Vertex Answer to an Intercept Question
A physical science class launches a model rocket from a platform 80 feet above the ground and records its height with a motion sensor. A quadratic model fits the data well: h(t) = -16t^2 + 64t + 80, where h is height in feet and t is seconds after launch. A student graphs the function, sees the vertex at (2, 144), and writes, 'The rocket's maximum height is 144 feet, and it is in the air for 2 seconds.' The teacher's actual question was, 'How long is the rocket in the air, and when does it pass 100 feet?' The student's graph shows a second x-intercept at t = -1, which she crossed out without comment. The tutor must decide which parts of the student's answer are right, how to treat the negative solution, and how to guide her through the part about passing 100 feet, which she has not started.
1. Find both x-intercepts and the vertex of h(t), and state the corrected answer to 'how long is the rocket in the air,' with justification.
Read the worked answer
Set h(t) = 0: -16t^2 + 64t + 80 = 0. Divide by -16: t^2 - 4t - 5 = 0, so (t - 5)(t + 1) = 0 and t = 5 or t = -1.
Vertex: t = -b/(2a) = -64/(-32) = 2, and h(2) = -64 + 128 + 80 = 144, so (2, 144).
Interpretation: t = -1 is outside the domain t ≥ 0 for this launch; it is the time the same parabola would have been at ground level if the motion had started earlier. Only t = 5 is physical, so the rocket lands at 5 seconds and is in the air for 5 seconds, not 2.
The student's maximum-height statement is correct but answers a different question. Her error is conflating the vertex (when the rocket is highest, and where its upward motion stops) with the root (when it reaches the ground). A quick check exposes the inconsistency: at t = 2 the model gives 144 feet, so the rocket is clearly not on the ground then.
2. The student asks why she cannot just read the answer off the vertex. Explain, using the structure of the equation, how the vertex and the intercepts each answer a different question.
Read the worked answer
Rewrite in vertex form by completing the square: h(t) = -16(t^2 - 4t) + 80 = -16(t - 2)^2 + 64 + 80 = -16(t - 2)^2 + 144.
This form displays the answers to two different questions at once. The squared term -16(t - 2)^2 is zero exactly when t = 2 and is negative otherwise, so 144 is the maximum height and t = 2 is when it occurs. The roots come from setting the whole expression to zero: -16(t - 2)^2 = -144, (t - 2)^2 = 9, t - 2 = ±3, t = 5 or t = -1.
So the vertex answers 'how high and when is it highest,' while the roots answer 'when is the height zero.' The question 'how long in the air' is a question about the positive root. A reliable habit: before answering, restate the question in terms of which feature of the graph (maximum, zero, or a specific height) it asks for, then match that feature to the algebraic form that reveals it.
3. Find the two times when the rocket is at 100 feet and explain what the pair of answers means, including any assumption you make.
Read the worked answer
Use vertex form: 100 = -16(t - 2)^2 + 144, so -16(t - 2)^2 = -44, (t - 2)^2 = 2.75, t - 2 = ±1.658, giving t ≈ 0.34 s and t ≈ 3.66 s.
Both are positive and inside the flight interval [0, 5], so both are physical. The smaller time is on the way up, the larger on the way down; between them the rocket is above 100 feet (about 3.3 seconds). Checking with the original function: h(0.34) ≈ -1.85 + 21.8 + 80 ≈ 99.9, and h(3.66) ≈ -214.3 + 234.2 + 80 ≈ 99.9, consistent within rounding.
Assumption: the model applies for 0 ≤ t ≤ 5 and the sensor data are not distorted near the launch platform. A useful check is to note that both answers are symmetric about t = 2, the vertex time, which must be true because the parabola is symmetric; an answer that is not symmetric about 2 signals an algebra error.