I'm Serena. For a science project, my friend Jack and I are launching three model rockets, one after another. We launch the first rocket, and then 3 seconds later, we launch the next one. And we're launching the final rocket three seconds after that, from a platform that is 20 feet high.

For our project, we need to predict the paths for all three rockets. We also need to estimate when they will all be in the air at the same time. [A graph that shows "Height of rocket (feet)" on the y-axis and "Time (seconds)" on the x-axis is shown. A red downturned parabola is shown and labeled "Path of the first rocket."]

We have calculated the path of the first rocket. It looks like this: a parabola that opens down. The y-axis is the height of the rocket in feet, and the x-axis is the time in seconds.

My friend Jack thinks we need to recalculate the graphs for the other two model rockets. But since the rockets are all the same, I think we can just shift the graph of the first rocket to find the graphs for the other two. [The graph is duplicated in green and shifts to the right, and then again in blue and shifts to the right and up. Then the rockets blast off again.]

What do you think? How can we use the graph of the first rocket to create the graphs of the second and third rockets? When will all three rockets be in the air at the same time?Evaluate the Conjectures:
2. Do you agree with Serena that you can draw the graphs for the other two rockets by shifting the functions? Or do you think that Jack is correct that you need to recalculate the other two? Explain. (2 points)

Analyzing the Data:
Suppose that the path of the first model rocket follows the equation
h(t) = −6 • (t − 3.7)2 + 82.14,
where t is the time in seconds (after the first rocket is launched), and h(t) is the height of each rocket, in feet.

Compare the equation with the graph of the function. Assume this graph is a transformation from f(t) = –6t2. What does the term –3.7 do to the rocket's graph? What does the value t = 3.7 represent in the science project? (What happens to the rocket?)

Again assuming a transformation from f(t) = –6t2, what does the term 82.14 do to the rocket's graph? What does the value h(t) = 82.14 represent in the science project? (What is happening to the rocket?) (2 points)

Serena and Jack launch the second rocket 3 seconds after the first one. How is the graph of the second rocket different from the graph of the first rocket? Describe in terms of the vertical and horizontal shift.

What is the equation of the second rocket?

They launch the third rocket 3 seconds after the second rocket and from a 20-foot-tall platform. What will the graph of the third rocket look like? Describe in terms of the vertical and horizontal shift.

What is the equation of the third rocket?

Answer the following questions about the three rockets. Refer to the graph of rocket heights and times shown above.

a. Approximately when is the third rocket launched?
b. Approximately when does the first rocket land?
c. What is the approximate interval during which all three rockets are in the air?

The Students' Conjectures: Serena and Jack are launching three identical model rockets, each at a different time. Jack says that they need to recalculate the graph each time, but Serena thinks they can just shift the function of the first graph.

1. Complete the table to summarize what you know about each rocket: (3 points: 1 point for each row of the chart)

First rocket

The first rocket is shot from 0 ft off the ground and reaches a vertex of 80 ft in 3.5 seconds before it lands again.

Second rocket

The second rocket is pretty much the same, but because it was fired 3 seconds after the first rocket, the function is shifted 3 units to the right.

Third rocket

The third rocket is shot from a height of 20 feet, 6 seconds after the first graph is shot. Because it landed below the point at which it was fired, it took more time to land, making the function appear slightly different, though it was the same.

Evaluate the Conjectures:

2. Do you agree with Serena that you can draw the graphs for the other two rockets by shifting the functions? Or do you think that Jack is correct that you need to recalculate the other two? Explain. (2 points)

I agree with serena. Because the rockets both have identical paths and times for ascent and descent, and ascend an equal distance, the functions can be shifted to show the exact path of each rocket.

Analyzing the Data:

Suppose that the path of the first model rocket follows the equation

h(t) = −6 • (t − 3.7)2 + 82.14,

where t is the time in seconds (after the first rocket is launched), and h(t) is the height of each rocket, in feet.

3. Compare the equation with the graph of the function. Assume this graph is a transformation from f(t) = –6t2. What does the term –3.7 do to the rocket's graph? What does the value t = 3.7 represent in the science project? (What happens to the rocket?) (2 points)

The value t=3.7 represents the time it took to for the rocket to reach its maximum height.

4. Again assuming a transformation from f(t) = –6t2, what does the term 82.14 do to the rocket's graph? What does the value h(t) = 82.14 represent in the science project? (What is happening to the rocket?) (2 points)

The value t=82.14 represents the maximum height reached by the rocket.

5. Serena and Jack launch the second rocket 3 seconds after the first one. How is the graph of the second rocket different from the graph of the first rocket? Describe in terms of the vertical and horizontal shift. (2 points)

The graph of the second rocket follows an identical path as the first rocket, but because the timer started when the first rocket was fired, and did not lap when the second rocket was fired, the graph for the two rockets were not in the exact same place. The graph for the second function displays the same path, but it is shifted over 3 places to the right.

6. What is the equation of the second rocket? (2 points)

h(t) = −6 • ((t-3) − 3.7)2 + 82.14

7. They launch the third rocket 3 seconds after the second rocket and from a 20-foot-tall platform. What will the graph of the third rocket look like? Describe in terms of the vertical and horizontal shift. (2 points)

The graph of the third rocket will start 6 seconds after the first, and 3 after the second, so on the graph t(time) will equal 6. H (height) starts at 20, so the function shape is identical to the first two rockets, but the function has shifted 3 places to the right from the graph of the second rocket, and 2.5 up.

8. What is the equation of the third rocket? (2 points)

h(t) = −6 • ((t-6) − 3.7)2 + 82.14+ 2.5

9. Answer the following questions about the three rockets. Refer to the graph of rocket heights and times shown above. (3 points: 1 point for each question)

a. Approximately when is the third rocket launched?

at 6 seconds

b. Approximately when does the first rocket land?

7.25 seconds

c. What is the approximate interval during which all three rockets are in the air?

At approximately 6-7 seconds, all three rockets are in the air.

juliestrella juliestrella · Answer 2 · 2021-03-16T15:07:26+01:00

Answer:

8. h(t) = -6 ((t-6) -3.7)^2 + 82.14 +20

9.b) At about 7.4 seconds.

Step-by-step explanation:

Yes, on #7 the function is located 3 places to the right and 2.5 up, but that's just counting the tiny squares. If you look at the number and problem, on #8, you can't add +2.5, because the rocket wasn't on a platform 2.5 feet high. Remember that outside the parentheses + means up and - means down. So add + 20(ft).

I see you already have things covered with another user, but for question 8, and 9. b), try these answers.