The function f is defined by \(\mathrm{f(x) = 4x^3}\). In the xy-plane, the graph of \(\mathrm{y = g(x)}\) is the...
GMAT Advanced Math : (Adv_Math) Questions
The function f is defined by \(\mathrm{f(x) = 4x^3}\). In the xy-plane, the graph of \(\mathrm{y = g(x)}\) is the result of stretching the graph of \(\mathrm{y = f(x)}\) vertically by a factor of 3. Which equation defines function g?
1. TRANSLATE the transformation description
- Given information:
- Original function: \(\mathrm{f(x) = 4x^3}\)
- Transformation: "stretched vertically by a factor of 3"
2. INFER what vertical stretching means mathematically
- Vertical stretching by factor k means: \(\mathrm{g(x) = k \cdot f(x)}\)
- Since our stretch factor is 3: \(\mathrm{g(x) = 3 \cdot f(x)}\)
3. SIMPLIFY by substituting and multiplying
- \(\mathrm{g(x) = 3 \cdot f(x)}\)
- \(\mathrm{g(x) = 3 \cdot (4x^3)}\)
- \(\mathrm{g(x) = 12x^3}\)
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students think "stretch by factor 3" means the coefficient becomes 3, not that we multiply the entire function by 3.
They reason: "The original has coefficient 4, stretching by 3 gives coefficient 3." This leads them to select Choice C (\(\mathrm{3x^3}\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Students confuse the direction of stretching and think "by a factor of 3" means dividing instead of multiplying.
They calculate \(\mathrm{g(x) = f(x)/3 = (4x^3)/3 = (4/3)x^3}\). This leads them to select Choice A (\(\mathrm{(4/3)x^3}\)).
The Bottom Line:
This problem tests whether students truly understand that vertical stretching multiplies the entire function output, not just changes one coefficient. The key insight is that stretching "by factor 3" means "3 times as tall," which translates to multiplying by 3.