On April 1, there were 233 views of an advertisement posted on a website. Every 2 days after April 1,...

1. TRANSLATE the problem information

• Given information:
  • April 1 starting point: 233 views
  • Growth pattern: Every 2 days, views increase by 70% of previous amount
  • Need: Function f(x) for views x days after April 1
• What this tells us: This is an exponential growth situation with a specific time interval

2. TRANSLATE the growth language

• "Increased by 70%" means:
  • New amount = Original amount + 70% of original amount
  • New amount = \(100\% + 70\% = 170\%\) of original
  • New amount = \(1.70 \times\) original amount

3. INFER the exponential pattern

• Since growth happens every 2 days:
  • Day 0: 233
  • Day 2: \(233 \times 1.70\)
  • Day 4: \(233 \times (1.70)^2\)
  • Day 6: \(233 \times (1.70)^3\)
• After x days, we've had \(\mathrm{x/2}\) growth periods
• Pattern: \(\mathrm{f(x) = 233(1.70)^{x/2}}\)

4. APPLY CONSTRAINTS to verify answer choices

• Looking at our derived function \(\mathrm{f(x) = 233(1.70)^{x/2}}\):
  • Base must be 1.70 (not 0.70)
  • Exponent must be \(\mathrm{x/2}\) (not \(\mathrm{2x}\))

Answer: C. \(\mathrm{f(x) = 233(1.70)^{x/2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "increased by 70%" as "becomes 70% of the original amount"

Students often think "increased by 70%" means the new amount is 70% of the old amount (0.70), when it actually means the new amount is 170% of the old amount (1.70). This fundamental language error leads to using 0.70 as the base instead of 1.70.

This may lead them to select Choice A (\(\mathrm{f(x) = 233(0.70)^{x/2}}\)) or Choice B (\(\mathrm{f(x) = 233(0.70)^{2x}}\)).

Second Most Common Error:

Poor INFER reasoning: Confusing the time interval in the exponent

Students correctly identify 1.70 as the growth factor but struggle with how "every 2 days" translates to the exponent. They might think the growth happens twice as fast rather than understanding that x days contains \(\mathrm{x/2}\) periods of 2-day intervals.

This may lead them to select Choice D (\(\mathrm{f(x) = 233(1.70)^{2x}}\)).

The Bottom Line:

This problem requires careful translation of percentage language and precise understanding of how time intervals work in exponential functions. The key insight is recognizing that "every 2 days" means the growth factor applies \(\mathrm{x/2}\) times over x days.