Exponents, Polynomials, and Polynomial Functions Chapter 6 Exponents, Polynomials, and Polynomial Functions

6.3 Polynomial Functions

6.3 Polynomial Functions Objectives Recognize and evaluate polynomial functions. Use a polynomial function to model data. Add and subtract polynomial functions. Graph basic polynomial functions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Definition of a Polynomial Function 6.3 Polynomial Functions Definition of a Polynomial Function Polynomial Function A polynomial function of degree n is defined by f (x) = an xn + an – 1 xn – 1 + · · · + a1 x + a0 , for real numbers an,an – 1, . . . , a1, and a0 , where an ≠ 0 and n is a whole number. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Evaluating Polynomial Functions EXAMPLE 1 Evaluating Polynomial Functions Let f(x) = 4x3 – 5x2 + 7. Find each value. (a) f(2) f(x) = 4x3 – 5x2 + 7 f(2) = 4 • 23 – 5 • 22 + 7 = 4 • 8 – 5 • 4 + 7 = 32 – 20 + 7 = 19 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Evaluating Polynomial Functions EXAMPLE 1 Evaluating Polynomial Functions Let f(x) = 4x3 – 5x2 + 7. Find each value. (b) f(–3) f(x) = 4x3 – 5x2 + 7 f(–3) = 4 • (–3)3 – 5 • (–3)2 + 7 = 4 • (–27) – 5 • 9 + 7 = –108 – 45 + 7 = –146 Copyright © 2010 Pearson Education, Inc. All rights reserved.

6.3 Polynomial Functions Functions While f is the most common letter used to represent functions, recall that other letters such as g and h are also used. The capital letter P is often used for polynomial functions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Using a Polynomial Model to Approximate 6.3 Polynomial Functions EXAMPLE 2 Using a Polynomial Model to Approximate Data The number of U.S. households estimated to see and pay at least one bill on-line each month during the years 2000 through 2006 can be modeled by the polynomial function defined by P(x) = 0.808x2 + 2.625x + 0.502, where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and so on, and P(x) is in millions. Use this function to approximate the number of households expected to pay at least one bill on-line each month in 2006. Since x = 6 corresponds to 2006, we must find P(6). P(x) = 0.808x2 + 2.625x + 0.502 P(6) = 0.808(6)2 + 2.625(6) + 0.502 Let x = 6. = 45.34 Evaluate. Thus, in 2006 about 45.34 million households are expected to pay at least one bill on-line each month. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Adding and Subtracting Functions 6.3 Polynomial Functions Adding and Subtracting Functions Adding and Subtracting Functions If f(x) and g(x) define functions, then (f + g) (x) = f (x) + g(x) Sum function and (f – g) (x) = f (x) – g(x). Difference function In each case, the domain of the new function is the intersection of the domains of f(x) and g(x). Copyright © 2010 Pearson Education, Inc. All rights reserved.

Adding and Subtracting Functions 6.3 Polynomial Functions EXAMPLE 3 Adding and Subtracting Functions For the polynomial functions defined by f(x) = 2x2 – 3x + 4 and g(x) = x2 + 9x – 5, find (a) the sum and (b) the difference. (a) (f + g) (x) = f (x) + g(x) Use the definition. = (2x2 – 3x + 4) + (x2 + 9x – 5) Substitute. = 3x2 + 6x – 1 Add the polynomials. (b) (f – g) (x) = f (x) – g(x) Use the definition. = (2x2 – 3x + 4) – (x2 + 9x – 5) Substitute. = (2x2 – 3x + 4) + (–x2 – 9x + 5) Change subtraction to addition. = x2 – 12x + 9 Add. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Adding and Subtracting Functions 6.3 Polynomial Functions EXAMPLE 4 Adding and Subtracting Functions For the polynomial functions defined by f(x) = 4x2 – x and g(x) = 3x, find each of the following. (a) (f + g) (5) (f + g) (5) = f (5) + g(5) Use the definition. = [4(5)2 – 5] + 3(5) Substitute. = 110 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Adding and Subtracting Functions 6.3 Polynomial Functions EXAMPLE 4 Adding and Subtracting Functions For the polynomial functions defined by f(x) = 4x2 – x and g(x) = 3x, find each of the following. (a) (f + g) (5) Alternatively, we could first find (f + g) (x). (f + g) (x) = f (x) + g(x) Use the definition. = (4x2 – x) + 3x Substitute. = 4x2 + 2x Then, (f + g) (5) = 4(5)2 + 2(5) = 110. The result is the same. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Adding and Subtracting Functions 6.3 Polynomial Functions EXAMPLE 4 Adding and Subtracting Functions For the polynomial functions defined by f(x) = 4x2 – x and g(x) = 3x, find each of the following. (b) (f – g) (x) and (f – g) (3) (f – g) (x) = f (x) – g(x) Use the definition. = (4x2 – x) – 3x Substitute. = 4x2 – 4x Combine like terms. Then, (f – g) (3) = 4(3)2 – 4(3) = 24. Substitute. Confirm that f (3) – g(3) gives the same result. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Basic Polynomial Functions The simplest polynomial function is the identity function, defined by f(x) = x. x y x f(x) = x –2 –1 1 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Basic Polynomial Functions The squaring function, is defined by f(x) = x2. x y x f(x) = x2 –2 –1 1 2 4 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Basic Polynomial Functions The cubing function, is defined by f(x) = x3. x y x f(x) = x3 –2 –1 1 2 –8 8 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Graphing Variations of the Identity Function 6.3 Polynomial Functions EXAMPLE 5 Graphing Variations of the Identity Function Graph the function by creating a table of ordered pairs. Give the domain and the range of the function by observing the graph. x y (a) f(x) = –2x. Range x f(x) = –2x –2 4 –1 2 Domain 1 –2 2 –4 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Graphing Variations of the Identity Function 6.3 Polynomial Functions EXAMPLE 5 Graphing Variations of the Identity Function Graph the function by creating a table of ordered pairs. Give the domain and the range of the function by observing the graph. x y (b) f(x) = x2 – 2. Range x f(x) = x2 – 2 –2 2 –1 –1 Domain –2 1 –1 2 2 Copyright © 2010 Pearson Education, Inc. All rights reserved.