The function h is defined by \(\mathrm{h(x) = (x + 3)(x - 1)(x - 4)}\). In the xy-plane, the graph...
GMAT Advanced Math : (Adv_Math) Questions
The function h is defined by \(\mathrm{h(x) = (x + 3)(x - 1)(x - 4)}\). In the xy-plane, the graph of \(\mathrm{y = k(x)}\) is the result of translating the graph of \(\mathrm{y = h(x)}\) down 3 units. What is the value of \(\mathrm{k(-1)}\)?
Enter your answer as an integer.
1. TRANSLATE the transformation information
- Given information:
- \(\mathrm{h(x) = (x + 3)(x - 1)(x - 4)}\)
- \(\mathrm{k(x)}\) is \(\mathrm{h(x)}\) translated down 3 units
- Need to find \(\mathrm{k(-1)}\)
- What "down 3 units" means mathematically: \(\mathrm{k(x) = h(x) - 3}\)
2. INFER the solution strategy
- To find \(\mathrm{k(-1)}\), I need to use the relationship \(\mathrm{k(-1) = h(-1) - 3}\)
- This means I must first evaluate \(\mathrm{h(-1)}\), then subtract 3
3. SIMPLIFY by evaluating h(-1)
- Substitute \(\mathrm{x = -1}\) into \(\mathrm{h(x)}\):
\(\mathrm{h(-1) = (-1 + 3)(-1 - 1)(-1 - 4)}\) - SIMPLIFY each factor:
- \(\mathrm{(-1 + 3) = 2}\)
- \(\mathrm{(-1 - 1) = -2}\)
- \(\mathrm{(-1 - 4) = -5}\)
- SIMPLIFY the multiplication:
\(\mathrm{h(-1) = (2)(-2)(-5) = 20}\)
4. Apply the transformation
- \(\mathrm{k(-1) = h(-1) - 3 = 20 - 3 = 17}\)
Answer: 17
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse "down 3 units" with "up 3 units" and write \(\mathrm{k(x) = h(x) + 3}\) instead of \(\mathrm{k(x) = h(x) - 3}\).
With this error, they would calculate \(\mathrm{k(-1) = h(-1) + 3 = 20 + 3 = 23}\). This leads to an incorrect answer of 23.
Second Most Common Error:
Poor SIMPLIFY execution: Students make sign errors when evaluating \(\mathrm{h(-1) = (2)(-2)(-5)}\).
Common mistakes include:
- Getting \(\mathrm{h(-1) = -20}\) (forgetting that negative × negative = positive)
- Making errors in the individual factor calculations
This leads to wrong values for \(\mathrm{h(-1)}\), resulting in incorrect final answers.
The Bottom Line:
This problem tests whether students can accurately translate verbal transformation descriptions into mathematical notation and execute multi-step function evaluation without arithmetic errors. The key insight is recognizing that vertical translations directly affect function outputs by adding or subtracting constants.