lecture 321 unit 4 lecture 32 quadratics applications & their graphs

Lecture 32Lecture 32 11

Unit 4 Lecture 32Unit 4 Lecture 32Quadratics Applications & Quadratics Applications &

Their GraphsTheir Graphs


ObjectivesObjectives

• Formulate a quadratic equation from a Formulate a quadratic equation from a problem situationproblem situation

• Solve a quadratic equation by using the Solve a quadratic equation by using the quadratic formulaquadratic formula

• Graph the quadratic equationGraph the quadratic equation

• Interpret the graph in terms of the problem Interpret the graph in terms of the problem situationsituation


Revenue:Revenue: R = 78x - xR = 78x - x22

Cost: Cost: C = xC = x2 2 –38 x + 400–38 x + 400

Find the equation for ProfitFind the equation for Profit

Profit = Profit = RevenueRevenue - - CostCost

Earl Black makes tea bags. The cost of making x Earl Black makes tea bags. The cost of making x million teas bags per month ismillion teas bags per month is C = xC = x2 2 –38 x+400. –38 x+400. The revenue from selling x million tea bags per The revenue from selling x million tea bags per month is R = 78x - xmonth is R = 78x - x22


The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost


P = ( 78x – xP = ( 78x – x22) – () – (xx2 2 –38 x + 400–38 x + 400))

Cost:Cost:C = xC = x2 2 –38 x + 400–38 x + 400




P = ( 78x – xP = ( 78x – x22) – () – (xx2 2 –38 x + 400–38 x + 400))

Cost:Cost:C = xC = x2 2 –38 x + 400–38 x + 400

P = 78x – xP = 78x – x22 – – xx2 2 +38 x - 400+38 x - 400




P = ( 78x – xP = ( 78x – x22) – () – (xx2 2 –38 x + 400–38 x + 400))

Cost:Cost:C = xC = x2 2 –38 x + 400–38 x + 400

P = 78x – xP = 78x – x22 – – xx2 2 + 38 x - 400+ 38 x - 400

P = P = – 2x– 2x22




P = ( 78x – xP = ( 78x – x22) – () – (xx2 2 –38 x + 400–38 x + 400))

Cost:Cost:C = xC = x2 2 –38 x + 400–38 x + 400

P = 78x – xP = 78x – x22 – – xx2 2 + 38 x - 400+ 38 x - 400

P = P = – 2x– 2x2 2 + + 116 x116 x




P = ( 78x – xP = ( 78x – x22) – () – (xx2 2 –38 x + 400–38 x + 400))

Cost:Cost:C = xC = x2 2 –38 x + 400–38 x + 400

P = 78x – xP = 78x – x22 – – xx2 2 + 38 x - 400+ 38 x - 400

P = P = – 2x– 2x2 2 + + 116 x116 x - 400- 400


Graph the Profit EquationGraph the Profit Equation

P = P = – 2x– 2x2 2 + 116 x - 400+ 116 x - 400



Vertex: Vertex: P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

2x

ab




2x

ab 116

294( 2)

112

6x




2x

ab 116

294( 2)

112

6x

P = P = – 2– 2(29)(29)2 2 + + 116116 (29) – 400=1282 (29) – 400=1282




2x

ab 116

294( 2)

112

6x

P = P = – 2– 2(29)(29)2 2 + + 116116 (29) – 400=1282 (29) – 400=1282

Vertex: (29,1282) Vertex: (29,1282)

x=29, P = $1282x=29, P = $1282



X-Intercepts: when P = 0, what is x?X-Intercepts: when P = 0, what is x?

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400




P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

0 = 0 = – 2– 2xx2 2 + + 116116 x x - 400- 400




P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

0 = 0 = – 2– 2xx2 2 + + 116116 x x - 400- 4002 4

2x

cb b a

a




P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

0 = 0 = – 2– 2xx2 2 + + 116116 x x - 400- 4002 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 4 )x



X-Intercepts: when P = 0, what is x?X-Intercepts: when P = 0, what is x?P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

2 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 4 )x

116 13456 3200

4x




2 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 4 )x

116 13456 3200

4x

116 10256

4x




2 4

2

( 2)

( 2

116 11

)

006 ( 4 )x

116 13456 3200

4x

116 10256

4x

116 101.3

4x


Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:

0 =0 = -2x -2x2 2 + 116x+ 116x - 400- 400116 101.3

4x

116 101.3 14.7

3.7116 101.34 4

4x



0 =0 = -2x -2x2 2 + 116x+ 116x - 400- 400116 101.3

4x

116 101.3 14.7

3.7116 101.3 4 4

4 116 101.3 217.354.3

4 4

x



0 =0 = -2x -2x2 2 + 116x+ 116x - 400- 400

116 101.3 14.73.7

116 101.3 4 44 116 101.3 217.3

54.34 4

x

Two solutions to make P = 0: Two solutions to make P = 0: x = 3.7 and x = 54.3 itemsx = 3.7 and x = 54.3 items

116 101.3

4x



P-Intercept: when x = 0, what is P?P-Intercept: when x = 0, what is P?

