The function p satisfies \(\mathrm{p(2) = 11}\). The graph of \(\mathrm{y = q(x)}\) is obtained by shifting the graph of...

1. TRANSLATE the transformation information

• Given information:
  • \(\mathrm{p(2) = 11}\)
  • Graph of \(\mathrm{q(x)}\) comes from shifting \(\mathrm{p(x)}\) upward 3 units, then downward 9 units
  • Need to find: \(\mathrm{q(2)}\)

• What the shifts mean mathematically:
  • "Upward 3 units" → add 3 to every y-value
  • "Downward 9 units" → subtract 9 from every y-value

2. INFER how to combine the shifts

• These transformations happen in sequence, so we can combine them
• Net vertical shift = \(+3 - 9 = -6\) units
• This means every point on \(\mathrm{p(x)}\) moves down 6 units to create \(\mathrm{q(x)}\)
• Therefore: \(\mathrm{q(x) = p(x) - 6}\)

3. SIMPLIFY to find the specific value

• Since \(\mathrm{q(x) = p(x) - 6}\), we have:
  \(\mathrm{q(2) = p(2) - 6}\)
  \(\mathrm{q(2) = 11 - 6}\)
  \(\mathrm{q(2) = 5}\)

Answer: B (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly understand that upward means +3 and downward means -9, but incorrectly think both shifts work in the same direction, treating the net shift as \(+3 + 9 = +12\) or calculating the net as +6 instead of -6.

If they use +6: \(\mathrm{q(2) = 11 + 6 = 17}\)
This may lead them to select Choice D (17)

Second Most Common Error:

Incomplete INFER reasoning: Students apply only one of the two shifts instead of combining them.

• Applying only the upward shift: \(\mathrm{q(2) = 11 + 3 = 14}\) → Choice C (14)
• Applying only the downward shift: \(\mathrm{q(2) = 11 - 9 = 2}\) → Choice A (2)

The Bottom Line:

This problem tests whether students can systematically handle sequential transformations. The key insight is recognizing that "upward then downward" creates a net effect that can be calculated as a single transformation.