The table below shows some values of x and the corresponding values of \(\mathrm{g(x)}\) for the exponential function g.x\(\mathrm{g(x)}\)0121826Which...

1. TRANSLATE the problem information

• Given information:
  • Table showing three \(\mathrm{(x, g(x))}\) coordinate pairs: \(\mathrm{(0, 12)}\), \(\mathrm{(1, 8)}\), \(\mathrm{(2, 6)}\)
  • Four possible exponential equations as answer choices
  • Need to find which equation produces these exact values
• What this tells us: The correct equation must work for ALL three given points

2. INFER the most efficient approach

• Strategy: Test each answer choice systematically using the given points
• Start with \(\mathrm{x = 0}\) since any number to the 0 power equals 1, making calculations simpler
• Use process of elimination - once a choice fails for any point, we can eliminate it

3. SIMPLIFY by testing \(\mathrm{x = 0}\) first (where \(\mathrm{g(0) = 12}\))

Remember that any base raised to the 0 power equals 1:

• (A) \(\mathrm{g(0) = 12(\frac{3}{4})^0 = 12(1) = 12}\) ✓
• (B) \(\mathrm{g(0) = 8(\frac{1}{2})^0 + 4 = 8(1) + 4 = 12}\) ✓
• (C) \(\mathrm{g(0) = 15(\frac{2}{3})^0 - 3 = 15(1) - 3 = 12}\) ✓
• (D) \(\mathrm{g(0) = 16(\frac{1}{2})^0 - 4 = 16(1) - 4 = 12}\) ✓

All four choices work for \(\mathrm{x = 0}\), so we need the next test point.

4. SIMPLIFY by testing \(\mathrm{x = 1}\) (where \(\mathrm{g(1) = 8}\))

Now any base to the first power equals itself:

• (A) \(\mathrm{g(1) = 12(\frac{3}{4})^1 = 12 \times \frac{3}{4} = 9 \neq 8}\) → Eliminate A
• (B) \(\mathrm{g(1) = 8(\frac{1}{2})^1 + 4 = 8 \times \frac{1}{2} + 4 = 4 + 4 = 8}\) ✓
• (C) \(\mathrm{g(1) = 15(\frac{2}{3})^1 - 3 = 15 \times \frac{2}{3} - 3 = 10 - 3 = 7 \neq 8}\) → Eliminate C
• (D) \(\mathrm{g(1) = 16(\frac{1}{2})^1 - 4 = 16 \times \frac{1}{2} - 4 = 8 - 4 = 4 \neq 8}\) → Eliminate D

Only choice B works for both \(\mathrm{x = 0}\) and \(\mathrm{x = 1}\).

5. SIMPLIFY by verifying with \(\mathrm{x = 2}\) (where \(\mathrm{g(2) = 6}\))

Let's confirm choice B works for the final point:

• \(\mathrm{g(2) = 8(\frac{1}{2})^2 + 4 = 8 \times \frac{1}{4} + 4 = 2 + 4 = 6}\) ✓

Perfect! Choice B satisfies all three given points.

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make arithmetic errors when computing exponential expressions, particularly with fractional bases and multiple operations.

For example, when testing choice B at \(\mathrm{x = 1}\), they might calculate:

\(\mathrm{g(1) = 8(\frac{1}{2})^1 + 4 = \frac{8}{2} + 4 = 6}\) (incorrect)

Instead of the correct: \(\mathrm{g(1) = 8(\frac{1}{2})^1 + 4 = 4 + 4 = 8}\)

These calculation errors lead them to eliminate the correct answer and potentially select Choice A (\(\mathrm{g(x) = 12(\frac{3}{4})^x}\)) or guess randomly.

Second Most Common Error:

Poor INFER reasoning: Students don't develop a systematic testing strategy and either test points randomly or give up after the first test (\(\mathrm{x = 0}\)) shows all choices work.

Without recognizing that they need to test additional points when multiple choices satisfy the first condition, they become confused about how to proceed and may guess among the remaining choices.

The Bottom Line:

This problem rewards systematic thinking and careful arithmetic. The key insight is that finding the y-intercept (\(\mathrm{x = 0}\)) is just the starting point - you must continue testing until only one choice remains.