Application ExampleMrs. Warren throws o footboji into the ojr with on jnJfioJ upward vel,oct/ u fee'

per second. Tho height h In foot of tho boll above the ground con be rrode(ed

by h(t) -1 6t' 98t 5.5, where t is the time in seconds offer Mts. Warren

the ball.

a. What is the height of the ball after 3 seconds?

b. When will the ball reach its maximum height?

a c -lip

c. What is the maximum height the ball will reach?

h(t) + 155 slr

d. What is the vertex? Describe the meaning of the vertex in the situation

ISS.Sc)reached cc nnoxrnum

e. When will the ball hit the ground? What time does not make sense? Exp4ain

why.

16t2 + 98t + 5.5

-32

—1.06 sec S sec

IAIarm CIP

Quadratic Model

Plot of projectile height v time125

100

S 75

.g' 50

25

0 1 2 34 56 7 8

time (s)

Problems:

You can fit the data for a projectile fired

from the ground to a quadratic function.

Time is on the x-axis and height on the y-

axis. The function can be fitted to

+125.

9 10

S. You are in the navy and fire a grenade straight up in the air. Using the plot and the function above answer the

following questions:

a. What is the maximum height will the grenade reach? 0")

b. How high is the grenade at 2 seconds?

c. How high is the grenade at S seconds?

d. When does the grenade hit the ground?

e. At what times is the grenade at a height of 25 meters?

f. At what height is the grenade at 15 seconds (use the function f (x) = +125

-SC IS-s)2 å

g. Does the solution of (f) make sense? Why or why not? NO/ b/c IS

m

ve

1 U3C Quadratic Applications Notes

Wil E. Coyote is catapulting a boulder offa cliff to hit the road runner. Let t represent the number of

seconds that the boulder catapults off the cliff and h(t) denote the height of the boulder, in feet, above the

base of the cliff. Ignoring air resistance, we can use the following formula to express the path of the

boulder: h(t) = -16t2 + 24t + 160

1. What does the x axis represent?

The y axis?

ul

2. What part of the graph is insignificant? Why?

Id—aKlS

3. What was the height of the boulder before it was launched?

What special point on the graph is associated with this

information?

4. If Wil E. Coyote simply pushed a boulder off the cliff, how would the graph look different?

5. How iong will it take before the boulder reaches the bottom of the cliff?

What special point on the graph is associated with this information? n

6. After how many seconds does the boulder change direction?

How high is the boulder when it changes direction?

What is this significant point called on the graph?

7. How high above the starting point does the boulder begin to change direction?

Name: Dote:

Quadratics Applications Practice

Philip is standing on a rock ledge 64 feet above a lake, and he tosses a rock with 0velocity of 48 feet per second. This graph and table represent the height above thowater, h(t), as a function of time, t, in seconds after Philip releases the rock.

100

.1 Time tIleight

o 64

96

2 96

3 644 O

2 3 4

Time (sec.)

l. What is the maximum height of the rock? | 06

2. After how many seconds does the rock change direc ion in the air?t 5 gcc (vevkb

3. How can you estimate the maximum height from the t 2

If Z SCCzJ4XlXp-h ace4hcne

4. When does the rock hit the surface of the lake? SCC

What is this point on the graph called?

How can you identify this from the table? COOK @/0)

5. Identify the vertex and the axis of symmetry.V'. I '5/ 100

AoS:X4 ISWhat do these represent in our story?) 115 CCC... VDCK

a be 100t+

6. At I sec, what direction is the rock moving?

At 2 sec, what direction is the rock moving?

TCM Applications of QuadraticsEx. A ball is thrown and follows the path shown in the graph. The x-axis represents time (inseconds) and the y-axis represents height (in feet).

Use the graph to answer the following questions:

a) What is the maximum height of the ball?60

b) How many seconds did it take to reach themaximum height?

X 2.5 Sec—

c) How long did it take for the ball to land onthe ground? z

Isec)Ex. 2 A rocket is launched at 19.6 meters per second (m/s) from a 58.8 meter tall platform. The graphfor the object's height (meters) and time (seconds) after launch is below.

ooUse the graph to answer the following questions:

a) When d es the r cket strike the ground?

b) What is the maximum height of the rocket?•o,

c) How many seconds did the rocket take toreach the maximum height? 2.0

kine )Ex. 3 Use the information on the graph to Graph of a Dolphin Jumping out of the Oceandetermine the key features listed below. Then 6explain what each of these means in the context

5of the given situation.

Vertex: 3

3

2

x-intercepts: 0123456> .5 SS see/ +oe dolphi nwaS .1 Time (seconds)

++c vva}d -2-3

APPLICATIONS WITH PARABOLIC FUNCTIONS

For 1-6, Use the graph at the right to answer the followingQuestions. The graph shows the height h in feet of a smallrocket t seconds after it is launched. The path of the rocketis given by the equation: h(t) = -16t2 + 128t.

1. How long is the rocket in the air? 8 SCC

2. What is the greatest height the rocket reaches?

3. About how high is the rocket after 1 second?

4. After 2 seconds,a. about how high is the rocket?

b. is the rocket going up or going down?

5. After 6 seconds,a. about how high is the rocket?

b. is the rocket going up or going down?

200

IOO

50

12345678time (seconds)

6. Do you think the rocket is traveling faster from 0 to 1 second or from 3 to 4 seconds? Explainyour answer. Oh l. 4) vee 4-0 I Sec PecaUt

(S slbW down from 3-6 Q-qe c7. After t seconds, a ball tossed in the air from the ground level reaches a height of h feet given bythe equation h(t) = -16t2 + 144t.

a. What is the height ofßhe ball after 3 second?-16( 9 feet

b. When will the ball reach its maximum height? x x

seconds

c. What is the maximum height the ball will reach?feet

d. What is the verte ? Describe the meaning of the vertex in the ituation abovH 4.rreo, pal/ hi) a mal • hagh+ 0+

e. When will the ball hit the ground? What time does not make sense? Explain why.h(t) = + 144t

Po/y9m/Pd/g

o'fnå 4 fe / beh marc sense.

(C

8. A rocket carrying fireworks is launched from a hill that is 80 feet above o lake, The rocket

will fall into lake after exploding at its maximum height. The rocket's height above the

surface of the lake is given by h(t) = -1612 + 64t + 80.

a. What is the height of th$ rocket after 1.5 second.5) = + + 80 feet

b. At what time does the rocket reach its maximum height? / (b

seconds

c. What is the maximum height reached by the rpcket?h(t) — —-16(_%_)2 + 64(4) + 80 feet

A YO/ /%1/4

e. When will the rocket land on the ground? What time does not make sense? Explain

why.

x= I sect b/c + me Can} bc nega/)te

9. A rock is thrown from the top of a tall building. The distance, in feet, between the

rock and the ground t seconds after it is thrown is given by d = -16t2+ 8 +382.

a. At what time does the rock reach its maximum height?

b. What is the maximum height reached by the rocket?d(t)

2a 26b

t = seconds

+382. -Y feet

c. W at is the v rtex? Describe the meaning of the vertex in the situati gbove.

L-f/Cj H *fie roc/< was a+ 41Ck

d. When will the rock hit the ground? What time does not make sense? Explain why.

h(t) = -16t2 - 48t + 382

x= d -3.4 SVC (+)'me caret be tit9Q4jvt)