A function v estimates the value of a machine that was $10,000 in 2015. Each year from 2015 to 2027,...

1. TRANSLATE the problem information

• Given information:
  • Initial value: $10,000 in 2015
  • Decreases by 8% each year
  • Need v(x) = value after x years

• What 'decreases by 8%' means: The machine retains \(\mathrm{100\% - 8\% = 92\%}\) of its value each year

2. INFER the mathematical approach

• This is an exponential decay situation
• General form: \(\mathrm{v(x) = initial\ value \times (growth\ factor)^x}\)
• Growth factor = retention rate = \(\mathrm{0.92}\) (since it keeps 92% each year)

3. TRANSLATE into final equation

• Initial value: 10,000
• Growth factor: 0.92
• Therefore: \(\mathrm{v(x) = 10,000(0.92)^x}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the decay rate with what belongs in the exponential function. They see 'decreases by 8%' and think the growth factor should be \(\mathrm{0.08}\) or just 8.

• Using 0.08: They think since it decreases by 8%, the factor is 0.08
• Using 8: They see '8%' and drop the decimal/percent conversion entirely

This may lead them to select Choice A (\(\mathrm{v(x) = 10,000(8)^x}\)) or Choice B (\(\mathrm{v(x) = 10,000(0.08)^x}\))

Second Most Common Error:

Conceptual confusion about growth vs. decay: Students misunderstand the direction of change and think '8% change' means the value grows by 8%, so they add 8% to 100%.

This reasoning leads to \(\mathrm{100\% + 8\% = 108\% = 1.08}\) as the growth factor.

This may lead them to select Choice C (\(\mathrm{v(x) = 10,000(1.08)^x}\))

The Bottom Line:

The key insight is distinguishing between the decay rate (8%) and the retention rate (92%). In exponential functions, you need the retention rate - what fraction of the value remains after each time period.