FINALThe function p is exponential, meaning \(\mathrm{p(x) = k \cdot r^x}\) for some positive constants k and r.It is given...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{p(x) = k \cdot r^x}\) (exponential function with positive k and r)
  • \(\mathrm{p(1) = 36}\)
  • \(\mathrm{p(3) = 324}\)
  • Find \(\mathrm{p(4)}\)
• What this tells us: We have two function values and need to find the constants k and r, then use them to find \(\mathrm{p(4)}\).

2. INFER the strategic approach

• Key insight: Rather than solve for k and r separately, we can use the ratio of function values to eliminate k
• Since \(\mathrm{\frac{p(3)}{p(1)} = \frac{k \cdot r^3}{k \cdot r} = r^2}\), this ratio gives us the base squared
• This is much more efficient than setting up a system of equations

3. SIMPLIFY to find the base r

• Calculate the ratio:
  \(\mathrm{\frac{p(3)}{p(1)} = \frac{324}{36} = 9}\)
• So \(\mathrm{r^2 = 9}\), which means \(\mathrm{r = 3}\) (taking positive root since \(\mathrm{r \gt 0}\))

4. INFER the most efficient path to p(4)

• Now that we know \(\mathrm{r = 3}\), we can find \(\mathrm{p(4)}\) using the exponential pattern
• Since \(\mathrm{p(4) = p(3) \cdot r}\), we have:
  \(\mathrm{p(4) = 324 \cdot 3 = 972}\)

Answer: 972

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve for k and r separately by setting up two equations (\(\mathrm{k \cdot r = 36}\) and \(\mathrm{k \cdot r^3 = 324}\)), then getting bogged down in algebraic manipulation instead of recognizing the elegant ratio approach.

This leads to unnecessarily complex algebra and potential arithmetic errors, causing students to abandon the systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{\frac{p(3)}{p(1)} = r^2 = 9}\), but then make computational errors such as calculating \(\mathrm{324/36}\) incorrectly, or making errors in the final multiplication \(\mathrm{324 \times 3}\).

This leads to getting an incorrect numerical answer even with the right approach.

The Bottom Line:

The key insight is recognizing that ratios of exponential function values eliminate the coefficient k, giving direct access to powers of the base r. Students who miss this strategic insight get trapped in more complex algebraic approaches that increase error opportunities.