The function h is defined by \(\mathrm{h(t) = -2(t - 5)^2 + 10}\). In the coordinate plane, the graph of...
GMAT Advanced Math : (Adv_Math) Questions
The function h is defined by \(\mathrm{h(t) = -2(t - 5)^2 + 10}\). In the coordinate plane, the graph of the function k, where \(\mathrm{y = k(t)}\), is the result of reflecting the graph of \(\mathrm{y = h(t)}\) across the horizontal axis (the t-axis). If the graph of \(\mathrm{y = k(t)}\) passes through the point \(\mathrm{(2, c)}\), what is the value of c?
1. TRANSLATE the transformation information
- Given information:
- Original function: \(\mathrm{h(t) = -2(t - 5)^2 + 10}\)
- New function \(\mathrm{k(t)}\) is \(\mathrm{h(t)}\) reflected across the horizontal axis
- Point (2, c) lies on the graph of \(\mathrm{y = k(t)}\)
- What reflection across horizontal axis means: \(\mathrm{k(t) = -h(t)}\)
2. TRANSLATE the point condition
- Since point (2, c) is on the graph of \(\mathrm{y = k(t)}\), this means when \(\mathrm{t = 2}\), \(\mathrm{y = c}\)
- In function notation: \(\mathrm{k(2) = c}\)
3. INFER the relationship to find c
- We know: \(\mathrm{k(t) = -h(t)}\) and \(\mathrm{k(2) = c}\)
- Therefore: \(\mathrm{c = k(2) = -h(2)}\)
- Strategy: Calculate \(\mathrm{h(2)}\) first, then apply the negative
4. SIMPLIFY to evaluate h(2)
- Substitute \(\mathrm{t = 2}\) into \(\mathrm{h(t) = -2(t - 5)^2 + 10}\):
- \(\mathrm{h(2) = -2(2 - 5)^2 + 10}\)
- \(\mathrm{h(2) = -2(-3)^2 + 10}\)
- \(\mathrm{h(2) = -2(9) + 10}\)
- \(\mathrm{h(2) = -18 + 10}\)
- \(\mathrm{h(2) = -8}\)
5. SIMPLIFY to find c
- \(\mathrm{c = -h(2) = -(-8) = 8}\)
Answer: 8
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students forget or misapply the reflection transformation rule. Instead of \(\mathrm{k(t) = -h(t)}\), they might think \(\mathrm{k(t) = h(t)}\) (no change) or apply some other incorrect transformation.
If they use \(\mathrm{k(t) = h(t)}\), then \(\mathrm{c = h(2) = -8}\), leading to the wrong answer of -8.
Second Most Common Error:
Poor SIMPLIFY execution: Students make sign errors when evaluating \(\mathrm{h(2)}\), particularly with \(\mathrm{(-3)^2 = 9}\) or when handling the multiple negative signs in the expression.
Common mistake: treating \(\mathrm{(-3)^2}\) as -9 instead of +9, or getting confused with the negative coefficient -2, leading to incorrect intermediate calculations and wrong final answers.
The Bottom Line:
This problem tests whether students truly understand function transformations (not just memorize rules) and can carefully track negative signs through multi-step algebraic evaluation. The reflection concept is straightforward, but the execution requires precision with signs.