The function h is defined by \(\mathrm{h(x) = 5(1/4)^{(x - 2)}}\). In the xy-plane, the function is graphed as \(\mathrm{y...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = 5(1/4)^{(x - 2)}}\)
  • Need to find: y-intercept of the graph \(\mathrm{y = h(x)}\)

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

2. INFER the approach

• To find the y-intercept, substitute \(\mathrm{x = 0}\) into the function
• We need to evaluate \(\mathrm{h(0)}\)

3. SIMPLIFY the function evaluation

• Substitute \(\mathrm{x = 0}\):
  \(\mathrm{h(0) = 5(1/4)^{(0 - 2)}}\)
  \(\mathrm{= 5(1/4)^{-2}}\)

• Apply the negative exponent rule:
  \(\mathrm{(1/4)^{-2} = (4/1)^2}\)
  \(\mathrm{= 4^2}\)
  \(\mathrm{= 16}\)

• Complete the calculation:
  \(\mathrm{h(0) = 5 × 16}\)
  \(\mathrm{= 80}\)

Answer: C (80)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that "y-intercept" means "set \(\mathrm{x = 0}\)"

Some students try to find where \(\mathrm{y = 0}\) instead of where \(\mathrm{x = 0}\), confusing y-intercept with x-intercept. This leads to trying to solve \(\mathrm{5(1/4)^{(x-2)} = 0}\), which has no solution since exponential functions are always positive. This causes confusion and typically leads to random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make errors when working with negative exponents

Students might incorrectly calculate \(\mathrm{(1/4)^{-2}}\) as \(\mathrm{1/16}\) instead of \(\mathrm{16}\), forgetting that the negative exponent "flips" the fraction. This would give \(\mathrm{h(0) = 5 × (1/16) = 5/16}\), leading them to select Choice A (5/16).

The Bottom Line:

This problem tests two fundamental skills: translating the concept of y-intercept into a mathematical procedure, and correctly handling negative exponents. Students who struggle with either concept will likely select an incorrect answer or abandon the systematic approach.