The function g is defined by \(\mathrm{g(x) = a\sqrt{x + 16} + b}\), where a and b are real constants....

1. TRANSLATE the problem conditions

• Given information:
  • Function: \(\mathrm{g(x) = a\sqrt{x + 16} + b}\)
  • Point on graph: \(\mathrm{(-16, 0)}\)
  • Additional condition: \(\mathrm{g(0) \lt 0}\)

• What this tells us:
  • We need to find relationships between constants a and b
  • The point condition gives us \(\mathrm{g(-16) = 0}\)
  • The inequality gives us information about the sign of \(\mathrm{g(0)}\)

2. INFER the strategic approach

• Use the point condition first since it will give us a direct equation
• Then use the inequality condition to get information about the remaining unknown
• Compare a and b once we know their values or signs

3. SIMPLIFY using the point condition

• Since \(\mathrm{(-16, 0)}\) lies on the graph: \(\mathrm{g(-16) = 0}\)
• Calculate: \(\mathrm{g(-16) = a\sqrt{-16 + 16} + b}\)
  \(\mathrm{= a\sqrt{0} + b}\)
  \(\mathrm{= a(0) + b}\)
  \(\mathrm{= b}\)
• Therefore: \(\mathrm{b = 0}\)

4. SIMPLIFY using the inequality condition

• We know \(\mathrm{g(0) \lt 0}\), so calculate \(\mathrm{g(0)}\):
• \(\mathrm{g(0) = a\sqrt{0 + 16} + b}\)
  \(\mathrm{= a\sqrt{16} + 0}\)
  \(\mathrm{= a(4)}\)
  \(\mathrm{= 4a}\)
• Since \(\mathrm{g(0) \lt 0}\): \(\mathrm{4a \lt 0}\)
• Therefore: \(\mathrm{a \lt 0}\)

5. INFER the final relationship

• We found: \(\mathrm{a \lt 0}\) and \(\mathrm{b = 0}\)
• Comparing these: \(\mathrm{a \lt 0 \lt b}\)
• Therefore: \(\mathrm{a \lt b}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly evaluate the square root expressions, particularly confusing \(\mathrm{\sqrt{0}}\) or making arithmetic errors with \(\mathrm{\sqrt{16}}\).

For example, they might think \(\mathrm{\sqrt{0} = \text{undefined}}\) or \(\mathrm{\sqrt{16} = 8}\), leading to wrong values for the constants. This creates confusion about the relationship between a and b, potentially leading them to select Choice A \(\mathrm{(a = b)}\) or Choice C \(\mathrm{(a \gt b)}\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "passes through the point \(\mathrm{(-16, 0)}\)" means, failing to set up the equation \(\mathrm{g(-16) = 0}\) correctly.

Without this crucial first step, they cannot determine that \(\mathrm{b = 0}\), making it impossible to establish the correct relationship between a and b. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires careful function evaluation and systematic use of given conditions. Students must recognize that each piece of information gives them specific equations or inequalities to work with, then combine these findings logically.