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400




P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2(0)(0)2 2 + + 116116 (0) (0) – 400 – 400 = - 400= - 400




P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2xx2 2 + + 116116 x - 400 x - 400

P = P = – 2– 2(0)(0)2 2 + + 116116 (0) (0) – 400 – 400 = - 400= - 400

P P = - 400= - 400



P = P = – 2x– 2x2 2 + 116 x - 400+ 116 x - 400

1.1. Because a = -2, parabola opens downBecause a = -2, parabola opens down

2.2. Vertex: (Vertex: ( 29,1282)29,1282) 3.3. X-Intercepts: (3.7, 0) and (54.3,0)X-Intercepts: (3.7, 0) and (54.3,0)

4.4. P-Intercept: (0,-400)P-Intercept: (0,-400)


10 20 30 40 50 60

400

800

1200

-400

-800

(29,1282)

Graph:Graph: P = P = – 2x– 2x2 2 + 116 x - 400+ 116 x - 400


10 20 30 40 50 60

400

800

1200

-400

-800

(29,1282)

(54.3,0)(3.7,0)



10 20 30 40 50 60

400

800

1200

-400

-800

(29,1282)

(54.3,0)(3.7,0)

(0,-400)



How many items must be made and sold How many items must be made and sold to generate $500 profit.to generate $500 profit.

P = P = – 2x– 2x2 2 + + 116 x116 x - 400- 400



P = P = – 2x– 2x2 2 + + 116 x116 x - 400- 400

500 =500 = -2x -2x2 2 + 116x+ 116x - 400- 400



P = P = – 2x– 2x2 2 + + 116 x116 x - 400- 400

500 =500 = -2x -2x2 2 + 116x+ 116x - 400- 400 Subtract 500 from Subtract 500 from both sidesboth sides



P = P = – 2x– 2x2 2 + + 116 x116 x - 400- 400

500 =500 = -2x -2x2 2 + 116x+ 116x - 400- 400

0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

Subtract 500 from Subtract 500 from both sidesboth sides



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

0 =0 = a axx22 ++ b bx +x + c c2 4

2x

cb b a

a



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

a = -2a = -2 b = 116b = 116 c = -900c = -900

0 =0 = a axx22 ++ b bx +x + c c2 4

2x

cb b a

a



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

a = -2a = -2 b = 116b = 116 c = -900c = -900

0 =0 = a axx22 ++ b bx +x + c c2 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 9 )x



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

a = -2a = -2 b = 116b = 116 c = -900c = -900

0 =0 = a axx22 ++ b bx +x + c c2 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 9 )x

116 13456 7200

4x



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

0 =0 = a axx22 ++ b bx +x + c c2 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 9 )x

116 13456 7200

4x

116 6256

4x



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

0 =0 = a axx22 ++ b bx +x + c c2 4

2x

cb b a

a

2 4

2

( 2)

( 2

116 11

)

006 ( 9 )x

116 13456 7200

4x

116 6256

4x

116 79.1

4x



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900

116 79.1 36.99.2116 79.1

4 44

x

116 79.1

4x



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900116 79.1

4x

116 79.1 36.9

9.2116 79.1 4 4

4 116 79.1 195.148.8

4 4

x



0 =0 = -2x -2x2 2 + 116x+ 116x - 900- 900116 79.1

4x

116 79.1 36.9

9.2116 79.1 4 4

4 116 79.1 195.148.8

4 4

x

Two solutions to make P = $500: Two solutions to make P = $500: x = 9.2 and x = 48.8 itemsx = 9.2 and x = 48.8 items


10 20 30 40 50 60

400

800

1200

-400

-800



10 20 30 40 50 60

400

800

1200

-400

-800


(9.2,500) (48.8,500)


An angry student stands on the top of a 250 foot cliff An angry student stands on the top of a 250 foot cliff and throws his book upward with a velocity of 46 feet and throws his book upward with a velocity of 46 feet per second. The height of the book from the ground is per second. The height of the book from the ground is given by the equation: h = -16tgiven by the equation: h = -16t22 + 46t + 250 + 46t + 250h is in feet and t is in seconds. Graph the equation.h is in feet and t is in seconds. Graph the equation.


