\(\mathrm{f(x) = 3|x + 7| - 4}\) The function g is defined by \(\mathrm{g(x) = f(x + 2)}\). For what...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 3|x + 7| - 4}\)
  • \(\mathrm{g(x) = f(x + 2)}\)
  • Need to find where \(\mathrm{g(x)}\) reaches its minimum

2. INFER the most direct approach

• To find \(\mathrm{g(x)}\) explicitly, I need to substitute \(\mathrm{(x + 2)}\) wherever I see x in \(\mathrm{f(x)}\)
• This will give me \(\mathrm{g(x)}\) in a form where I can easily identify the minimum

3. SIMPLIFY to find g(x)

• \(\mathrm{g(x) = f(x + 2) = 3|(x + 2) + 7| - 4}\)
• \(\mathrm{g(x) = 3|x + 9| - 4}\)

4. INFER where the minimum occurs

• For absolute value functions of the form \(\mathrm{a|x + b| + c}\) where \(\mathrm{a \gt 0}\):
  • The minimum occurs when \(\mathrm{|x + b| = 0}\)
  • This happens when \(\mathrm{x + b = 0}\), so \(\mathrm{x = -b}\)
• For \(\mathrm{g(x) = 3|x + 9| - 4}\):
  • Minimum occurs when \(\mathrm{x + 9 = 0}\)
  • Therefore \(\mathrm{x = -9}\)

5. Verify the minimum

• At \(\mathrm{x = -9}\):
  \(\mathrm{g(-9) = 3|(-9) + 9| - 4}\)
  \(\mathrm{= 3(0) - 4}\)
  \(\mathrm{= -4}\)

Answer: (A) -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusion about horizontal shift direction in function transformations

Students often think that \(\mathrm{g(x) = f(x + 2)}\) means "shift the graph of \(\mathrm{f(x)}\) to the RIGHT by 2 units" because they see the "+2" and associate addition with moving right. However, \(\mathrm{f(x + 2)}\) actually shifts LEFT by 2 units.

If they know \(\mathrm{f(x)}\) has its minimum at \(\mathrm{x = -7}\) but incorrectly shift right: \(\mathrm{-7 + 2 = -5}\)

This may lead them to select Choice (C) (-5)

Second Most Common Error:

Inadequate SIMPLIFY execution: Errors in algebraic substitution

Students might make mistakes when substituting \(\mathrm{(x + 2)}\) into \(\mathrm{f(x)}\), potentially getting expressions like \(\mathrm{3|x + 2 + 7|}\) but then incorrectly combining to get \(\mathrm{3|x + 5|}\) instead of \(\mathrm{3|x + 9|}\).

This could lead to finding the minimum at \(\mathrm{x = -5}\) instead of \(\mathrm{x = -9}\).

This may lead them to select Choice (C) (-5)

The Bottom Line:

The key insight is recognizing that function transformations follow counterintuitive rules - \(\mathrm{f(x + c)}\) shifts LEFT, not right. Students need to either master these transformation rules or use the reliable algebraic substitution method to avoid directional confusion.