = years after 2015

2. INFER the mathematical model needed

• This describes exponential decay - the population decreases by the same percentage each year
• Key insight: If enrollment decreases by 2% each year, then each year it retains 98% of the previous year's value
• This means each year we multiply by 0.98

3. INFER the exponential decay formula structure

• General form: \(\mathrm{f(x) = a(1-r)^x}\)
  • \(\mathrm{a}\) = initial value
  • \(\mathrm{r}\) = decay rate (as decimal)
  • \(\mathrm{x}\) = time periods after initial

4. TRANSLATE our specific values into the formula

• \(\mathrm{a = 3{,}000}\) (initial enrollment)
• \(\mathrm{r = 0.02}\) (2% decay rate)
• \(\mathrm{(1-r) = 1 - 0.02 = 0.98}\)

5. Build the final function

• \(\mathrm{f(x) = 3{,}000(0.98)^x}\)

Answer: D. \(\mathrm{f(x) = 3{,}000(0.98)^x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret what "2% decrease" means mathematically. They think that if something decreases by 2%, you multiply by 0.02 rather than understanding that you multiply by (1 - 0.02) = 0.98.

Their reasoning: "It decreases by 2%, so I multiply by 0.02."

This leads them to select Choice C (\(\mathrm{f(x) = 3{,}000(0.02)^x}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify 0.98 as the multiplier but get confused about the structure of exponential functions, placing the base and coefficient in wrong positions.

This may lead them to select Choice B (\(\mathrm{f(x) = 0.98(3{,}000)^x}\)) where they've swapped the positions of the initial value and the decay factor.

The Bottom Line:

The key insight is understanding that a percentage decrease means you keep the remaining percentage, not multiply by the decrease percentage itself. This problem tests whether students truly understand exponential decay versus just memorizing formulas.