Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of the function concept wikipedia , lookup
Big O notation wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Elementary mathematics wikipedia , lookup
Horner's method wikipedia , lookup
Vincent's theorem wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Section 2-2 Polynomial Functions End Behavior Objectives • I can determine if an equation is a polynomial in one variable • I can find the degree of a polynomial • I can use the Leading Coefficient Test for end behavior in Limit Notation 2 A polynomial function is a function of the form f ( x ) a n x n a n 1 x n 1 a1 x a 0 , a n 0 where n is a nonnegative integer and each ai (i = 0, , n) is a real number. The polynomial function has a leading coefficient an and degree n. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function f ( x ) 2 x 5 3 x 3 5 x 1 Leading Coefficient Degree –2 5 f ( x) x3 6 x 2 x 7 1 3 f ( x ) 14 14 0 3 Complex Numbers Real Numbers Rationals Imaginary Numbers Irrational 4 Polynomial Functions/Equations: A polynomial function in one variable may look like this. f x 5x 2 x 4 x x 3 5 4 3 2 A.The coefficients are complex numbers (real or imaginary). B. Exponents must be a non-negative integer (zero or positive). C. The leading coefficient (the coefficient of the variable with greatest degree) may not be zero. 5 Not a polynomial Polynomials f x 3x x 1 2 f x 3 x 3x 1/ 2 the exp. is not an integer f x 3x 5 3 f x 4 x 3x 1 4 x x the exp. is not non-negative 2 3x f x 1 5 x 5 f x 2 x 4 denominator has a variable factor 6 EX: Determine if each expression is a polynomial in one variable f x x 2x 4x 4 2 f s s s 5s 6s 4 2 5 3 f y 3y 9 y 2 YES 3 YES No 7 Practice 1: Given the following equations determine the following: 1. Determine if the equation a polynomial. Why or why not? 2. If the equation is a polynomial what is the degree of each term, of the polynomial. A. f x 3x2 x 1 Yes, notice powers on the x are positive integers and coefficients are real numbers. C. f x 3 x No, notice power on the x is the fraction 1/2 xx 1 2 B. f x 3 4x x No, notice power on x is -1 D. 3 3x 1 x 3x 2 f x 1 5 Yes, notice powers on the x are positive integers and coefficients are real numbers. 8 Group Exploration Directions: Divide into groups of 2 Open your text to page 141. Read the Exploration exercise 10 minutes!! Answers on next slides Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 a. b. f ( x) x 2 x x 1 3 2 The leading coefficient is + 1 The degree of the function is 3, that is, f(x) is a cubic. c. d. f ( x) as x - f(x) as x e. f. g. 10 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 b. f ( x) 2 x 5 2 x 2 5 x 1 c. The leading coefficient is + 2 The degree is 5 and odd. d. e. f. f ( x) as x - f(x) as x g. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 c. d. f ( x) 2 x x 5 x 3 5 2 The leading coefficient is - 2 The degree is 5 and odd. e. f. g. f ( x) as x f(x) as x 12 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 d. e. f ( x) x 5 x 2 3 The leading coefficient is - 1 The degree is 3 and odd. f. g. f ( x) as x f(x) as x 13 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 e. f. f ( x) 2 x 2 3x 4 The leading coefficient is + 2 The degree is 2 and even. g. f(x) + as x - f(x) + as x + 14 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 f. g. f ( x) x 3 x 2 x 1 4 2 The leading coefficient is + 1 The degree is 4 and even. f(x) + as x - f(x) + as x + 15 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 g. f ( x) x 3x 2 2 The leading coefficient is + 1 The degree is 2 and even. f(x) + as x - f(x) + as x + Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Leading Coefficient Test As x grows positively or negatively without bound, the value f (x) of the polynomial function f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an 0) grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n is odd or even. y y an positive x x n odd an negative Copyright © by Houghton Mifflin Company, Inc. All rights reserved. n even 17 Example: Describe the right-hand and left-hand behavior for the graph of f(x) = –2x3 + 5x2 – x + 1. Degree Leading Coefficient 3 Odd -2 Negative As x , f ( x ) and as x , f ( x ) y x f (x) = –2x3 + 5x2 – x + 1 18 Closure: 19 A real number a is a zero of a function y = f (x) if and only if f (a) = 0. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. (a, 0) is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. (x – a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x). 20 Solution or Root Zero or X-intercept Factor x4 x 2 (4, 0) ( x 4) ( 2, 0) ( x 2) 2 x 3 2 ( , 0) 3 (3x 2) 21 Homework • WS 3-3 22