\(\mathrm{f(x) = 4x^2 + 64x + 262}\) The function g is defined by \(\mathrm{g(x) = f(x + 5)}\). For what...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 4x^2 + 64x + 262}\)
  • \(\mathrm{g(x) = f(x + 5)}\)
  • Need to find where \(\mathrm{g(x)}\) reaches its minimum

2. INFER the approach needed

• The notation \(\mathrm{g(x) = f(x + 5)}\) means: wherever we see \(\mathrm{x}\) in \(\mathrm{f(x)}\), replace it with \(\mathrm{(x + 5)}\)
• To find where \(\mathrm{g(x)}\) is minimized, we need the explicit form of \(\mathrm{g(x)}\) first
• Since \(\mathrm{g(x)}\) will be quadratic with positive leading coefficient, it will have a minimum at its vertex

3. SIMPLIFY to find g(x) explicitly

• Substitute \(\mathrm{(x + 5)}\) for \(\mathrm{x}\) in \(\mathrm{f(x)}\):
  \(\mathrm{g(x) = 4(x + 5)^2 + 64(x + 5) + 262}\)

• Expand \(\mathrm{(x + 5)^2}\):
  \(\mathrm{(x + 5)^2 = x^2 + 10x + 25}\)

• Substitute:
  \(\mathrm{g(x) = 4(x^2 + 10x + 25) + 64(x + 5) + 262}\)

• Distribute:
  \(\mathrm{g(x) = 4x^2 + 40x + 100 + 64x + 320 + 262}\)

• Combine like terms:
  \(\mathrm{g(x) = 4x^2 + 104x + 682}\)

4. INFER how to find the minimum

• For quadratic \(\mathrm{ax^2 + bx + c}\) with \(\mathrm{a \gt 0}\), minimum occurs at \(\mathrm{x = \frac{-b}{2a}}\)
• Here: \(\mathrm{a = 4}\), \(\mathrm{b = 104}\), so minimum at:
  \(\mathrm{x = \frac{-104}{2 \times 4}}\)
  \(\mathrm{= \frac{-104}{8}}\)
  \(\mathrm{= -13}\)

Answer: A. -13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students see that \(\mathrm{f(x)}\) has its minimum at:
\(\mathrm{x = \frac{-64}{2 \times 4}}\)
\(\mathrm{= -8}\)
and mistakenly think this is what the problem is asking for.

They don't properly process that \(\mathrm{g(x) = f(x + 5)}\) creates a different function entirely. They might think "the problem mentions \(\mathrm{f(x)}\), so I'll just find where \(\mathrm{f(x)}\) is minimized" without recognizing that the question specifically asks for \(\mathrm{g(x)}\)'s minimum.

This leads them to select Choice B (-8).

Second Most Common Error:

Conceptual confusion about function transformations: Students might confuse \(\mathrm{g(x) = f(x + 5)}\) with \(\mathrm{g(x) = f(x - 5)}\), thinking the transformation shifts in the opposite direction.

If they incorrectly work with \(\mathrm{f(x - 5)}\) instead of \(\mathrm{f(x + 5)}\), their algebra will be wrong from the start, and they might end up with a different vertex location altogether.

This confusion combined with calculation errors may lead them to select Choice D (-3).

The Bottom Line:

The key challenge is correctly interpreting the function composition notation and understanding that \(\mathrm{g(x) = f(x + 5)}\) creates an entirely new quadratic function that must be expanded and analyzed separately from the original \(\mathrm{f(x)}\).