For x geq 0, functions p and q are defined by the equations below.Which of the following equations displays, as...

1. INFER the function behavior

• Given information:
  • I. \(\mathrm{p(x) = 50(0.80)^x}\)
  • II. \(\mathrm{q(x) = 50(0.80)^{(x+3)}}\)
  • Domain: \(\mathrm{x ≥ 0}\)

• Since \(\mathrm{0.80 \lt 1}\), both functions are exponential decay functions that decrease as x increases.

2. INFER where maximum occurs

• For decreasing functions on the interval \(\mathrm{[0, ∞)}\), the maximum value occurs at the left endpoint \(\mathrm{x = 0}\).
• We need to find the maximum value for each function, then check if it appears as a constant or coefficient in the original equation.

3. SIMPLIFY to find maximum values

• For I: \(\mathrm{p(0) = 50(0.80)^0}\)
  \(\mathrm{= 50(1)}\)
  \(\mathrm{= 50}\)
• For II: \(\mathrm{q(0) = 50(0.80)^{(0+3)}}\)
  \(\mathrm{= 50(0.80)^3}\)
  \(\mathrm{= 50(0.512)}\)
  \(\mathrm{= 25.6}\)

4. INFER which values are "displayed"

• Key insight: An equation "displays" a value when that exact number appears explicitly as a constant or coefficient in the given form.

• For I: The maximum value is 50, which appears explicitly as the coefficient 50 in \(\mathrm{p(x) = 50(0.80)^x}\) ✓
• For II: The maximum value is 25.6, but this number does not appear anywhere in \(\mathrm{q(x) = 50(0.80)^{(x+3)}}\) ✗

Answer: A (I only)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret what "displays as a constant or coefficient" means, thinking that any value they can calculate from the equation counts.

For function II, they calculate \(\mathrm{q(0) = 25.6}\) and think "well, I can get 25.6 from this equation, so it displays the maximum." They don't realize the question requires the maximum value to appear explicitly as a written number in the original equation form.

This may lead them to select Choice C (I and II).

The Bottom Line:

This problem tests whether students can distinguish between values that are embedded or calculable versus values that are explicitly visible in an equation's given form. The key insight is recognizing that "displaying" a value means it appears as a literal constant or coefficient, not just that it can be derived through calculation.