The function f is defined by \(\mathrm{f(x) = (x - 7)(x - 1)(x + 1)}\). In the xy-plane, the graph...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{f}\) is defined by \(\mathrm{f(x) = (x - 7)(x - 1)(x + 1)}\). In the xy-plane, the graph of \(\mathrm{y = h(x)}\) is the result of reflecting the graph of \(\mathrm{y = f(x)}\) across the x-axis. What is the value of \(\mathrm{h(3)}\)?
1. TRANSLATE the transformation description
- Given information:
- \(\mathrm{f(x) = (x - 7)(x - 1)(x + 1)}\)
- \(\mathrm{h(x)}\) is \(\mathrm{f(x)}\) reflected across the x-axis
- Need to find \(\mathrm{h(3)}\)
- What "reflection across x-axis" means mathematically: If the original graph has point \(\mathrm{(x, y)}\), the reflected graph has point \(\mathrm{(x, -y)}\). Therefore: \(\mathrm{h(x) = -f(x)}\)
2. INFER the solution strategy
- To find \(\mathrm{h(3)}\), I need to use the relationship \(\mathrm{h(3) = -f(3)}\)
- This means I first calculate \(\mathrm{f(3)}\), then apply the negative sign
3. SIMPLIFY to evaluate f(3)
- Substitute \(\mathrm{x = 3}\) into \(\mathrm{f(x) = (x - 7)(x - 1)(x + 1)}\):
\(\mathrm{f(3) = (3 - 7)(3 - 1)(3 + 1)}\)
\(\mathrm{f(3) = (-4)(2)(4)}\)
\(\mathrm{f(3) = -32}\)
4. SIMPLIFY to find the final answer
- Apply the reflection relationship:
\(\mathrm{h(3) = -f(3) = -(-32) = 32}\)
Answer: 32
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not correctly interpret "reflecting across the x-axis" and might think it means \(\mathrm{h(x) = f(-x)}\) (reflection across y-axis) instead of \(\mathrm{h(x) = -f(x)}\).
If they use \(\mathrm{h(x) = f(-x)}\), they would calculate \(\mathrm{f(-3) = (-3-7)(-3-1)(-3+1) = (-10)(-4)(-2) = -80}\), leading to an incorrect answer.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify \(\mathrm{h(3) = -f(3)}\) but make sign errors in the final calculation, getting \(\mathrm{h(3) = -(-32) = -32}\) instead of +32.
This leads them to select a negative answer if available among choices, or causes confusion about the correct sign.
The Bottom Line:
This problem tests whether students can correctly translate geometric transformations into algebraic relationships. The key insight is recognizing that "across the x-axis" affects the y-values (making them negative), not the x-values.