\(\mathrm{f(x) = (x + 7)^2 + 4}\)The function f is defined by the given equation. For what value of x...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{f(x) = (x + 7)^2 + 4}\)
The function f is defined by the given equation. For what value of x does \(\mathrm{f(x)}\) reach its minimum?
1. TRANSLATE the function into recognizable form
- Given: \(\mathrm{f(x) = (x + 7)^2 + 4}\)
- This matches vertex form: \(\mathrm{f(x) = a(x - h)^2 + k}\)
- Rewrite to make the pattern clear: \(\mathrm{f(x) = (x - (-7))^2 + 4}\)
2. INFER the vertex form parameters
- From \(\mathrm{f(x) = (x - (-7))^2 + 4}\), we identify:
- \(\mathrm{a = 1}\) (coefficient of the squared term)
- \(\mathrm{h = -7}\) (the value that makes the parentheses equal zero)
- \(\mathrm{k = 4}\) (the constant term)
3. INFER the direction and type of extreme value
- Since \(\mathrm{a = 1 \gt 0}\), the parabola opens upward
- Upward-opening parabolas have a minimum value (not maximum)
- The minimum occurs at the vertex
4. APPLY CONSTRAINTS to find the minimum location
- For vertex form \(\mathrm{f(x) = a(x - h)^2 + k}\), the vertex is at \(\mathrm{x = h}\)
- Therefore, the minimum occurs at \(\mathrm{x = -7}\)
Answer: -7
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students see \(\mathrm{(x + 7)^2}\) and incorrectly think the vertex occurs at \(\mathrm{x = 7}\) instead of \(\mathrm{x = -7}\).
They reason: "The vertex is where the expression inside the parentheses equals zero, so \(\mathrm{x + 7 = 0}\), which means \(\mathrm{x = -7}\)... wait, that doesn't feel right. Maybe it's \(\mathrm{x = 7}\)?"
This sign confusion leads them to answer 7 instead of -7.
Second Most Common Error:
Missing conceptual knowledge: Students don't recognize this as vertex form and attempt to find the minimum by taking the derivative or completing the square unnecessarily.
This causes them to get stuck in complex algebra when a simple vertex form recognition would solve the problem immediately, leading to confusion and guessing.
The Bottom Line:
The key insight is recognizing vertex form immediately and being careful with the sign when \(\mathrm{(x + 7)^2}\) means \(\mathrm{x - (-7)}\), making \(\mathrm{h = -7}\), not +7.