Lecture 321 Unit 4 Lecture 32 Quadratics Applications & Their Graphs

Lecture 322 Objectives Formulate a quadratic equation from a problem situationFormulate a quadratic equation from a problem situation Solve a quadratic equation by using the quadratic formulaSolve a quadratic equation by using the quadratic formula Graph the quadratic equationGraph the quadratic equation Interpret the graph in terms of the problem situationInterpret the graph in terms of the problem situation

Lecture 323 Revenue: R = 78x - x 2 Cost: C = x 2 –38 x Find the equation for Profit Profit = Revenue - Cost Earl Black makes tea bags. The cost of making x million teas bags per month is C = x 2 –38 x+400. The revenue from selling x million tea bags per month is R = 78x - x 2

Lecture 324 The equation for profit is, Profit = Revenue - Cost Revenue: R = 78x - x 2 P = ( 78x – x 2 ) – (x 2 –38 x + 400) Cost: C = x 2 –38 x + 400

Lecture 325 The equation for profit is, Profit = Revenue - Cost Revenue: R = 78x - x 2 P = ( 78x – x 2 ) – (x 2 –38 x + 400) Cost: C = x 2 –38 x P = 78x – x 2 – x x - 400

Lecture 326 The equation for profit is, Profit = Revenue - Cost Revenue: R = 78x - x 2 P = ( 78x – x 2 ) – (x 2 –38 x + 400) Cost: C = x 2 –38 x P = 78x – x 2 – x x P = – 2x 2

Lecture 327 The equation for profit is, Profit = Revenue - Cost Revenue: R = 78x - x 2 P = ( 78x – x 2 ) – (x 2 –38 x + 400) Cost: C = x 2 –38 x P = 78x – x 2 – x x P = – 2x x

Lecture 328 The equation for profit is, Profit = Revenue - Cost Revenue: R = 78x - x 2 P = ( 78x – x 2 ) – (x 2 –38 x + 400) Cost: C = x 2 –38 x P = 78x – x 2 – x x P = – 2x x - 400

Lecture 329 Graph the Profit Equation P = – 2x x - 400

Lecture 3210 Graph the Profit Equation Vertex: P = – 2x x - 400

Lecture 3211 Graph the Profit Equation Vertex: P = – 2x x - 400

Lecture 3212 Graph the Profit Equation Vertex: P = – 2x x P = – 2(29) (29) – 400=1282

Lecture 3213 Graph the Profit Equation Vertex: P = – 2x x P = – 2(29) (29) – 400=1282 Vertex: (29,1282) x=29, P = $1282 x=29, P = $1282

Lecture 3214 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x - 400

Lecture 3215 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x = – 2x x - 400

Lecture 3216 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x = – 2x x - 400

Lecture 3217 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x = – 2x x - 400

Lecture 3218 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x - 400

Lecture 3219 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x - 400

Lecture 3220 Graph the Profit Equation X-Intercepts: when P = 0, what is x? P = – 2x x - 400

Lecture 3221 Use the Quadratic Formula to solve: 0 = -2x x - 400

Lecture 3222 Use the Quadratic Formula to solve: 0 = -2x x - 400

Lecture 3223 Use the Quadratic Formula to solve: 0 = -2x x Two solutions to make P = 0: x = 3.7 and x = 54.3 items x = 3.7 and x = 54.3 items

Lecture 3224 Graph the Profit Equation P-Intercept: when x = 0, what is P? P = – 2x x - 400

Lecture 3225 Graph the Profit Equation P-Intercept: when x = 0, what is P? P = – 2x x P = – 2(0) (0) – 400 = - 400

Lecture 3226 Graph the Profit Equation P-Intercept: when x = 0, what is P? P = – 2x x P = – 2(0) (0) – 400 = P = - 400

Lecture 3227 Graph the Profit Equation P = – 2x x Because a = -2, parabola opens down 2.Vertex: ( 29,1282) 3.X-Intercepts: (3.7, 0) and (54.3,0) 4.P-Intercept: (0,-400)

Lecture (29,1282) Graph: P = – 2x x - 400

Lecture (29,1282) (54.3,0) (3.7,0) Graph: P = – 2x x - 400

Lecture (29,1282) (54.3,0) (3.7,0) (0,-400) Graph: P = – 2x x - 400

Lecture (29,1282) (54.3,0) (3.7,0) (0,-400) Graph: P = – 2x x - 400

Lecture 3232 How many items must be made and sold to generate $500 profit. P = – 2x x - 400

Lecture 3233 How many items must be made and sold to generate $500 profit. P = – 2x x = -2x x - 400

Lecture 3234 How many items must be made and sold to generate $500 profit. P = – 2x x = -2x x Subtract 500 from both sides

