\(\mathrm{y = 4(81)^{x/2 + 1}}\) The graph of the given equation in the xy-plane has a y-intercept at \(\mathrm{(0, s)}\)....

1. TRANSLATE the problem requirements

• Given: \(\mathrm{y = 4(81)^{(x/2 + 1)}}\) with y-intercept at \(\mathrm{(0, s)}\)
• Need: Equivalent equation that shows s as a coefficient or constant
• This means: Find s first, then rewrite the equation to display that value

2. TRANSLATE how to find the y-intercept

• Y-intercept occurs when \(\mathrm{x = 0}\)
• Substitute \(\mathrm{x = 0}\) into the original equation:
  \(\mathrm{y = 4(81)^{(0/2 + 1)}}\)
  \(\mathrm{= 4(81)^1}\)
  \(\mathrm{= 4(81)}\)
  \(\mathrm{= 324}\)
• So \(\mathrm{s = 324}\)

3. INFER the strategy needed

• We know \(\mathrm{s = 324}\), but none of the answer choices show "s"
• We need to manipulate \(\mathrm{y = 4(81)^{(x/2 + 1)}}\) to show 324 as a visible coefficient
• Look for forms like "324 × (something)^x"

4. SIMPLIFY using exponent rules

• Start with: \(\mathrm{y = 4(81)^{(x/2 + 1)}}\)
• Apply rule \(\mathrm{a^{(b+c)} = a^b \times a^c}\):
  \(\mathrm{y = 4 \times 81^{(x/2)} \times 81^1}\)
• Multiply the constants: \(\mathrm{y = 324 \times 81^{(x/2)}}\)
• Apply rule \(\mathrm{a^{(bc)} = (a^b)^c}\) to get: \(\mathrm{81^{(x/2)} = (81^{(1/2)})^x}\)
• Since \(\mathrm{81^{(1/2)} = \sqrt{81} = 9}\): \(\mathrm{y = 324(9)^x}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find \(\mathrm{s = 324}\) correctly but don't understand they need to rewrite the original equation to display this value. They look for answer choices containing "s" or get confused about what "displays the value of s" means.

This leads to confusion and guessing among the choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt to rewrite the equation but make errors with exponent rules, particularly confusing \(\mathrm{a^{(b+c)} = a^b \times a^c}\) or incorrectly simplifying \(\mathrm{81^{(1/2)}}\). They might incorrectly conclude that \(\mathrm{81^{(1/2)} = 3}\) instead of 9.

This may lead them to select Choice C (\(\mathrm{324(3)^x}\)).

The Bottom Line:

This problem requires both computational accuracy and conceptual understanding that "displaying s as a coefficient" means algebraically manipulating the equation to make the y-intercept value visible in the new form.