4.3.4 journal: shifting functions
scenario: model rocket path
instructions:
view the video found on page 1 of this journal activity.
using the information provided in the video, answer the questions below.
show your work for all calculations.
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

second rocket

third rocket

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.

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)

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)

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)

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

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)

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

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?

b. approximately when does the first rocket land?

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