The function p is linear. The following values are known: \(\mathrm{p(0) = 6}\), \(\mathrm{p(1) = 3}\), and \(\mathrm{p(2) = 0}\)....

1. INFER the solution strategy

• Given: p is linear with specific values, need to find \(\mathrm{q(x) = 2p(x-1) + 3}\)
• Strategy: First find the explicit formula for \(\mathrm{p(x)}\), then work through the composition

2. SIMPLIFY to find p(x) using point-slope method

• Using points \(\mathrm{(0, 6)}\) and \(\mathrm{(1, 3)}\):
  • Slope: \(\mathrm{m = \frac{3-6}{1-0} = -3}\)
  • From \(\mathrm{p(0) = 6}\): \(\mathrm{b = 6}\)
  • Therefore: \(\mathrm{p(x) = -3x + 6}\)

• Verify with third point: \(\mathrm{p(2) = -3(2) + 6 = 0}\) ✓

3. SIMPLIFY to find p(x-1)

• Substitute \(\mathrm{(x-1)}\) for x in \(\mathrm{p(x) = -3x + 6}\):
  \(\mathrm{p(x-1) = -3(x-1) + 6}\)
• Distribute: \(\mathrm{p(x-1) = -3x + 3 + 6 = -3x + 9}\)

4. SIMPLIFY the final transformation

• \(\mathrm{q(x) = 2p(x-1) + 3 = 2(-3x + 9) + 3}\)
• Distribute: \(\mathrm{q(x) = -6x + 18 + 3 = -6x + 21}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students may try to work directly with \(\mathrm{q(x) = 2p(x-1) + 3}\) using the given point values without first finding the explicit formula for \(\mathrm{p(x)}\). They might substitute specific values like \(\mathrm{q(0) = 2p(-1) + 3}\) but can't find \(\mathrm{p(-1)}\) from the given information.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find \(\mathrm{p(x) = -3x + 6}\) but make algebraic errors when finding \(\mathrm{p(x-1)}\). A common mistake is computing \(\mathrm{p(x-1) = -3x - 3 + 6 = -3x + 3}\) (sign error when distributing the negative), leading to \(\mathrm{q(x) = 2(-3x + 3) + 3 = -6x + 9}\).

This may lead them to select Choice A (-6x + 9).

The Bottom Line:

This problem tests whether students can systematically work through function composition by first establishing the explicit form of the given function, then carefully executing the algebraic transformations without sign errors.