Each year, the value of an investment increases by 0.49% of its value the previous year. Which of the following...

1. TRANSLATE the problem information

• Given information:
  • Each year, the investment increases by \(0.49\%\) of its value from the previous year
• What this tells us: If the current value is V, next year's value = \(\mathrm{V + 0.49\% \text{ of } V}\)

2. INFER what type of mathematical relationship this creates

• When something increases by a fixed percentage each year, we multiply by the same factor each year
• Next year's value = \(\mathrm{V \times (1 + 0.0049)}\) = \(\mathrm{V \times 1.0049}\)
• This creates a pattern: \(\mathrm{V_0, V_0(1.0049), V_0(1.0049)^2, V_0(1.0049)^3, ...}\)

3. INFER the function type and behavior

• The general form is \(\mathrm{V(t) = V_0(1.0049)^t}\) - this is exponential (variable in the exponent)
• Since \(1.0049 \gt 1\), each year the value gets larger - the function is increasing
• Therefore: increasing exponential function

Answer: C. Increasing exponential

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse exponential and linear growth patterns

Many students think: "The investment increases each year by some amount, so it must be linear growth." They fail to recognize that percentage-based growth (where you add a percent of the current value) creates exponential relationships, while fixed-amount growth (adding the same dollar amount each year) creates linear relationships.

This may lead them to select Choice D (Increasing linear)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "\(0.49\%\) increase" means

Some students might think a small percentage like \(0.49\%\) means the function is decreasing, not understanding that any positive percentage growth creates an increasing function.

This may lead them to select Choice A (Decreasing exponential)

The Bottom Line:

The key insight is recognizing that percentage-based growth always creates exponential relationships. When something grows by a fixed percentage each period, you're multiplying by the same factor repeatedly - that's the definition of exponential growth.