Quadratic Applications
Warm Up
Look at both categories. Summarize what you found. What characteristic of the equation
determines whether the parabola opens up or opens down?
Trajectory Motion
The function 𝑔 𝑡 = −16𝑡 ! + 140𝑡 + 20 describes the height of the Angry Bird in feet over
time in seconds.
1. Graph on your calculator. Find an appropriate window!
2. What is the height of the Angry Bird at 5 seconds?
3. What is the maximum height of the model rocket? At what time does the rocket reach
this height?
4. When is the Angry Bird exactly 216 feet above the ground?
5. When does the Angry Bird hit the ground?
6. What is the y-intercept? What does it represent?
7. What is the initial upward velocity of the Angry Bird?
Quadratic Application Facts:
ü The points on the graph are (
,
ü If you’re asked to find the…
1.) Time for maximum height –
2.) Maximum height –
) or (
,
)
GENERAL TRAJECTORY MOTION:
𝑓 (𝑡) = −16𝑡 ! + 𝑣! 𝑡 + ℎ! ,
where 𝑣! is the initial velocity and
3.) When the object hit the ground –
ℎ! is the initial height.
4.) Random height given a time –
5.) Random time given a height –
Write a function that represents the trajectory motion described in each problem situation.
1. A catapult hurls a watermelon from a height of 36 feet at an initial velocity of 82 feet per
second.
2. A catapult hurls a cantaloupe from a height of 12 feet at an initial velocity of 47 feet per
second.
3. A basketball is thrown from a height of 7 feet at an initial velocity of 58 feet per second.
4. A football is thrown from a height of 6 feet at an intial velocity of 74 feet per second.
Write a problem situation describing the trajectory motion of the object given the equation.
5. ℎ 𝑡 = −16𝑡 ! + 46𝑡 + 25
6. ℎ 𝑡 = −16𝑡 ! + 110𝑡 + 49
Project Requirements:
ü Choose your equation and create a story around the circumstance. Tell the story by either
creating a children’s book, comic strip, or interactive poster.
ü Answers to the following questions:
o What is the initial velocity of your object?
o What is the initial height of your object?
o When does your object reach a maximum? What is the height?
o When does your object hit the ground?
o What is the domain of your problem situation?
o What is the range of your problem situation?
o Write two extension questions and answer.
• Given a time, find a height.
• Given a height, find a time.
ü Provide a graph of the entire “flight” of your object with clearly labeled axes and scaling.
Your group’s equation:
General scenario:
o
What is the initial velocity of your object?
o
What is the initial height of your object?
o
When does your object reach a maximum?
o
What is the height?
o
When does your object hit the ground?
o
What is the domain of your problem situation?
o
What is the range of your problem situation?
o
Write two extension questions and answer.
o
•
Given a time, find a height.
•
Given a height, find a time.
GRAPH:
Practice / HW
1. A young girl standing on a cliff is throwing stones up into the air so that they land in the ocean below. The height h, in meters, of each stone above the ocean is related to the time t, in seconds, after it has been thrown by the function h = -­‐2t2 + 2t + 40. How many seconds after it is thrown will a stone strike the ocean? 2. Kali is jumping on her super trampoline, getting as high as she possibly can. The equations to represent her height is h = −5t ! + 20t, where h = distance in feet and t = the time in seconds. How high is she after 2 seconds? 3. The function 𝑓 𝑡 = −10𝑡 ! + 40𝑡 + 120 models the approximate height of an object t seconds after it is launched. What was the initial height of the object?
4. A worker finds a ball on the roof of a building as he is doing some repairs. He tosses the ball up and off the roof so that its height h, in meters, above the ground is related to time t, in seconds, after it has been tossed by the function h = -­‐t2 + 4t + 32. After how many seconds will the ball return to the ground? 5. A football’s height after kickoff is given by the function ℎ 𝑡 = −𝑡 ! + 15𝑡 where t is seconds after kickoff and h is the height of the ball in feet. No one catches the ball. How long does it take for the ball to hit the ground?
6. The length of a rectangle is 4 more than twice the width. The area of the rectangle is 160 𝑓𝑡 ! . Find the width of the rectangle.