The function \(\mathrm{g(t)} = -\frac{1}{4}(\mathrm{t} - 5)^2 + 9\) models the height \(\mathrm{g(t)}\), in feet, of a fireworks rocket t...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{g(t)} = -\frac{1}{4}(\mathrm{t} - 5)^2 + 9\) models the height \(\mathrm{g(t)}\), in feet, of a fireworks rocket \(\mathrm{t}\) seconds after launch, where \(0 \leq \mathrm{t} \leq 10\). Which of the following is the best interpretation of the vertex of the graph of \(\mathrm{y} = \mathrm{g(t)}\) in the ty-plane?
- The rocket's maximum height was 9 feet above the ground.
- The rocket's maximum height was 5 feet above the ground.
- The rocket's height was 9 feet above the ground when it was launched.
- The rocket's height was 5 feet above the ground when it was launched.
1. TRANSLATE the function structure
- Given: \(\mathrm{g(t) = -\frac{1}{4}(t - 5)^2 + 9}\)
- This is in vertex form: \(\mathrm{g(t) = a(t - h)^2 + k}\)
- TRANSLATE the components:
- \(\mathrm{a = -\frac{1}{4}}\)
- \(\mathrm{h = 5}\)
- \(\mathrm{k = 9}\)
- Therefore, the vertex is at \(\mathrm{(h, k) = (5, 9)}\)
2. INFER the parabola direction
- Since \(\mathrm{a = -\frac{1}{4} \lt 0}\), the parabola opens downward
- For downward-opening parabolas, the vertex represents the maximum point
- This means \(\mathrm{(5, 9)}\) is the highest point on the graph
3. TRANSLATE vertex coordinates to real-world meaning
- The vertex \(\mathrm{(5, 9)}\) means:
- At \(\mathrm{t = 5}\) seconds: the time when maximum height occurs
- At \(\mathrm{g(t) = 9}\) feet: the maximum height achieved
- TRANSLATE to context: "The rocket reaches its maximum height of 9 feet at \(\mathrm{t = 5}\) seconds"
4. INFER which answer choice matches
- We need the interpretation of what the vertex tells us
- The vertex \(\mathrm{(5, 9)}\) directly tells us the maximum height is 9 feet
- This matches choice (A)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Confusing the coordinates of the vertex \(\mathrm{(5, 9)}\)
Students often mix up which number represents time and which represents height. They might think "the vertex is \(\mathrm{(5, 9)}\), so the maximum height is 5 feet" because they associate the first coordinate with the output rather than the input.
This may lead them to select Choice B (5 feet maximum height)
Second Most Common Error:
Poor TRANSLATE reasoning: Confusing maximum height with initial height
Students correctly identify that 9 is the height value from the vertex, but then misinterpret what this means in context. They think "9 feet" must refer to when the rocket was launched \(\mathrm{(t = 0)}\) rather than the maximum point.
This may lead them to select Choice C (9 feet at launch)
The Bottom Line:
Success requires carefully TRANSLATING between mathematical coordinates and their real-world meanings. The vertex form makes finding the maximum easy, but interpreting what those coordinates represent in the rocket context is where students struggle most.