The function f is defined by \(\mathrm{f(x) = 12(4)^x}\). The function h is a transformation of f such that \(\mathrm{h(x)...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 12(4)^x}\)
  • \(\mathrm{h(x) = f(x - 1/2)}\) (horizontal shift transformation)
  • Need y-intercept of \(\mathrm{h(x)}\)

• What this tells us: The y-intercept occurs where the graph crosses the y-axis, which is at \(\mathrm{x = 0}\)

2. INFER the approach

• To find the y-intercept, I need to evaluate \(\mathrm{h(0)}\)
• Since \(\mathrm{h(x) = f(x - 1/2)}\), this means \(\mathrm{h(0) = f(0 - 1/2) = f(-1/2)}\)
• I need to substitute \(\mathrm{x = -1/2}\) into the original function f

3. TRANSLATE the function evaluation

• \(\mathrm{h(0) = f(-1/2) = 12(4)^{-1/2}}\)
• Now I need to evaluate this exponential expression

4. SIMPLIFY the exponential expression

• First, handle the negative exponent: \(\mathrm{4^{-1/2} = \frac{1}{4^{1/2}}}\)
• Next, evaluate the fractional exponent: \(\mathrm{4^{1/2} = \sqrt{4} = 2}\)
• Therefore: \(\mathrm{4^{-1/2} = \frac{1}{2}}\)

5. SIMPLIFY the final calculation

• \(\mathrm{f(-1/2) = 12 \times \frac{1}{2} = 6}\)
• So \(\mathrm{h(0) = 6}\), meaning the y-intercept is \(\mathrm{(0, 6)}\)

Answer: B (0, 6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what \(\mathrm{h(x) = f(x - 1/2)}\) means and try to evaluate \(\mathrm{h(0)}\) by substituting \(\mathrm{x = 0}\) directly into \(\mathrm{f(x)}\) instead of recognizing they need \(\mathrm{f(-1/2)}\).

This leads them to calculate \(\mathrm{f(0) = 12(4)^0 = 12(1) = 12}\), making them select Choice C (0, 12).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need \(\mathrm{f(-1/2)}\) but make errors when evaluating \(\mathrm{4^{-1/2}}\). They might forget the negative exponent rule or incorrectly evaluate the square root, potentially getting \(\mathrm{4^{-1/2} = -2}\) or some other incorrect value.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests your ability to work with function transformations and negative/fractional exponents. The key insight is recognizing that the transformation shifts the input, so finding \(\mathrm{h(0)}\) requires evaluating the original function at a shifted input value.