Lecture 3235 How many items must be made and sold to generate $500 profit. P = – 2x x = -2x x = -2x x Subtract 500 from both sides

Lecture 3236 Use the Quadratic Formula to solve: 0 = -2x x = ax 2 + bx + c

Lecture 3237 Use the Quadratic Formula to solve: 0 = -2x x a = -2 b = 116 c = = ax 2 + bx + c

Lecture 3238 Use the Quadratic Formula to solve: 0 = -2x x a = -2 b = 116 c = = ax 2 + bx + c

Lecture 3239 Use the Quadratic Formula to solve: 0 = -2x x a = -2 b = 116 c = = ax 2 + bx + c

Lecture 3240 Use the Quadratic Formula to solve: 0 = -2x x = ax 2 + bx + c

Lecture 3241 Use the Quadratic Formula to solve: 0 = -2x x = ax 2 + bx + c

Lecture 3242 Use the Quadratic Formula to solve: 0 = -2x x - 900

Lecture 3243 Use the Quadratic Formula to solve: 0 = -2x x - 900

Lecture 3244 Use the Quadratic Formula to solve: 0 = -2x x - 900

Lecture 3245 Use the Quadratic Formula to solve: 0 = -2x x Two solutions to make P = $500: x = 9.2 and x = 48.8 items x = 9.2 and x = 48.8 items

Lecture Graph: P = – 2x x - 400

Lecture Graph: P = – 2x x - 400

Lecture Graph: P = – 2x x (9.2,500) (48.8,500)

Lecture 3249 An angry student stands on the top of a 250 foot cliff and throws his book upward with a velocity of 46 feet per second. The height of the book from the ground is given by the equation: h = -16t t h is in feet and t is in seconds. Graph the equation.

Lecture 3250Graph:Vertex: h = -16t t + 250

Lecture 3251Graph:Vertex: h = -16t t + 250

Lecture 3252Graph:Vertex: h = -16t t h = – 16(1.4) (1.4) = 283

Lecture 3253Graph:Vertex: h = -16t t h = – 16(1.4) (1.4) = 283 Vertex: (1.4, 283) x=1.4 sec, h = 283 feet

Lecture 3254Graph: t-Intercepts: when h = 0, what is t? h = -16t t + 250

Lecture 3255Graph: t-Intercepts: when h = 0, what is t? h = – 16t t h = -16t t + 250

Lecture 3256Graph: t-Intercepts: when h = 0, what is t? h = – 16t t h = -16t t = – 16t t + 250

Lecture 3257Graph: t-Intercepts: when h = 0, what is t? h = – 16t t h = -16t t = – 16t t + 250

Lecture 3258Graph: t-Intercepts: when h = 0, what is t? h = – 16t t h = -16t t = – 16t t + 250

Lecture 3259Graph: t-Intercepts: when P = 0, what is t? h = -16t t + 250

Lecture 3260Graph: t-Intercepts: when P = 0, what is t? h = -16t t + 250

Lecture 3261Graph: t-Intercepts: when P = 0, what is t? h = -16t t + 250

Lecture 3262Graph: t-Intercepts: when P = 0, what is t? h = -16t t + 250

Lecture 3263Graph: t-Intercepts: when P = 0, what is t? h = -16t t + 250

Lecture 3264 Use the Quadratic Formula to solve: 0 = -16t t + 250

Lecture 3265 Use the Quadratic Formula to solve: 0 = -16t t + 250

Lecture 3266 Use the Quadratic Formula to solve: 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 sense t = -2.8 does not make sense 0 = -16t t + 250

Lecture 3267Graph: h-Intercept: when t = 0, what is h? h = -16t t + 250

Lecture 3268Graph: h-Intercept: when t = 0, what is h? h = -16t t h = – 16t t + 250

Lecture 3269Graph: h-Intercept: when t = 0, what is h? h = – 16(0) (0) h = -16t t h = – 16t t + 250

Lecture 3270Graph: h-Intercept: when t = 0, what is h? h = – 16(0) (0) h = -16t t h = – 16t t h = 250 feet

Lecture 3271 Graph: 1.Because a = -16, parabola opens down 2.Vertex: ( 1.4, 283) 3.t-Intercept: (5.6, 0) 4.h-Intercept: (0, 250) h = -16t t + 250

Lecture (1.4,283) Graph: h = -16t t Time (sec) Height (feet)

Lecture (1.4,283) (5.6,0) Graph: h = -16t t Time (sec) Height (feet)

Lecture (1.4,283) (5.6,0) (0,250) Graph: h = -16t t Time (sec) Height (feet)

Lecture (1.4,283) (5.6,0) (0,250) Graph: h = -16t t Time (sec) Height (feet)

Lecture 3276 Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the formula for the area of the chicken coops. Wall L WWW

Lecture 3277 Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the formula for the area of the chicken coops. Wall L WW WidthLengthArea X W

