Question:The function f is defined by \(\mathrm{f(x) = p \cdot 3^{(x + 2)} - 1}\), where p is a constant....
GMAT Advanced Math : (Adv_Math) Questions
The function f is defined by \(\mathrm{f(x) = p \cdot 3^{(x + 2)} - 1}\), where p is a constant. The graph of \(\mathrm{y = f(x)}\) in the xy-plane is translated 3 units to the right and 4 units up to produce the graph of \(\mathrm{y = g(x)}\). Which equation defines g?
1. TRANSLATE the transformation description
- Given information:
- Original function: \(\mathrm{f(x) = p \cdot 3^{(x + 2)} - 1}\)
- Transform: 3 units right and 4 units up
- What this tells us:
- Right translation means replace \(\mathrm{x}\) with \(\mathrm{(x - 3)}\)
- Up translation means add 4 to the result
- So \(\mathrm{g(x) = f(x - 3) + 4}\)
2. SIMPLIFY by computing f(x - 3)
- Substitute \(\mathrm{(x - 3)}\) everywhere we see \(\mathrm{x}\) in the original function:
- \(\mathrm{f(x - 3) = p \cdot 3^{((x - 3) + 2)} - 1}\)
- \(\mathrm{f(x - 3) = p \cdot 3^{(x - 3 + 2)} - 1}\)
- \(\mathrm{f(x - 3) = p \cdot 3^{(x - 1)} - 1}\)
3. Complete the transformation
- Add the vertical translation:
- \(\mathrm{g(x) = f(x - 3) + 4}\)
- \(\mathrm{g(x) = [p \cdot 3^{(x - 1)} - 1] + 4}\)
- \(\mathrm{g(x) = p \cdot 3^{(x - 1)} + 3}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students confuse the direction of horizontal translations, thinking "3 units right" means \(\mathrm{g(x) = f(x + 3) + 4}\) instead of \(\mathrm{g(x) = f(x - 3) + 4}\).
When they compute \(\mathrm{f(x + 3) = p \cdot 3^{((x + 3) + 2)} - 1 = p \cdot 3^{(x + 5)} - 1}\), then add 4 to get \(\mathrm{g(x) = p \cdot 3^{(x + 5)} + 3}\).
This may lead them to select Choice A (\(\mathrm{p \cdot 3^{(x + 5)} + 3}\)).
Second Most Common Error:
Incomplete SIMPLIFY execution: Students correctly set up \(\mathrm{g(x) = f(x - 3) + 4}\) and find \(\mathrm{f(x - 3) = p \cdot 3^{(x - 1)} - 1}\), but make an arithmetic error when adding the vertical shift, getting \(\mathrm{g(x) = p \cdot 3^{(x - 1)} - 3}\) instead of \(\mathrm{g(x) = p \cdot 3^{(x - 1)} + 3}\).
This may lead them to select Choice C (\(\mathrm{p \cdot 3^{(x - 1)} - 3}\)).
The Bottom Line:
The trickiest part is remembering that horizontal translations work "backwards" - moving right means subtracting from \(\mathrm{x}\). Students who memorize the transformation pattern \(\mathrm{g(x) = f(x - h) + k}\) avoid this confusion.