Graph:Graph:

Vertex: Vertex:

h = -16th = -16t22 + 46t + 250 + 46t + 250

2x

ab


Graph:Graph:

Vertex: Vertex:

h = -16th = -16t22 + 46t + 250 + 46t + 250

2x

ab 46

1.432( 16)

462

t


Graph:Graph:

Vertex: Vertex:

h = -16th = -16t22 + 46t + 250 + 46t + 250

2x

ab 46

1.432( 16)

462

t

h = h = – 16– 16(1.4)(1.4)2 2 + + 4646 (1.4) + 250 = 283 (1.4) + 250 = 283


Graph:Graph:

Vertex: Vertex:

h = -16th = -16t22 + 46t + 250 + 46t + 250

2x

ab 46

1.432( 16)

462

t

h = h = – 16– 16(1.4)(1.4)2 2 + + 4646 (1.4) + 250 = 283 (1.4) + 250 = 283

Vertex: (1.4, 283) Vertex: (1.4, 283)

x=1.4 sec, h = 283 feetx=1.4 sec, h = 283 feet


Graph:Graph:

t-Intercepts: when h = 0, what is t?t-Intercepts: when h = 0, what is t?

h = -16th = -16t22 + 46t + 250 + 46t + 250


Graph:Graph:

t-Intercepts: when h = 0, what is t?t-Intercepts: when h = 0, what is t?h = h = – 16– 16tt2 2 + + 4646 t + 250 t + 250

h = -16th = -16t22 + 46t + 250 + 46t + 250


Graph:Graph:


h = -16th = -16t22 + 46t + 250 + 46t + 250

0 = 0 = – 16– 16tt2 2 + + 4646 t + 250 t + 250


Graph:Graph:


2 4

2x

cb b a

a

h = -16th = -16t22 + 46t + 250 + 46t + 250

0 = 0 = – 16– 16tt2 2 + + 4646 t + 250 t + 250


Graph:Graph:


2 4

2x

cb b a

a

2 ( 16)(4

2

46

( 16

)

)

46 250t

h = -16th = -16t22 + 46t + 250 + 46t + 250

0 = 0 = – 16– 16tt2 2 + + 4646 t + 250 t + 250


Graph:Graph:

t-Intercepts: when P = 0, what is t?t-Intercepts: when P = 0, what is t?2 ( 16)(4

2

46

( 16

)

)

46 250t

h = -16th = -16t22 + 46t + 250 + 46t + 250


Graph:Graph:

t-Intercepts: when P = 0, what is t?t-Intercepts: when P = 0, what is t?

46 2116 16000

32t

2 ( 16)(4

2

46

( 16

)

)

46 250t

h = -16th = -16t22 + 46t + 250 + 46t + 250


Graph:Graph:


46 2116 16000

32t

46 18116

32t

2 ( 16)(4

2

46

( 16

)

)

46 250t

h = -16th = -16t22 + 46t + 250 + 46t + 250


Graph:Graph:


46 2116 16000

32t

46 18116

32t

46 134.6

32t

2 ( 16)(4

2

46

( 16

)

)

46 250t

h = -16th = -16t22 + 46t + 250 + 46t + 250


46 134.6 88.62.846 134.6

32 3232

t


0 = -16t0 = -16t22 + 46t + 250 + 46t + 25046 134.6

32t


46 134.6 88.62.8

46 134.6 32 3232 46 134.6 180.6

5.632 32

t

Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:0 = -16t0 = -16t22 + 46t + 250 + 46t + 250

46 134.6

32t



Two solutions to make h = 0: Two solutions to make h = 0: t = -2.8 and t = 5.6 seconds t = -2.8 and t = 5.6 seconds t = -2.8 does not make senset = -2.8 does not make sense

0 = -16t0 = -16t22 + 46t + 250 + 46t + 25046 134.6

32t

46 134.6 88.62.8

46 134.6 32 3232 46 134.6 180.6

5.632 32

t


Graph:Graph:

h-Intercept: when t = 0, what is h?h-Intercept: when t = 0, what is h?

