Chapter 3: Polynomial Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs 3.3 Quadratic Equations and Inequalities 3.4 Further Applications of Quadratic Functions and Models 3.5 Higher Degree Polynomial Functions and Graphs 3.6 Topics in the Theory of Polynomial Functions (I) 3.7 Topics in the Theory of Polynomial Functions (II) 3.8 Polynomial Equations and Inequalities; Further Applications and Models

Two numbers add up to 20. What are the two numbers that maximize the product? Let 1st # = x Let 2nd # = 20-x What is the product of these two numbers?

3.4 Applications of Quadratic Functions and Models Example A farmer wishes to enclose a rectangular region. He has 120 feet of fencing, and plans to use one side of his barn as part of the enclosure. Let x represent the length of one side of the fencing. Find a function A that represents the area of the region in terms of x. What are the restrictions on x? Graph the function in a viewing window that shows both x-intercepts and the vertex of the graph. What is the maximum area the farmer can enclose? Solution Area = width  length, so Since x represents length, x > 0. Also, 120 – 2x > 0, or x < 60. Putting these restrictions together gives 0 < x < 60.

3.4 Applications: Area of a Rectangular Region (d) Maximum value occurs at the vertex. Figure 38 pg 3-60a

Among all rectangles having a perimeter of 10 ft, find the dimensions (length and width) that maximizes the area. Need the formula for perimeter Need the formula for area We will solve one equation for a variable, and then substitute back into the other equation. How do you know which equation to solve for? Well what information are you given??? That should help.

Most cars get their best gas mileage when traveling at a modest speed Most cars get their best gas mileage when traveling at a modest speed. The gas mileage M for a certain new car is modeled by the function. where s is the speed in mi/h and M is measured in mi/gal. What is the car’s best gas mileage, and at what speed is it attained? mi/g Speed

A rancher with 750 ft of fencing wants to enclose a rectangular area then divide it into four pens with fencing parallel to one side of the rectangle. Find the largest possible area for the 4 pens, find the area of one pen if all 4 are equal in size.

3.4 Finding the Volume of a Box Example A machine produces rectangular sheets of metal satisfying the condition that the length is 3 times the width. Furthermore, equal size squares measuring 5 inches on a side can be cut from the corners so that the resulting piece of metal can be shaped into an open box by folding up the flaps. Determine a function V that expresses the volume of the box in terms of the width x of the original sheet of metal. What restrictions must be placed on x? If specifications call for the volume of such a box to be 1435 cubic inches, what should the dimensions of the original piece of metal be? (d) What dimensions of the original piece of metal will assure a volume greater than 2000 but less than 3000 cubic inches? Solve graphically.

3.4 Finding the Volume of a Box Using the drawing, we have Volume = length  width  height, Dimensions must be positive, so 3x – 10 > 0 and x – 10 > 0, or Both conditions are satisfied when x > 10, so the theoretical domain is (10,). (c) Only 17 satisfies x > 10. The original dimensions : 17 inches by 3(17) = 51 inches.

3.4 Finding the Volume of a Box Set y1 = 15x2 – 200x + 500, y2 = 2000, and y3 = 3000. Points of intersection are approximately (18.7, 2000) and (21.2,3000) for x > 10. Therefore, the dimensions should be between 18.7 and 21.2 inches, with the corresponding length being 3(18.7)  56.1 and 3(21.2)  63.6 inches.

3.4 A Problem Requiring the Pythagorean Theorem The longer leg of a right triangular lot is approximately 20 meters longer than twice the length of the shorter leg. The hypotenuse is approximately 10 meters longer than the length of the longer leg. Estimate the lengths of the sides of the triangular lot. Analytic Solution Let s = the length of the shorter leg in meters. Then 2s + 20 is the length of the longer leg, and (2s + 20) + 10 = 2s + 30 is the length of the hypotenuse. The approximate lengths of the sides are 50 m, 120 m, and 130 m.

3.4 A Problem Requiring the Pythagorean Theorem Graphing Calculator Solution Replace s with x and find the x-intercepts of the graph of y1 – y2 = 0 where y1 = x2 + (2x + 20)2 and y2 = (2x + 30)2. We use the x-intercept method because the y-values are very large. The x-intercept is 50, supporting the analytic solution. Figure 44 pg 3-64a

3.4 Quadratic Models The percent of Americans 65 and older for selected years is shown in the table. Plot data, letting x = 0 correspond to 1900. Find a quadratic function, f (x) = a(x – h)2 + k, that models the data by using the vertex (0,4.1). Graph f in the same viewing window as the data. Use the quadratic regression feature of a graphing calculator to determine the quadratic function g that provides the best fit. Year, x 1900 1920 1940 1960 1980 2000 2020 2040 % 65 and older, y 4.1 4.7 6.8 9.3 11.3 12.4 16.5 20.6

3.4 Quadratic Models L1 = x-list and L2 = y-list Substituting 0 for h and 4.1 for k, and choose the data point (2040,20.6) that corresponds to the values x = 140, y = f (140) = 20.6, we can solve for a.

3.4 Quadratic Models (c) (d) This is a pretty good fit, especially for the later years. Note that choosing other second points would produce other models. Figure 46a pg 3-67a