The function g is defined by \(\mathrm{g(x) = k \cdot a^x + b}\), where k, a, and b are positive...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = k \cdot a^x + b}\) (k, a, b positive)
  • Horizontal asymptote at \(\mathrm{y = 3}\) as \(\mathrm{x \to -\infty}\)
  • y-intercept of 15
  • Point \(\mathrm{(2, 51)}\) on the graph
  • Need to find \(\mathrm{ab}\)

2. INFER what the horizontal asymptote tells us

• Key insight: As x approaches negative infinity, what happens to \(\mathrm{a^x}\)?
• Since \(\mathrm{a \gt 0}\), when \(\mathrm{x \to -\infty}\), we get \(\mathrm{a^x \to 0}\)
• This means \(\mathrm{g(x) \to k \cdot 0 + b = b}\)
• Therefore, the horizontal asymptote \(\mathrm{y = 3}\) tells us \(\mathrm{b = 3}\)

3. TRANSLATE the y-intercept condition

• Y-intercept means the point where \(\mathrm{x = 0}\)
• \(\mathrm{g(0) = k \cdot a^0 + b = k \cdot 1 + b = k + b = 15}\)
• Since we found \(\mathrm{b = 3}\): \(\mathrm{k + 3 = 15}\)

4. SIMPLIFY to find k

\(\mathrm{k + 3 = 15}\)
\(\mathrm{k = 12}\)

5. TRANSLATE the point condition and SIMPLIFY to find a

• Point \(\mathrm{(2, 51)}\) means \(\mathrm{g(2) = 51}\)
• \(\mathrm{g(2) = k \cdot a^2 + b = 12 \cdot a^2 + 3 = 51}\)
• \(\mathrm{12 \cdot a^2 = 48}\)
• \(\mathrm{a^2 = 4}\)
• \(\mathrm{a = 2}\) (since a must be positive)

6. Calculate the final answer

\(\mathrm{ab = 2 \times 3 = 6}\)

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the behavior as \(\mathrm{x \to -\infty}\) versus \(\mathrm{x \to +\infty}\) for exponential functions. They might think that as \(\mathrm{x \to -\infty}\), the exponential term \(\mathrm{a^x}\) grows large rather than approaches zero.

This leads them to incorrectly conclude that the horizontal asymptote involves both k and b, making the problem much more complex than necessary. Without properly identifying that \(\mathrm{b = 3}\), they cannot systematically solve for the other parameters and end up guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify \(\mathrm{b = 3}\) and \(\mathrm{k = 12}\), but make algebraic errors when solving \(\mathrm{12 \cdot a^2 + 3 = 51}\). Common mistakes include forgetting to subtract 3 first, or incorrectly taking the square root.

This may lead them to find an incorrect value for a, resulting in a wrong value for \(\mathrm{ab}\). They might select Choice A (3) if they use \(\mathrm{a = 1}\), or Choice C (12) if they confuse the product \(\mathrm{ab}\) with just the value of k.

The Bottom Line:

Success on this problem requires understanding how horizontal asymptotes work with exponential functions—specifically that the exponential term vanishes as x approaches negative infinity, leaving only the constant term.