1. Chapter 2 Functions and Graphs
Section 4
Polynomial and Rational Functions

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Cubic Regression Lake Trout Length (in.) Weight (oz.) 10 5 14 12 18 2622 56 96 30
152 34
226 38
326 44
536
Barnett/Ziegler/Byleen Business Calculus 12e

24. 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) = oz.
The weight of a lake trout that is 46 inches long is about oz.
Barnett/Ziegler/Byleen Business Calculus 12e