h = -16th = -16t22 + 46t + 250 + 46t + 250


Graph:Graph:


h = -16th = -16t22 + 46t + 250 + 46t + 250

h = h = – 16– 16tt2 2 + + 4646 t + 250 t + 250


Graph:Graph:


h = h = – 16– 16(0)(0)2 2 + + 4646 (0) + 250 (0) + 250

h = -16th = -16t22 + 46t + 250 + 46t + 250

h = h = – 16– 16tt2 2 + + 4646 t + 250 t + 250


Graph:Graph:


h = h = – 16– 16(0)(0)2 2 + + 4646 (0) + 250 (0) + 250

h = -16th = -16t22 + 46t + 250 + 46t + 250

h = h = – 16– 16tt2 2 + + 4646 t + 250 t + 250

h = h = 250 feet250 feet


Graph:Graph:


2.2. Vertex: (Vertex: ( 1.4, 283)1.4, 283) 3.3. t-Intercept: (5.6, 0) t-Intercept: (5.6, 0)

4.4. h-Intercept: (0, 250)h-Intercept: (0, 250)

h = -16th = -16t22 + 46t + 250 + 46t + 250


1 2 3 4 5 6

250

200

300

100

50

(1.4,283)

Graph:Graph: h = -16th = -16t22 + 46t + 250 + 46t + 250

150

Time (sec)

Height (feet)


1 2 3 4 5 6

250

200

300

100

50

(1.4,283)

(5.6,0)


150

Time (sec)

Height (feet)


1 2 3 4 5 6

250

200

300

100

50

(1.4,283)

(5.6,0)

(0,250)


150

Time (sec)

Height (feet)


Ms. Piggie wants to enclose two adjacent chicken Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the make the chicken coup as large as possible. Find the formula for the area of the chicken coops.formula for the area of the chicken coops. Wall

L

WW W



L

WWWidthWidth LengthLength AreaArea

55

1010

1515

XX

W



L


55 66-3(66-3(55)=56)=56 5(51)=2515(51)=251

1010

1515

XX

W



L


55 66-3(66-3(55)=56)=56 5(51)=2515(51)=251

1010 66-3(66-3(1010)=36)=36 10(36)=36010(36)=360

1515

XX

W



L


55 66-3(66-3(55)=56)=56 5(51)=2515(51)=251

1010 66-3(66-3(1010)=36)=36 10(36)=36010(36)=360

1515 66-3(66-3(1515)=21)=21 15(21)=31515(21)=315

XX

W



L


55 66-3(66-3(55)=56)=56 5(51)=2515(51)=251

1010 66-3(66-3(1010)=36)=36 10(36)=36010(36)=360

1515 66-3(66-3(1515)=21)=21 15(21)=31515(21)=315

XX 66-3(66-3(XX)) X(66-3X)=X(66-3X)=

66X-3X66X-3X22

W


Graph:Graph:

Vertex: Vertex:

A = - 3XA = - 3X22 + + 66X66X

2x

ab


Graph:Graph:

Vertex: Vertex:

A = - 3XA = - 3X22 + + 66X66X

2x

ab 66

116( 3)

662

x


Graph:Graph:

Vertex: Vertex:

A = - 3XA = - 3X22 + + 66X66X

2x

ab 66

116( 3)

662

x

A = A = – 3– 3(11)(11)2 2 + + 6666 (11) = 363 (11) = 363


Graph:Graph:

Vertex: Vertex:

A = - 3XA = - 3X22 + + 66X66X

2x

ab 66

116( 3)

662

x

A = A = – 3– 3(11)(11)2 2 + + 6666 (11) = 363 (11) = 363

Vertex: (11, 363) Vertex: (11, 363)

x=11 feet, A = 363 sq. feetx=11 feet, A = 363 sq. feet


Graph:Graph:

x-Intercepts: when A = 0, what is x?x-Intercepts: when A = 0, what is x?

A = - 3XA = - 3X22 + + 66X66X


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X

0 = - 3X0 = - 3X22 + + 66X66X


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X

0 = - 3X0 = - 3X22 + + 66X66X

0 = - 3X(X - 22)0 = - 3X(X - 22)


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X

0 = - 3X0 = - 3X22 + + 66X66X

0 = - 3X(X - 22)0 = - 3X(X - 22)

0 = - 3X or0 = - 3X or


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X

0 = - 3X0 = - 3X22 + + 66X66X

0 = - 3X(X - 22)0 = - 3X(X - 22)

0 = - 3X or0 = - 3X or 0 = (X - 22)0 = (X - 22)


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X

0 = - 3X0 = - 3X22 + + 66X66X

0 = - 3X(X - 22)0 = - 3X(X - 22)

0 = - 3X or0 = - 3X or 0 = (X - 22)0 = (X - 22)

0 = X or X = 220 = X or X = 22


Graph:Graph:

A-Intercept: when x = 0, what is A?A-Intercept: when x = 0, what is A?

