1. 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)

2. 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?

3. 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?

4. 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) = and checked as follows: If the width is 18 feet, the length is 120 − 2(18) = 84 feet, and 18 × 84 = 1512.

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

6. 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.

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

8. Finding the Volume of a Box
Set y1 = 15x2 − 200x + 500, y2 = 2000, and y3 = 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)  inches.

9. 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 is the length of the longer leg, and (2s + 20) + 10 = 2s + 30 is the length of the hypotenuse.

10. 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.

11. 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

12. 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

13. Extrema

14. 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.

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

16. Determining End Behavior

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

18. 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.

19. 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.

20. 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.

21. 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 (−, ).

22. 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.

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

24. 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.

25. 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 − 36x x2 − 256

26. 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 − 36x x2 − 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].