Question: \(\mathrm{q(x) = \frac{128}{2^x}}\) Which table gives three values of x and their corresponding values of q(x) for function q?...

1. TRANSLATE the function into specific calculations

• Given: \(\mathrm{q(x) = \frac{128}{2^x}}\)
• Need to find: \(\mathrm{q(-1)}\), \(\mathrm{q(0)}\), and \(\mathrm{q(1)}\)
• This means substituting each x-value into the function

2. INFER the strategy for handling different exponents

• We need to evaluate \(\mathrm{2^x}\) for three different exponents: -1, 0, and 1
• Each will require different exponent rules
• Start with the most straightforward case first

3. SIMPLIFY each calculation systematically

For x = 0 (easiest case):

\(\mathrm{q(0) = \frac{128}{2^0}}\)
\(\mathrm{= \frac{128}{1}}\)
\(\mathrm{= 128}\)

For x = 1:

\(\mathrm{q(1) = \frac{128}{2^1}}\)
\(\mathrm{= \frac{128}{2}}\)
\(\mathrm{= 64}\)

For x = -1 (requires negative exponent rule):

\(\mathrm{q(-1) = \frac{128}{2^{-1}}}\)
Since \(\mathrm{2^{-1} = \frac{1}{2^1} = \frac{1}{2}}\)
\(\mathrm{q(-1) = 128 \div \frac{1}{2}}\)
\(\mathrm{= 128 \times 2}\)
\(\mathrm{= 256}\)

4. TRANSLATE results back to table format

The three values are: 256, 128, and 64
This matches the pattern in choice D.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Mishandling negative exponents by treating \(\mathrm{2^{-1}}\) as -2 instead of 1/2

Students often think that a negative exponent just makes the result negative, so they calculate:

\(\mathrm{q(-1) = \frac{128}{2^{-1}}}\)
\(\mathrm{= \frac{128}{-2}}\)
\(\mathrm{= -64}\)

This leads them to select Choice A (-64, 128, 64) since the other two calculations would be correct.

Second Most Common Error:

Missing conceptual knowledge: Not remembering that any number to the zero power equals 1

Students might think \(\mathrm{2^0 = 0}\) or \(\mathrm{2^0 = 2}\), leading to incorrect calculations like:

\(\mathrm{q(0) = \frac{128}{0}}\) (undefined) or \(\mathrm{q(0) = \frac{128}{2} = 64}\)

This creates confusion about which table could be correct and often leads to guessing.

The Bottom Line:

This problem tests whether students truly understand exponent rules, particularly negative exponents and the zero exponent rule. The key insight is recognizing that division by a fraction (like 1/2) is the same as multiplication by its reciprocal (2).