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)

3.4 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. Determine 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. Interpret (18, 1512). What is the maximum area the farmer can enclose?

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. Determine 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. Interpret (18, 1512). What is the maximum area the farmer can enclose?

Applications: Area of a Rectangular Region d) (18, 1512) If each parallel side of fencing measures 18 feet, the area of the enclosure is 1512 square feet. This can be written A(18) = 1512 and checked as follows: If the width is 18 feet, the length is 120 − 2(18) = 84 feet, and 18 × 84 = 1512.

Applications: Area of a Rectangular Region Therefore, the farmer can enclose a maximum of 1800 square feet if the parallel sides of fencing each measure 30 feet.

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? What dimensions of the original piece of metal will assure a volume greater than 2000 but less than 3000 cubic inches? Solve graphically.

Finding the Volume of a Box Only 17 satisfies x > 10. The original dimensions : 17 inches by 3(17) = 51 inches.

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.

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.

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.

3.5 Higher Degree Polynomial Functions and Graphs where each ai is real, an  0, and n is a whole number. an is called the leading coefficient anxn is called the dominating term a0 is called the constant term P(0) = a0 is the y-intercept of the graph of P

Extrema Turning points – where the graph of a function changes from increasing to decreasing or vice versa Local maximum point – highest point or “peak” in an interval function values at these points are called local maxima Local minimum point – lowest point or “valley” in an interval function values at these points are called local minima Extrema – plural of extremum, includes all local maxima and local minima

Extrema

Number of Local Extrema A linear function has degree 1 and no local extrema. A quadratic function has degree 2 with one extreme point. A cubic function has degree 3 with at most two local extrema. A quartic function has degree 4 with at most three local extrema. Extending this idea: Number of Turning Points The number of turning points of the graph of a polynomial function of degree n  1 is at most n − 1.

End Behavior Let axn be the dominating term of a polynomial function P.

Determining End Behavior

Determining End Behavior Solution: f matches C, g matches A, h matches B, k matches D.

x-Intercepts (Real Zeros) Number of x-Intercepts (Real Zeros) of a Polynomial Function The graph of a polynomial function of degree n will have at most n x-intercepts (real zeros). Example Find the x-intercepts of P(x) = x3 + 5x2 + 5x − 2.

x-Intercepts (Real Zeros) Number of x-Intercepts (Real Zeros) of a Polynomial Function The graph of a polynomial function of degree n will have at most n x-intercepts (real zeros). Example Find the x-intercepts of P(x) = x3 + 5x2 + 5x − 2. Solution By using the graphing calculator in a standard viewing window, the x-intercepts (real zeros) are −2, approximately −3.30, and approximately 0.30.

Analyzing a Polynomial Function P(x) = x5 + 2x4 − x3 + x2 − x − 4 State the domain. Determine its range. Use its graph to find approximations of local extrema. Use its graph to find the approximate and/or exact x- intercepts.

Analyzing a Polynomial Function P(x) = x5 + 2x4 − x3 + x2 − x − 4 State the domain. Determine its range. Use its graph to find approximations of local extrema. Use its graph to find the approximate and/or exact x- intercepts. Solution Since P is a polynomial, its domain is (−, ). Because it is of odd degree, its range is (−, ).

Analyzing a Polynomial Function Two extreme points that we approximate using a graphing calculator: local maximum point (−2.02, 10.01), and local minimum point (0.41, −4.24). Looking Ahead to Calculus The derivative gives the slope of f at any value in the domain. The slope at local extrema is 0 since the tangent line is horizontal.

Analyzing a Polynomial Function We use calculator methods to find that the x-intercepts are −1 (exact), 1.14(approximate), and −2.52 (approximate).

Comprehensive Graphs The most important features of the graph of a polynomial function are: intercepts, extrema, end behavior. A comprehensive graph of a polynomial function will exhibit the following features: all x-intercepts (if any), the y-intercept, all extreme points (if any), enough of the graph to reveal the correct end behavior.

Determining the Appropriate Graphing Window The window [−1.25, 1.25] by [−400, 50] is used in the following graph. Is this a comprehensive graph? P(x) = x6 − 36x4 + 288x2 − 256

Determining the Appropriate Graphing Window The window [−1.25, 1.25] by [−400, 50] is used in the following graph. Is this a comprehensive graph? P(x) = x6 − 36x4 + 288x2 − 256 Solution Since P is a sixth degree polynomial, it can have up to 6 x-intercepts. Try a window of [−8, 8] by [−1000, 600].