Lecture 3278 Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the formula for the area of the chicken coops. Wall L WW WidthLengthArea (5)=56 5(51)= X W

Lecture 3279 Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the formula for the area of the chicken coops. Wall L WW WidthLengthArea (5)=56 5(51)= (10)=36 10(36)= X W

Lecture 3280 Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the formula for the area of the chicken coops. Wall L WW WidthLengthArea (5)=56 5(51)= (10)=36 10(36)= (15)=21 15(21)=315 X W

Lecture 3281 Ms. Piggie wants to enclose two adjacent chicken coops of equal size against the hen house wall. She has 66 feet of chicken-wire fencing and would like to make the chicken coup as large as possible. Find the formula for the area of the chicken coops. Wall L WW WidthLengthArea (5)=56 5(51)= (10)=36 10(36)= (15)=21 15(21)=315 X 66-3(X) X(66-3X)= 66X-3X 2 W

Lecture 3282Graph:Vertex: A = - 3X X

Lecture 3283Graph:Vertex: A = - 3X X

Lecture 3284Graph:Vertex: A = - 3X X A = – 3(11) (11) = 363

Lecture 3285Graph:Vertex: A = - 3X X A = – 3(11) (11) = 363 Vertex: (11, 363) x=11 feet, A = 363 sq. feet

Lecture 3286Graph:Vertex: A = - 3X X A = – 3(11) (11) = 363 Vertex: (11, 363) x=11 feet, A = 363 sq. feet

Lecture 3287Graph: x-Intercepts: when A = 0, what is x? A = - 3X X

Lecture 3288Graph: x-Intercepts: when A = 0, what is x? A = - 3X X

Lecture 3289Graph: x-Intercepts: when A = 0, what is x? A = - 3X X 0 = - 3X X

Lecture 3290Graph: x-Intercepts: when A = 0, what is x? A = - 3X X 0 = - 3X X 0 = - 3X(X - 22)

Lecture 3291Graph: x-Intercepts: when A = 0, what is x? A = - 3X X 0 = - 3X X 0 = - 3X(X - 22) 0 = - 3X or

Lecture 3292Graph: x-Intercepts: when A = 0, what is x? A = - 3X X 0 = - 3X X 0 = - 3X(X - 22) 0 = - 3X or 0 = (X - 22)

Lecture 3293Graph: x-Intercepts: when A = 0, what is x? A = - 3X X 0 = - 3X X 0 = - 3X(X - 22) 0 = - 3X or 0 = (X - 22) 0 = X or X = 22

Lecture 3294Graph: A-Intercept: when x = 0, what is A? A = - 3X X

Lecture 3295Graph: A-Intercept: when x = 0, what is A? A = - 3X X

Lecture 3296Graph: A-Intercept: when x = 0, what is A? A = – 3(0) (0) A = - 3X X

Lecture 3297Graph: A-Intercept: when x = 0, what is A? A = – 3(0) (0) A = 0 sq. feet A = - 3X X

Lecture 3298 Graph: 1.Because a = -3, parabola opens down 2.Vertex: ( 11, 363) 3.x-Intercepts: (0, 0) and (22, 0) 4.A-Intercept: (0, 0) A = - 3X X

Lecture (11,363) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X

Lecture (11,363) (22,0)(0,0) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X

Lecture (11,363) (22,0)(0,0) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X

Lecture (11,363) (22,0)(0,0) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x?

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X Subtract 360 from both sides

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X Subtract 360 from both sides 0 = - 3X X - 360

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X Subtract 360 from both sides 0 = - 3X X = - 3(X 2 – 22X + 120)

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X Subtract 360 from both sides 0 = - 3X X = - 3(X 2 – 22X + 120) 0 = - 3(X – 10)(X – 12)

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X 0 = (X - 10) or Subtract 360 from both sides 0 = - 3X X = - 3(X 2 – 22X + 120) 0 = - 3(X – 10)(X – 12)

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X 0 = (X - 10) or 0 = (X - 12) Subtract 360 from both sides 0 = - 3X X = - 3(X 2 – 22X + 120) 0 = - 3(X – 10)(X – 12)

Lecture Find the dimensions of the coop if the area can only be 360 sq feet. When A = 360, what is x? A = - 3X X 360 = - 3X X 0 = (X - 10) or 0 = (X - 12) X =10 or X = 12 Subtract 360 from both sides 0 = - 3X X = - 3(X 2 – 22X + 120) 0 = - 3(X – 10)(X – 12)

Lecture (22,0)(0,0) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X (11,363)

Lecture (12,360) (22,0)(0,0) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X (11,363) (10,360)

Lecture (12,360) (22,0)(0,0) Graph: 150 Width (feet) Area (sq. feet) A = - 3X X (11,363) (10,360)

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