Quadratic Functions…Quadratic Functions…

and their and their applications!applications!

For a typical basketball shot, the ball’s For a typical basketball shot, the ball’s height (in feet) will be a function of time height (in feet) will be a function of time in flight (in seconds), modeled by an in flight (in seconds), modeled by an equation such as h = -16tequation such as h = -16t22 +40 t +6. +40 t +6.

a) What is the maximum height of the ball?

b) When will the shot reach the height of the basket? (10 feet)

c) When will the ball hit the floor, if it missed the basket entirely?

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

Put it in your calculator!Put it in your calculator!

Answer: The maximum height of the ball is 31 feet!

Use your zooms and change your Use your zooms and change your window until you see the maximum.window until you see the maximum.

Find the maximum!Find the maximum!

b) When will the shot reach the b) When will the shot reach the height of the basket? (10 feet)height of the basket? (10 feet)

Key words to highlight:Key words to highlight:

Put 10 in for y2 and find the…Put 10 in for y2 and find the… INTERSECTION!

Answer: 2.4 seconds!

When (When (so we are looking for our xso we are looking for our x))Height of the basket (10 feet)Height of the basket (10 feet)

c) When will the ball hit the floor, if c) When will the ball hit the floor, if it missed the basket entirely?it missed the basket entirely?

What do we put in for y2?What do we put in for y2?

y2 = 0 Now find the intersection!Now find the intersection!

Answer: The ball will hit the floor after 2.64 seconds!

YOU DO:YOU DO:

The height, H metres, of a rocket t The height, H metres, of a rocket t seconds after it is fired vertically upwards seconds after it is fired vertically upwards is given by is given by How long does it take for the rocket to reach How long does it take for the rocket to reach

its maximum height? its maximum height? What is the maximum height reached by the What is the maximum height reached by the

rocket? rocket? How long does it take for the rocket to fall How long does it take for the rocket to fall

back to earth? back to earth?

0,5080)( 2 ttttH

Mrs. Holst (who loves to swim!) is putting in a swimming pool next to her house. She wants to put a nice, rectangular privacy fence around it, but she can only afford to pay for 50 feet of fencing. If she does not need a fence on the part adjacent to her house, what are the dimensions of the fence with the largest area she could have for her pool?

My house!

My pool will go here! My future

fence!

Help me get the most space for my money!

x ft.

x ft. y ft.

2x + y = 50y = 50 - 2x

Area = x y50 – 2x

A = x(50 – 2x) A = 50x – 2x2

Now graph it!

Maximum Area

050

100150200250300350

0 10 20 30

Length

Are

a

Put it in your calculator and

find the what??? MAXIMUM

Do we need the x value or the y

value?

x value!x = 12.5 ft.

thus y = 50 – 2(12.5)y = 25

Dimensions of the Fence:

25 ft x 12.5 ft

A farmer wants to build two A farmer wants to build two rectangular pens of the same size rectangular pens of the same size next to a river so they are separated next to a river so they are separated by one fence. If she has 240 meters by one fence. If she has 240 meters of fencing and does not fence the of fencing and does not fence the side next to the river, what are the side next to the river, what are the dimensions of the largest area dimensions of the largest area enclosed? What is the largest area?enclosed? What is the largest area?

Step 1: Draw a figure!

x m x m x m

y m

Step 2: Set up your equations!

3x + y = 240A = xy

y = 240 – 3x

Perimeter equation

Area equation

Solve for y!

Substitute y into the area equation A = x(240 – 3x)

A = 240x – 3x2Distribute the x.

Now what type of function do we have????

So graph it!

Step 3: Graph it!Step 3: Graph it!Remember: There are two questions in the problem.

1. What are the dimensions of the largest area enclosed?

2. What is the largest area?

So when we graph and find the maximum, are we looking for the x or y for number 1?

So when we graph and find the maximum, are we looking for the x or y for number 2?

x!

y!

The Chesapeake Bay

Average Monthly Temperatures of Average Monthly Temperatures of the Chesapeake Baythe Chesapeake Bay

MonthMonth JanJan FebFeb MarMar AprApr MayMay JunJun JulJul AugAug SepSep OctOct NovNov DecDecTempTemp 3131 3434 4444 5454 6464 7272 7676 7575 6868 5757 4747 3636

1. Turn on your STAT PLOT and Diagnostics (2nd 0 x-1)

2. Enter your data in L1 and L2

3. Look at the data you have entered. What is the temperature doing? Now let’s actually look at the STAT PLOT (Zoom 9).4. Which function that we’ve studied would best model the data?

Do a quadratic regression!STAT CALC 5

What is the rWhat is the r22 value? value?

r2 = .927 This tells us that 92.7% of the time, the model is a good predictor, and the closer this value is to 1, the closer the data is to the model.

AnalysisAnalysis According to the model, what month does According to the model, what month does

the maximum temperature occur?the maximum temperature occur?

According to the model, during what According to the model, during what months would the temperature be 50months would the temperature be 50°?°?

June!

March and October

Darryl is standing on top of the Darryl is standing on top of the bleachers and throws a football across bleachers and throws a football across

the field. The data that follows gives the the field. The data that follows gives the height of the ball in feet versus the height of the ball in feet versus the seconds since the ball was thrown.seconds since the ball was thrown.

TimeTime 0.20.2 0.60.6 11 1.21.2 1.51.5 22 2.52.5 2.82.8 3.43.4 3.83.8 4.54.5Ht.Ht. 9292 110110 130130 134134 142142 144144 140140 132132 112112 9090 4444

a. Show a scatter plot of the data. What is the independent variable, and what is the dependent variable?

b. What prediction equation (mathematical model) describes this data?

c. When will the ball be at a height of 150 feet?

d. When will the ball be at a height of 100 feet?

e. At what times will the ball be at a height greater than 100 feet?

f. When will the ball be at a height of 40 feet?

g. When will the ball hit the ground?

a. Show a scatter plot of the data. What is a. Show a scatter plot of the data. What is the independent variable, and what is the the independent variable, and what is the

dependent variable?dependent variable?

Independent variable (x): Time! (always!)

Dependent variable (y): Height

b. What prediction equation b. What prediction equation (mathematical model) describes (mathematical model) describes

this data?this data?

QUADRATIC!!QUADRATIC!!

c. When will the ball be at a height c. When will the ball be at a height of 150 feet?of 150 feet?

Height (y)Height (y)Put 150 in y2.Put 150 in y2.

What happened?!? Explain. What happened?!? Explain.

d. When will the ball be at a height d. When will the ball be at a height of 100 feet?of 100 feet?

Put 100 in y2 and find the intersection!Put 100 in y2 and find the intersection!

.34 seconds .34 seconds and and

3.65 seconds3.65 seconds

e. At what times will the ball be at a e. At what times will the ball be at a height greater than 100 feet?height greater than 100 feet?

65334 .x.

f. When will the ball be at a height f. When will the ball be at a height of 40 feet?of 40 feet?

4.53 seconds4.53 seconds

g. When will the ball hit the g. When will the ball hit the ground?ground?

Put 0 in y2 and find the intersection!Put 0 in y2 and find the intersection!

4.98 seconds

Now try it on Now try it on your own!your own!