For t geq 0, the function p is defined as follows: \(\mathrm{p(t) = 200 \times (1.03)^t}\) Which of the following...

1. TRANSLATE the function form

• Given: \(\mathrm{p(t) = 200 \times (1.03)^t}\)
• This matches the exponential function pattern: \(\mathrm{f(x) = a \cdot b^x}\)
  • Initial value \(\mathrm{(a) = 200}\)
  • Base \(\mathrm{(b) = 1.03}\)
  • Variable in the exponent = t

2. INFER the function behavior

• Since the base \(\mathrm{1.03 \gt 1}\), this is exponential growth
• As t increases, \(\mathrm{(1.03)^t}\) gets larger, so \(\mathrm{p(t)}\) increases
• This eliminates "decreasing" options (A) and (B)

3. INFER exponential vs linear distinction

• Exponential: variable in the exponent (like our function)
• Linear: variable multiplied by a constant rate
• Since t is in the exponent, this is definitely exponential
• This eliminates "linear" option (D)

Answer: C (Increasing exponential)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not connect that "base > 1" means "increasing function"

They might see 1.03 and think "that's close to 1, so maybe it's not really growing much" and incorrectly select a decreasing option. Or they might not remember the rule that exponential functions with base > 1 are always increasing.

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

Second Most Common Error:

Conceptual confusion about function types: Students confuse exponential and linear functions

They might focus on the coefficient 200 and think "this looks like it could be linear" without paying attention to where the variable appears. They don't recognize that having the variable in the exponent is the defining characteristic of exponential functions.

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

The Bottom Line:

Success requires recognizing both the structural form (exponential vs linear) AND the behavioral implication (increasing vs decreasing based on the base value). Many students can identify one aspect but miss the other.