\(\mathrm{g(x) = 11(\frac{1}{12})^x}\). If the given function g is graphed in the xy-plane, where \(\mathrm{y = g(x)}\), what is the...

Step-by-Step Solution

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = 11(1/12)^x}\) graphed in the xy-plane where \(\mathrm{y = g(x)}\)
• Need to find: The y-intercept of the graph

2. INFER what y-intercept means

• The y-intercept is the point where the graph crosses the y-axis
• This happens when the x-coordinate is 0
• Therefore, I need to find the point \(\mathrm{(0, g(0))}\)

3. SIMPLIFY by substituting x = 0

• Substitute \(\mathrm{x = 0}\) into the function:
  \(\mathrm{g(0) = 11(1/12)^0}\)

• Apply the zero exponent rule: Any nonzero number to the 0th power equals 1
  \(\mathrm{(1/12)^0 = 1}\)

• Complete the calculation:
  \(\mathrm{g(0) = 11 \times 1 = 11}\)

4. Express as coordinate point

• The y-intercept is the point \(\mathrm{(0, 11)}\)

Answer: A. \(\mathrm{(0, 11)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Zero exponent rule

Students who don't remember or understand that any nonzero number raised to the 0th power equals 1 might:

• Think \(\mathrm{(1/12)^0 = 1/12}\), leading to \(\mathrm{g(0) = 11 \times (1/12) \approx 0.92}\)
• Think \(\mathrm{(1/12)^0 = 12}\), leading to \(\mathrm{g(0) = 11 \times 12 = 132}\)

This may lead them to select Choice B \(\mathrm{(0, 132)}\) or get confused and guess.

Second Most Common Error:

Weak TRANSLATE skill: Misunderstanding y-intercept

Students might not clearly understand what "y-intercept" means and try various approaches:

• Confuse y-intercept with x-intercept and try to solve \(\mathrm{g(x) = 0}\)
• Think they need to find where \(\mathrm{y = 1}\) instead of where \(\mathrm{x = 0}\)

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students understand the fundamental definition of y-intercept and remember the zero exponent rule - two basic but crucial concepts that must be solid for success with exponential functions.