The function \(\mathrm{f(x) = a(2.2^x + 2.2^b)}\), where a and b are integer constants and 0 lt a lt b....

1. TRANSLATE the problem requirements

• Given information:
  • \(\mathrm{f(x) = a(2.2^x + 2.2^b)}\) where a and b are integers, \(\mathrm{0 \lt a \lt b}\)
  • \(\mathrm{g(x) = a(2.2^x + k)}\) and \(\mathrm{h(x) = a(2.2)^x + m}\) are equivalent to f
  • Need to determine which displays the y-intercept as a "constant or coefficient"

• What this means: We need to find the y-intercept value and see if it appears directly as one of the individual constants (k, m) or coefficients (a) in the equivalent forms

2. INFER the approach and find the y-intercept

• Strategy: Find f(0), then analyze each equivalent form
• Y-intercept occurs when x = 0:
  \(\mathrm{f(0) = a(2.2^0 + 2.2^b)}\)
  \(\mathrm{= a(1 + 2.2^b)}\)
  \(\mathrm{= a + a(2.2^b)}\)

3. INFER the equivalence relationships

• Since g is equivalent to f: \(\mathrm{g(x) = a(2.2^x + k)}\) must equal \(\mathrm{f(x) = a(2.2^x + 2.2^b)}\)
• Therefore: \(\mathrm{k = 2.2^b}\)
• Since h is equivalent to f: \(\mathrm{h(x) = a(2.2)^x + m}\) must equal \(\mathrm{f(x) = a(2.2^x) + a(2.2^b)}\)
• Therefore: \(\mathrm{m = a(2.2^b)}\)

4. INFER whether each function displays the y-intercept

• For function I: \(\mathrm{g(x) = a(2.2^x + k)}\)
  • Y-intercept: \(\mathrm{a + a(2.2^b)}\)
  • Coefficient 'a': just a ≠ \(\mathrm{a + a(2.2^b)}\)
  • Constant 'k': \(\mathrm{2.2^b}\) ≠ \(\mathrm{a + a(2.2^b)}\)
  • Neither displays the y-intercept value

• For function II: \(\mathrm{h(x) = a(2.2)^x + m}\)
  • Y-intercept: \(\mathrm{a + a(2.2^b)}\)
  • Coefficient 'a': just a ≠ \(\mathrm{a + a(2.2^b)}\)
  • Constant 'm': \(\mathrm{a(2.2^b)}\) ≠ \(\mathrm{a + a(2.2^b)}\)
  • Neither displays the y-intercept value

Answer: D. Neither I nor II

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly find the y-intercept as \(\mathrm{a + a(2.2^b)}\) but then incorrectly conclude that since the constant \(\mathrm{m = a(2.2^b)}\) is "part of" the y-intercept, function II displays the y-intercept as a constant.

They think: "The y-intercept is \(\mathrm{a + a(2.2^b)}\), and m equals \(\mathrm{a(2.2^b)}\), so m shows the y-intercept." This misses the crucial point that the y-intercept value must appear as a single constant or coefficient, not as part of a sum.

This may lead them to select Choice B (II only).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "displays the y-intercept as a constant or coefficient" means. They might think it just means the y-intercept can be calculated from the constants/coefficients, rather than requiring the y-intercept value to literally appear as one of those terms.

This conceptual confusion about the question's requirement leads them to incorrectly analyze both functions and may result in selecting Choice C (I and II).

The Bottom Line:

This problem tests precise interpretation of mathematical language. The y-intercept value must literally appear as a single constant or coefficient in the function form, not just be derivable from those terms.