Function g is defined by \(\mathrm{g(x) = a(x - h)^2 + k}\), where a, h, and k are constants. In...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = a(x - h)^2 + k}\) (vertex form)
  • \(\mathrm{y = g(x + 3)}\) has vertex at \(\mathrm{(1, -8)}\)
  • \(\mathrm{hk = -32}\)
• What this tells us: We need to find how the transformation \(\mathrm{g(x + 3)}\) affects the vertex location

2. INFER the approach

• Since \(\mathrm{g(x + 3)}\) represents a horizontal transformation, we need to rewrite it in vertex form
• Strategy: Substitute \(\mathrm{(x + 3)}\) into the original function and rearrange to identify the new vertex

3. SIMPLIFY the transformation

• Start with \(\mathrm{g(x) = a(x - h)^2 + k}\)
• Substitute: \(\mathrm{g(x + 3) = a((x + 3) - h)^2 + k}\)
• Rearrange: \(\mathrm{g(x + 3) = a(x + 3 - h)^2 + k = a(x - (h - 3))^2 + k}\)

4. TRANSLATE the vertex form

• The function \(\mathrm{g(x + 3) = a(x - (h - 3))^2 + k}\) is now in vertex form
• Comparing to standard form \(\mathrm{f(x) = a(x - p)^2 + q}\), we have \(\mathrm{p = h - 3}\) and \(\mathrm{q = k}\)
• Therefore, the vertex is at \(\mathrm{(h - 3, k)}\)

5. APPLY CONSTRAINTS to find the values

• Given vertex at \(\mathrm{(1, -8)}\):
  • \(\mathrm{h - 3 = 1}\), so \(\mathrm{h = 4}\)
  • \(\mathrm{k = -8}\)
• Verify with constraint: \(\mathrm{hk = 4(-8) = -32}\) ✓

Answer: D) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what \(\mathrm{g(x + 3)}\) means and incorrectly assume the vertex of \(\mathrm{g(x + 3)}\) is the same as the vertex of \(\mathrm{g(x)}\).

They might think: "If \(\mathrm{g(x)}\) has vertex at \(\mathrm{(h, k)}\), then \(\mathrm{g(x + 3)}\) also has vertex at \(\mathrm{(h, k)}\)" and set \(\mathrm{h = 1}\), \(\mathrm{k = -8}\). Then checking the constraint: \(\mathrm{hk = 1(-8) = -8 \neq -32}\), which doesn't work. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly understand that \(\mathrm{g(x + 3)}\) represents a transformation but make algebraic errors when expanding \(\mathrm{a((x + 3) - h)^2}\).

They might incorrectly get \(\mathrm{g(x + 3) = a(x - (h + 3))^2 + k}\) instead of \(\mathrm{a(x - (h - 3))^2 + k}\), leading them to think the vertex is at \(\mathrm{(h + 3, k)}\). Setting \(\mathrm{h + 3 = 1}\) gives \(\mathrm{h = -2}\), and with \(\mathrm{k = -8}\), they get \(\mathrm{hk = (-2)(-8) = 16 \neq -32}\). This may lead them to select Choice B (-2) despite the constraint not working.

The Bottom Line:

This problem requires careful attention to how horizontal transformations affect vertex coordinates. The key insight is that \(\mathrm{g(x + 3)}\) shifts the original function, and students must track how this shift changes the vertex location relative to the original parameters.