Question:The exponential function h is defined by \(\mathrm{h(x) = 12 \cdot b^x}\), where b is a positive constant. If \(\mathrm{h(2)...

1. TRANSLATE the given information into a solvable equation

• Given information:
  • Function form: \(\mathrm{h(x) = 12 \cdot b^x}\)
  • Known point: \(\mathrm{h(2) = 48}\)
  • \(\mathrm{b}\) is positive

• TRANSLATE this into: \(\mathrm{48 = 12 \cdot b^2}\)

2. INFER the solution strategy

• To find \(\mathrm{h(3)}\), we first need to determine what \(\mathrm{b}\) is
• Once we know \(\mathrm{b}\), we can substitute any \(\mathrm{x}\)-value into the complete function

3. SIMPLIFY to solve for b

• Start with: \(\mathrm{48 = 12 \cdot b^2}\)
• Divide both sides by 12: \(\mathrm{b^2 = 4}\)
• Take the square root: \(\mathrm{b = 2}\) (positive since \(\mathrm{b \gt 0}\))

4. INFER the complete function and calculate h(3)

• Now we know: \(\mathrm{h(x) = 12 \cdot 2^x}\)
• Substitute \(\mathrm{x = 3}\): \(\mathrm{h(3) = 12 \cdot 2^3}\)
• SIMPLIFY: \(\mathrm{h(3) = 12 \cdot 8 = 96}\)

Answer: 96

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to connect "\(\mathrm{h(2) = 48}\)" with the function definition to create the equation \(\mathrm{48 = 12 \cdot b^2}\). They might recognize that \(\mathrm{h(2)}\) means "substitute 2 for \(\mathrm{x}\)" but fail to set up the equation properly.

Without this crucial first step, they get stuck immediately and resort to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors. Common mistakes include:

• Calculating \(\mathrm{48 \div 12}\) incorrectly
• Forgetting that \(\mathrm{2^3 = 8}\) (not 6)
• Making errors in the final multiplication \(\mathrm{12 \times 8}\)

These arithmetic slips can lead to various incorrect answers that seem reasonable.

The Bottom Line:

This problem tests whether students can work backwards from a known point on an exponential function to determine the base, then use that base to find another point. The key insight is recognizing that finding the unknown parameter comes first.