Chapter 2 Functions and Graphs Section 4 Polynomial and Rational Functions

Learning Objectives for Section 2.4 Polynomial and Rational Functions The student will be able to graph and identify properties of polynomial functions. The student will be able to calculate polynomial regression using a calculator. The student will be able to graph and identify properties of rational functions. The student will be able to solve applications of polynomial and rational functions. Barnett/Ziegler/Byleen Business Calculus 12e

Polynomial Functions A polynomial function is a function that can be written in the form for n a nonnegative integer, called the degree of the polynomial. The domain of a polynomial function is the set of all real numbers. A polynomial of degree 0 is a constant. A polynomial of degree 1 is a linear function. A polynomial of degree 2 is a quadratic function. Barnett/Ziegler/Byleen Business Calculus 12e

Odd degree, but NOT an odd polynomial. Odd Polynomials A polynomial is called odd if it only contains odd powers of x. The graphs of all odd functions are symmetric with respect to the origin. (Test: If you can rotate the graph around the origin and get the same picture in less than 360, then it’s odd) Odd degree, but NOT an odd polynomial. 𝑦= 𝑥 3 𝑦= −0.2𝑥 5 +4𝑥 𝑦= 𝑥 3 +2 𝑥 2 Barnett/Ziegler/Byleen Business Calculus 12e

Even degree, but NOT an even polynomial. Even Polynomials A polynomial is called even if it only contains even powers of x. The graphs of all even functions are symmetric with respect to the y-axis. Even degree, but NOT an even polynomial. 𝑦= 𝑥 2 𝑦= 𝑥 4 𝑦= 𝑥 2 −4𝑥+3 𝑦=− 𝑥 4 +3 𝑥 2 Barnett/Ziegler/Byleen Business Calculus 12e

Shapes of Polynomials As we look at the shapes of some polynomials of various degrees, observe some of the following properties: Number of x-intercepts Number of local maxima/minima End behavior What happens as x approaches +∞ or -∞? Barnett/Ziegler/Byleen Business Calculus 12e

Graphs of Polynomials Degree = 1 One x-intercept No maxima/minima End behavior: As x  + , y  +  As x  - , y  -  Barnett/Ziegler/Byleen Business Calculus 12e

Graphs of Polynomials Degree = 3 Three x-intercepts 1 local max, 1 local min. End behavior: As x  + , y  +  As x  - , y  -  Barnett/Ziegler/Byleen Business Calculus 12e

Graphs of Polynomials Degree = 5 Five x-intercepts 2 local max, 2 local min. End behavior: As x  + , y  +  As x  - , y  -  Barnett/Ziegler/Byleen Business Calculus 12e

Observations Odd Degree Polynomials For an odd degree polynomial: The graph starts negative, ends positive, or vice versa, depending on whether the leading coefficient is positive or negative Either way, a polynomial of degree n crosses the x axis at least once, at most n times. 𝑦=− 𝑥 3 +4 𝑥 2 Barnett/Ziegler/Byleen Business Calculus 12e

Graphs of Polynomials Degree = 2 Zero x-intercepts 1 absolute min. End behavior: As x  + , y  +  As x  - , y  +  Barnett/Ziegler/Byleen Business Calculus 12e

Graphs of Polynomials Degree = 4 Two x-intercepts 1 local max, 1 local min., 1 absolute min. End behavior: As x  + , y  +  As x  - , y  +  Barnett/Ziegler/Byleen Business Calculus 12e

Graphs of Polynomials Degree = 6 Four x-intercepts 2 local max, 2 local min., 1 absolute min. End behavior: As x  + , y  +  As x  - , y  +  Barnett/Ziegler/Byleen Business Calculus 12e

Observations Even Degree Polynomials For an even degree polynomial: the graph starts negative, ends negative, or starts and ends positive, depending on whether the leading coefficient is positive or negative either way, a polynomial of degree n crosses the x axis at most n times. It may or may not cross at all. 𝑦=− 𝑥 2 +2𝑥−2 Barnett/Ziegler/Byleen Business Calculus 12e

