# lesson 2-4 finding maximums and minimums of polynomial functions

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Lesson 2-4 Finding Maximums and Minimums of Polynomial Functions Slide 2 Objective: Slide 3 Objective: To write a polynomial function for a given situation and to find the maximum or minimum value of the function. Slide 4 : For quadratic functions: Slide 5 To find the minimum or maximum: Slide 6 : For quadratic functions: To find the minimum or maximum: 1 st : Find x. Slide 7 : For quadratic functions: To find the minimum or maximum: 1 st : Find x. 2 nd : Substitute x back into the equation to find y. Slide 8 : For quadratic functions: To find the minimum or maximum: This y value is the minimum or maximum. Slide 9 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. Slide 10 1. Draw a picture. Slide 11 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. Slide 12 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 = ?? Slide 13 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 = ?? Slide 14 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) = ? Slide 15 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) = ? Slide 16 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) Slide 17 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 ?? Slide 18 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 ?? Slide 19 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. The metal is folded to make an open-top box. What is the maximum volume of such a box? Slide 20 1. Draw a picture. Slide 21 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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. Slide 22 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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 Slide 23 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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. Slide 24 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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 Slide 25 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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) Slide 26 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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. Slide 27 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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. Slide 28 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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: Slide 29 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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: Slide 30 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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: Slide 31 Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions 6x10. 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 Slide 32 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. Slide 33 Assignment: Pgs. 70 71 C.E. 1-4 all W.E. 1, 3, 9, 10, 11

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