5.1 Polynomial Functions

Transcription

1 5.1 Polynomial Functions In this section, we will study the following topics: Identifying polynomial functions and their degree Determining end behavior of polynomial graphs Finding real zeros of polynomial functions and their multiplicity Analyzing the graph of a polynomial function 1

2 Definition of Polynomial Function In polynomial functions, THE EXPONENTS ON THE VARIABLE CANNOT BE FRACTIONS AND CANNOT BE NEGATIVE. Definition of Polynomial Function Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real numbers. The following function is called a polynomial function of x with degree n. n f ( x) = a x + a x a x + a x+ a n n 1 2 n

3

4 Graphs of Polynomial Functions 4 out of 5 math students surveyed preferred polynomial functions over the next leading brand of function (because their properties make them easy to analyze). One of the most important features of polynomial functions is the fact that their GRAPHS are SMOOTH and CONTINUOUS. (This fact is very important in Calculus.) For now, this fact helps us to find the zeros of polynomial functions and to sketch their graphs fairly well by hand. 4

5 5

6 Classifying Polynomials Polynomials are often classified by their degree. The degree of a polynomial is the highest degree of its terms. Degree Name of Polynomial Function Zero f(x) = -3 Example First f(x) = 2x + 5 Second f(x) = 3x 2 5x + 2 Third f(x) = x 3 2x 1 Fourth f(x) = x 4 3x 3 +7x-6 Fifth f(x) = 2x 5 + 3x 4 x 3 +x 2 6

7 Graphs of Polynomial Functions You have already looked at the graphs of two polynomial functions in some detail: linear and quadratic. It is VERY important to have a general idea as to the shape of the basic polynomial functions. We can then apply transformations to the graphs to obtain new functions. 7

8 The Leading Coefficient Test Just as we were able to tell if a parabola would open up or down by looking at the sign of a of the quadratic function, we can use the LEADING COEFFICIENT of higher degree polynomials to determine if the left and right ends of the graph rise or fall. We call this the END BEHAVIOR of the graph of the function. As x, f ( x)? As x +, f ( x)? 8

9 Using the Leading Coefficient to Describe End Behavior: Degree is EVEN f ( x) 4 4 = x f ( x) = x If the degree of the polynomial is even and the leading coefficient is positive, both ends. As x, f ( x) As x +, f ( x) If the degree of the polynomial is even and the leading coefficient is negative, both ends. As x, f ( x) As x +, f ( x) 9

10 Using the Leading Coefficient to Describe End Behavior: Degree is ODD f ( x) = x 5 f ( x) = x 5 If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the and rises to the. As x, f ( x) As x +, f ( x) If the degree of the polynomial is odd and the leading coefficient is negative, the graph rises to the and falls to the. As x, f ( x) As x +, f ( x) 10

11 The Leading Coefficient Test Example Use the leading coefficient test to describe the end behavior of the graphs of the following functions: f ( x) ( 3) = x+ g( x) = 4x x + 3x 2x 11

12 Zeros of Polynomial Function A nifty observation about polynomials: 1. A polynomial of degree n has at most n real zeros 2. A polynomial of degree n has at most n 1 turning points. For instance, a quartic polynomial (4 th degree) can have AT MOST 4 real zeros (it may have fewer than 4 real zeros, as we will see in a later section). Also, it can have AT MOST 3 turning points. Take a look: ow! This quartic function has 4 real zeros and 3 turning points. 12

13 Zeros of Polynomial Function VERY IMPORTANT INFORMATION ABOUT REAL ZEROS! Real Zeros of Polynomial Functions If f is a polynomial function and r is a real number then the following statements are equivalent. 1. x = r is a zero or root of the function. 2. x = r is a solution of the equation f(x) = (x - r) is a factor of the polynomial f(x) 4. (r, 0) is an x-intercept of the graph of f. 13

14 Zeros of Polynomial Function f From the graph, we can conclude that: 1. r 1, r 2, r 3, r 4 are of f r 1 r 2 r 3 r 4 x 2. x=r 1, x=r 2, x=r 3, x=r 4 are of f(x) = (x-r 1 ), (x-r 2 ), (x-r 3 ), (x-r 4 ) are of f(x). 4. (r 1,0), (r 2,0), (r 3,0), (r 4,0) are of f 14

15 Zeros of Polynomial Function Example 3 2 f ( x) = x x 20x Find all real zeros of algebraically. Use your graphing calculator to verify your answers. 15

16 Repeated Zeros If a polynomial function has a factor in the form (x r) m where m is a number greater than one, then we say that x = r is a repeated zero of multiplicity m. If m is EVEN, the graph touches the x-axis at x = r, but does not cross through. (The graph turns back around at that point the sign of f(x) does not change.) ( x ) 2 f ( x) = 4 This function has a double zero at x = 4. 16

17 Repeated Zeros If m is ODD, the graph crosses through the x-axis at x = r (it flattens out a little bit around x = r...sign of f(x) changes.) ( x ) 3 f ( x) = 4 This function has a triple zero at x = 4. 17

18 Is the degree of this polynomial even or odd? Is the leading coefficient positive or negative? What is the minimum degree of this polynomial?

19 For the polynomial, list all zeros and their multiplicities. 3 4 f x = 2 x 2 x+ 1 x 3 ( ) ( )( ) ( ) 19

20 Using the Zeros to Write a Polynomial Function We can work backwards to find a polynomial function with given zeros. Example: Write a cubic function with zeros: 3, 0, and 5 For each zero, write in the form of a factor (x r) f(x) = Multiply out the factors and write the polynomial in simplest form (in descending order) 20

21 Many correct answers For example, there are an infinite number of polynomials of degree 3 whose zeros are -4, -2, and 3. They can be expressed in the form: f x = a x+ 4 x+ 2 x 3 ( ) ( )( )( ) ( ) = ( + 4)( + 2)( 3) f x x x x ( ) = 2( + 4)( + 2)( 3) f x x x x ( ) = ( + 4)( + 2)( 3) f x x x x 21

22 Using the Zeros to Write a Polynomial Function Example Find a cubic polynomial function with the given zeros: (Verify on calculator) 2 2,,

23 Example Using the Zeros to Write a Polynomial Function Find a cubic polynomial function with the given zeros: -5; 1 multiplicity 2 (Verify on calculator) 23

24 Sketching Polynomial Functions by Hand Follow these steps when sketching polynomial functions: 1. Determine the BASIC SHAPE of the graph. You can tell by the degree of the polynomial. 2. Use the leading coefficient to determine the END BEHAVIOR of the graph (whether graph falls or rises as x approaches and - ). 3. Find the REAL ZEROS of the polynomial, if possible. These will be the x-intercepts of the graph. Watch out for repeated zeros. 4. Plot a few ADDITIONAL POINTS by substituting a few different values of x into the original function. Include some points close to the x-intercepts. 5. SKETCH the graph, keeping in mind that graphs of polynomials are smooth and continuous. 24

25 25

26 (Continued) 26

27 Sketching Polynomial Functions by Hand Example f ( x) = 2x + 2x + 12x Sketch the graph of by hand. 27

28 Sketching Polynomial Functions by Hand f ( x) = 2x + 2x + 12x y-axis: count by 4 28

29 Using the graph to write a polynomial function Example Write the equation for the cubic function shown in the graph below. (0, 5) 29

30 Using the graph to write a polynomial function Example Write the equation for the quartic function shown in the graph below. (0, -3) 30

31 End of Section