A function h is defined by \(\mathrm{h(x) = (x - 5)^2 + 2}\), which is written in vertex form. A...

1. TRANSLATE the given information

• Given information:
  • \(\mathrm{h(x) = (x - 5)^2 + 2}\) (vertex form)
  • \(\mathrm{g(x) = h(x + 4)}\)
  • Need x-coordinate of vertex of g
• What this tells us: We have a quadratic in vertex form and need to analyze a horizontal transformation

2. INFER the approach

• Since \(\mathrm{h(x)}\) is in vertex form, we can easily identify its vertex
• The transformation \(\mathrm{g(x) = h(x + 4)}\) will affect the vertex position
• We can either: expand \(\mathrm{g(x)}\) directly, or use transformation rules

3. TRANSLATE the vertex of h(x)

• For \(\mathrm{h(x) = (x - 5)^2 + 2}\), comparing to vertex form \(\mathrm{f(x) = a(x - p)^2 + q}\)
• The vertex is at \(\mathrm{(p, q) = (5, 2)}\)
• So the x-coordinate of h's vertex is 5

4. SIMPLIFY to find g(x)

• \(\mathrm{g(x) = h(x + 4) = ((x + 4) - 5)^2 + 2}\)
• \(\mathrm{g(x) = (x + 4 - 5)^2 + 2 = (x - 1)^2 + 2}\)

5. TRANSLATE the vertex of g(x)

• Since \(\mathrm{g(x) = (x - 1)^2 + 2}\) is in vertex form
• The vertex is at \(\mathrm{(1, 2)}\)
• The x-coordinate of g's vertex is 1

6. APPLY CONSTRAINTS to select the correct interval

• Check \(\mathrm{x = 1}\) against each option:
  • A. \(\mathrm{-2 \lt 1 \lt 0}\)? False
  • B. \(\mathrm{0 \lt 1 \lt 2}\)? True ✓
  • C. \(\mathrm{3 \lt 1 \lt 5}\)? False
  • D. \(\mathrm{8 \lt 1 \lt 10}\)? False

Answer: B. \(\mathrm{0 \lt x \lt 2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{g(x) = h(x + 4)}\) represents a horizontal transformation, so they get confused about how to proceed. They might try to substitute specific values or work backwards from the answer choices without a clear strategy.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Transformation direction confusion: Students remember that \(\mathrm{f(x + c)}\) represents a horizontal shift but get the direction wrong, thinking \(\mathrm{h(x + 4)}\) shifts the vertex 4 units to the right instead of left. They calculate the new x-coordinate as \(\mathrm{5 + 4 = 9}\).

This may lead them to select Choice D (\(\mathrm{8 \lt x \lt 10}\)) since 9 falls in that interval.

The Bottom Line:

This problem tests whether students can work with function transformations systematically. The key insight is recognizing that \(\mathrm{g(x) = h(x + 4)}\) creates a predictable change in the vertex position, and there are multiple valid approaches (direct substitution or transformation rules) to find the answer.