Characteristics of Polynomials Graphs are continuous. Their graphs can be sketched without lifting up your pencil. Graphs have no sharp corners. Graphs usually have turning points, which is a point that separates an increasing portion of the graph from a decreasing portion. A polynomial of degree n can have at most n linear factors. Therefore, the graph of a polynomial function of positive degree n will have at most n roots/zeros. The real roots are the x-intercepts of the graph. The imaginary roots come in pairs and are not visible on its graph. Barnett/Ziegler/Byleen Business Calculus 12e

Example 1 State the degree of the polynomial, determine the y-intercept, determine the x-intercepts. 𝑓(𝑥)= 𝑥 3 −2 𝑥 2 −24𝑥 𝐴) 𝐷𝑒𝑔𝑟𝑒𝑒=3 𝐵) 𝑓 0 =0 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡= 0 𝐶) 0=𝑥( 𝑥 2 −2𝑥−24) 0=𝑥(𝑥+4)(𝑥−6) 𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑎𝑟𝑒:0, −4, & 6 Barnett/Ziegler/Byleen Business Calculus 12e

Example 2 State the degree of the polynomial, determine the y-intercept, determine the x-intercepts. 𝑓 𝑥 =(2−4𝑥)(𝑥+5)(3𝑥+9)(𝑥−1) 𝐴) 𝐷𝑒𝑔𝑟𝑒𝑒=4 𝐵) 𝑓 0 =2∙5∙9∙−1 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡=−90 𝐶) 0=(2−4𝑥)(𝑥+5)(3𝑥+9)(𝑥−1) 𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑎𝑟𝑒: 1 2 , −5,−3, & 1 Barnett/Ziegler/Byleen Business Calculus 12e

Example 3 State the degree of the polynomial, determine the y-intercept, determine the x-intercepts. 𝑓 𝑥 = 𝑥 2 +9 𝐴) 𝐷𝑒𝑔𝑟𝑒𝑒=2 𝐵) 𝑓 0 =9 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡=9 𝐶) 0= 𝑥 2 +9 𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠:𝑁𝑜𝑛𝑒; 𝑟𝑜𝑜𝑡𝑠 𝑎𝑟𝑒 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑥 2 =−9 𝑥=±3𝑖 Barnett/Ziegler/Byleen Business Calculus 12e

Example 4 Write the lowest-degree polynomial that has x-intercepts 2, -3, and -1. 𝑓 𝑥 =(𝑥−2)(𝑥+3)(𝑥+1) 𝑓 𝑥 =( 𝑥 2 +𝑥−6)(𝑥+1) 𝑓 𝑥 = 𝑥 3 + 𝑥 2 −6𝑥+ 𝑥 2 +𝑥−6 𝑓 𝑥 = 𝑥 3 + 2𝑥 2 −5𝑥−6 Barnett/Ziegler/Byleen Business Calculus 12e

Determining the Minimum Degree of a Polynomial A polynomial of degree n will have, at most, n – 1 “bumps”. 𝑦= 𝑥 4 𝑦=− 𝑥 2 +2𝑥−2

Example 5 What is the minimum degree of a polynomial that could have these graphs? Is the leading coefficient positive or negative? Odd behavior, 4 𝑏𝑢𝑚𝑝𝑠 →𝑀𝑖𝑛. 𝑑𝑒𝑔𝑟𝑒𝑒 5 and leading coeff. is positive Even behavior, 3 𝑏𝑢𝑚𝑝𝑠 →𝑀𝑖𝑛. 𝑑𝑒𝑔𝑟𝑒𝑒 4 and leading coeff. is negative Barnett/Ziegler/Byleen Business Calculus 12e

Cubic Regression Using cubic regression, Lake Trout Length (in.) Weight (oz.) 10 5 14 12 18 26 22 56 96 30 152 34 226 38 326 44 536 Using cubic regression, predict the weight of a lake trout that is 46 inches long. Barnett/Ziegler/Byleen Business Calculus 12e

Cubic Regression Lake Trout Length (in.) Weight (oz.) 10 5 14 12 18 26 22 56 96 30 152 34 226 38 326 44 536 Barnett/Ziegler/Byleen Business Calculus 12e

Question Estimate the weight of a lake trout that is 46 inches long. Select TRACE Press Up Arrow to select the graph of y1 Enter 46 f(46) = 619.6 oz. The weight of a lake trout that is 46 inches long is about 619.6 oz. Barnett/Ziegler/Byleen Business Calculus 12e