A = - 3XA = - 3X22 + + 66X66X


Graph:Graph:


A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X


Graph:Graph:


A = A = – 3– 3(0)(0)2 2 + + 6666 (0) (0)

A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X


Graph:Graph:


A = A = – 3– 3(0)(0)2 2 + + 6666 (0) (0)

A = A = 0 sq. feet0 sq. feet

A = - 3XA = - 3X22 + + 66X66X

A = - 3XA = - 3X22 + + 66X66X


Graph:Graph:


2.2. Vertex: (Vertex: ( 11, 363)11, 363) 3.3. x-Intercepts: (0, 0) and (22, 0)x-Intercepts: (0, 0) and (22, 0)

4.4. A-Intercept: (0, 0)A-Intercept: (0, 0)

A = - 3XA = - 3X22 + + 66X66X


3 6 9 12 15 18

250

200

300

100

50

(11,363)

Graph:Graph:

150

Width (feet)

Area (sq. feet)

A = - 3XA = - 3X22 + + 66X66X

21

350


3 6 9 12 15 18

250

200

300

100

50

(11,363)

(22,0)(0,0)

Graph:Graph:

150

Width (feet)

Area (sq. feet)

A = - 3XA = - 3X22 + + 66X66X

21

350


Find the dimensions of the coop if the area can Find the dimensions of the coop if the area can only be 360 sq feet.only be 360 sq feet.

When A = 360, what is x?When A = 360, what is x?



When A = 360, what is x?When A = 360, what is x?A = - 3XA = - 3X22 + + 66X66X




360 = - 3X360 = - 3X22 + + 66X66X




360 = - 3X360 = - 3X22 + + 66X66XSubtract 360 from Subtract 360 from both sidesboth sides





0 = - 3X0 = - 3X22 + + 66X - 36066X - 360





0 = - 3X0 = - 3X22 + + 66X - 36066X - 360

0 = - 3(X0 = - 3(X22 –– 22X + 120)22X + 120)





0 = - 3X0 = - 3X22 + + 66X - 36066X - 360

0 = - 3(X0 = - 3(X22 –– 22X + 120)22X + 120)0 = - 3(X0 = - 3(X –– 10)(X – 12)10)(X – 12)




360 = - 3X360 = - 3X22 + + 66X66X

0 = (X - 10) or0 = (X - 10) or


0 = - 3X0 = - 3X22 + + 66X - 36066X - 360

0 = - 3(X0 = - 3(X22 –– 22X + 120)22X + 120)0 = - 3(X0 = - 3(X –– 10)(X – 12)10)(X – 12)




360 = - 3X360 = - 3X22 + + 66X66X

0 = (X - 10) or0 = (X - 10) or 0 = (X - 12)0 = (X - 12)


0 = - 3X0 = - 3X22 + + 66X - 36066X - 360

0 = - 3(X0 = - 3(X22 –– 22X + 120)22X + 120)0 = - 3(X0 = - 3(X –– 10)(X – 12)10)(X – 12)




360 = - 3X360 = - 3X22 + + 66X66X

0 = (X - 10) or0 = (X - 10) or 0 = (X - 12)0 = (X - 12)

X =10 or X = 12X =10 or X = 12


0 = - 3X0 = - 3X22 + + 66X - 36066X - 360

0 = - 3(X0 = - 3(X22 –– 22X + 120)22X + 120)0 = - 3(X0 = - 3(X –– 10)(X – 12)10)(X – 12)


3 6 9 12 15 18

250

200

300

100

50(22,0)(0,0)

Graph:Graph:

150

Width (feet)

Area (sq. feet)

A = - 3XA = - 3X22 + + 66X66X

21

350

(11,363)


3 6 9 12 15 18

250

200

300

100

50

(12,360)

(22,0)(0,0)

Graph:Graph:

150

Width (feet)

Area (sq. feet)

A = - 3XA = - 3X22 + + 66X66X

21

350

(11,363)

(10,360)

lecture 321 unit 4 lecture 32 quadratics applications & their graphs

Documents

x x2p

x x2cost

x x2the equation

profit equationvertex

profit equationxintercepts

profit equationpintercept

revenue costrevenue

xthe equation