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Lesson 2-4 Finding Maximums and Minimums of Polynomial Functions

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Objective:

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Objective: To write a polynomial function for a given situation and to find the maximum or minimum value of the function.

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: For quadratic functions:

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To find the minimum or maximum:

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: For quadratic functions: To find the minimum or maximum: 1 st : Find x.

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: For quadratic functions: To find the minimum or maximum: 1 st : Find x. 2 nd : Substitute x back into the equation to find y.

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: For quadratic functions: To find the minimum or maximum: This y value is the minimum or maximum.

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area.

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1. Draw a picture.

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 1. Draw a picture. 2. Let x = length (in meters) of one on the sides perpendicular to the barn.

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 1. Draw a picture. 2. Let x = length (in meters) of one on the sides perpendicular to the barn. 3. Determine the length of the other sides in terms of x. length = x width = ??

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ??

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ?? 5. Therefore, A(x) = x( ? ) A(x) = ?

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ?? 5. Therefore, A(x) = x( ? ) A(x) = ?

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. 4. Recall that the area of a rectangle : A = ?? 5. Therefore, A(x) = x( ? ) A(x) = ? Therefore, the maximum occurs when x = ?? (Note: this is not the maximum, just where it occurs)

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. So, the dimensions would have to be ??

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A rectangular dog pen is constructed using a barn wall as one side and 60m of fence for the other three sides. Find the dimensions of the pen that give the greatest possible area. So, the dimensions would have to be ?? So, the maximum area to enclose would be ??

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box?

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1. Draw a picture.

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 1. Draw a picture. 2. Identify the dimensions in terms of x.

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 1. Draw a picture. 2. Identify the dimensions in terms of x. Height = x Length = 10 – 2x Width = 6 – 2x

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula.

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h Therefore: V(x) = (10-2x)(6-2x)(x)

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h Therefore: V(x) = (10-2x)(6-2x)(x) So, 10 – 2x = 0 and 6 – 2x = 0 and x = 0.

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? 3. Come up with a formula. Recall the volume of a rectangular solid: V = l x w x h Therefore: V(x) = (10-2x)(6-2x)(x) So, 10 – 2x = 0 and 6 – 2x = 0 and x = 0. When solved; x = 5, x = 3, and x = 0.

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: Constant term:

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: 4x 3 a = 4 Constant term:

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: 4x 3 a = 4 (graph rises) Constant term:

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Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6”x10”. The metal is folded to make an open-top box. What is the maximum volume of such a box? Plot these points on a graph, to find: Leading term: 4x 3 a = 4 (graph rises) Constant term: 0 y-intercept

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Now, make a sketch and rationalize the local max. This occurs between the roots of 0 and 3. Now, use your grapher to find the local max. It is when x = 1.21 and y =32.84. Therefore, the maximum volume is 32.84 in 3. Answer problems from class exercises #1-5 in class.

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Assignment: Pgs. 70 – 71 C.E. 1-4 all W.E. 1, 3, 9, 10, 11

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© 2008 Pearson Addison-Wesley. All rights reserved 8-5-1 Chapter 1 Section 8-5 Quadratic Functions, Graphs, and Models.

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