Right Behavior. Left Behavior. Right Behavior

Transcription

1 U n i t 3 P a r t P a g e 1 Math 3 Unit 3 Part Day 1 Graphing Polynomial Functions Expression 9 x- 3x x + 4x 3 + x + x + 1 5x 4 + x + 10 X 5 + x + 5 3c + 4c /c Type of Function Left Behavior: Right Behavior: Graphs of Polynomial Functions A. B. C. Type of Function Degree (even or odd) Number of Zeros Left Behavior Type of Function Degree (even or odd) Number of Zeros Left Behavior Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior Right Behavior Right Behavior D. E. F. Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior

2 U n i t 3 P a r t P a g e Summary of Behavior: Odd Degree Even Degree Positive vs. Negative Positive vs. Negative End Behavior Examples: Is the degree even or odd? Is the leading coefficient greater than or less than 0 (positive or negative? Determine the end behaviors. Degree? Degree? Degree? Degree? Leading Coefficient? Leading Coefficient? Leading Coefficient? Leading Coefficient? Left Behavior? Left Behavior? Left Behavior? Left Behavior? Right Behavior? Right Behavior? Right Behavior? Right Behavior? Multiplicity Double Root Triple Root y ( x ) y ( x ) 3 y ( x 3) y ( x 3) 3 y ( x 1) y ( x 1) 3 Double Root means: Triple Root means: What does the graph look like? What does the graph look like?

3 U n i t 3 P a r t P a g e 3 Graphing Polynomial Functions Sketch the graphs of each equation. Step 1: Plot the x intercepts and indicate if any roots have multiplicity. Step : Determine the degree and whether the function is an even or an odd degree function. Step 3: Determine the end behaviors. 1. f ( x) ( x )( x 3)( x 4). f x x x x 3. ( ) ( 4) ( 1)( 3) f ( x) ( x 5)( x 4) ( x 1) 3 X Intercepts: X Intercepts X Intercepts Degree Degree Degree Left Behavior Left Behavior Left Behavior Right Behavior Right Behavior Right Behavior 4. f x x x x 5. ( ) ( )( 4) (1 ) f x x x x x 6. 3 ( ) ( )( 4) ( 1)( ) f ( x) ( x ) ( x 1) 3 3 X Intercepts: X Intercepts X Intercepts Degree Degree Degree Left Behavior Left Behavior Left Behavior Right Behavior Right Behavior Right Behavior

4 U n i t 3 P a r t P a g e 4 Math 3 Unit 3 Part Day Even and Odd FUNCTIONS Even and Odd Functions (Not the same as even and odd degree) Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither. The characterization of a function as even, odd, or neither can be made either by using the equation of the function or by looking at its graph. Even Functions Equation Example: f ( x) x 1 X F(x) Equation Example: X F(x) f ( x) x Odd Functions Graph the function above: Graph the function above: ***Very Important Trick*** EVEN functions should look the same when you take your paper and. Algebraically: Replace x with x and compare the result to f(x). If f ( x) f ( x) the function is EVEN. ***Very Important Trick*** ODD functions should look the same when you take your paper and. Algebraically: Replace x with x and compare the result to f(x). If f ( x) f ( x) the function is ODD. Example: f x ( ) x 1 Example: f ( x) x Steps: 1. Substitute x with x. Simplify and compare result with given function (they should be the same) Steps: 1. Substitute x with x. Simplify and compare result with given function (they should be opposites)

5 U n i t 3 P a r t P a g e 5 Practice: For Questions 1-3, classify the functions as even, odd, or neither based on the table of values or the list of ordered pairs. 1.. x 3. r( x ) {(1,3),( 1, 3),(4,11),( 4, 11)} x sx ( ) For question 4, classify as even, odd, or neither based on the given graph.

6 U n i t 3 P a r t P a g e 6 For questions 5-8, determine algebraically whether the function is even, odd, or neither. g x 6x x 4x 4 5. f x 1x 3x k x x x h x 1x 9x 8. Operations with Polynomials A. Adding B. Subtracting 1. ( x xy y ) ( 3x 4xy 3 y ) 1. (5m mp 6 p ) ( 3m 5 mp p ) C. Distributive Property D. Multiplying a w( a w aw ) 1. (3y )(5y 4). ( x 1)( x x 3)

7 U n i t 3 P a r t P a g e 7 Math 3 Unit 3 Part Day 3 Operations with Polynomials Continued E. Dividing Placeholders: 1. x 3 x. a a x x x 5 Method 1 - Long Division Review Example: Steps: Steps to Dividing a Polynomial using Long Division: 1. Divide - Divide the first term of the outside into first term of the inside. Multiply - Multiply that term by the entire outside 3. Subtract - Change all the signs 4. Bring Down - Bring down enough terms so that the number of terms at the bottom is the same as the number of terms on the outside Write your answer as a fraction IF there is a remainder. Ex 1: x x x 4 4 Ex : x x x

8 U n i t 3 P a r t P a g e 8 Ex 3: Ex 4: 4 y 1 4y 5y y 4 x x x x Method : Synthetic Division Synthetic division is used when the divisor is linear and has a leading coefficient of 1. What is meant by linear? Ex 1: 3 1 (x 4x 6)( x 3) Ex : x 3 1 x 6

9 U n i t 3 P a r t P a g e 9 Ex 3: 5 (3m m 1) ( m 1) Ex 4: ( a 8) ( a 5) Find the value of K so that the remainder is 5. Ex 1: ( x 8 x k) ( x 5) Ex : Ex 3: ( x 1x 40) ( x k) ( x kx 44) ( x 4)

10 U n i t 3 P a r t P a g e 10 Math 3 Unit 3 Part Day 4 The Remainder and Factor Theorems I. Remainder Theorem A. Evaluate 3 f ( x) 3x 8x 5x 7 at x. B. Evaluate 4 3 f ( x) 3x 18x 0x 4x at x 5. In Summary the answer is. II. Factor Theorem A. Definition A polynomial function has a factor of x c if and only if P ( c) 0. That is, x c is a factor if and only if c is a zero of the function P. P(x) Ex 1: If f (4) 0 this means. Ex : If f ( 7) 0 this means is a zero is a factor is a zero is a factor Ex 3: If f 0 this means 3 is a zero Ex 4: If f () 5 this means is a factor

11 U n i t 3 P a r t P a g e 11 Practice: Decide whether the given x value is a zero of the function and if so determine the factor of the polynomial. 1. f x x x x x x 4 3 ( ) 5 8 4; 1. ( 3 f ( x) x x 5x 6 ; x = -3 B. The Factor Theorem to factor polynomials. A. Given f x x x x and 3 ( ) x 3 is a zero (which means ), factor f(x). B. Given that x is a factor of h x x x x, use the factor theorem to factor. 3 ( ) 13 6 C. Find the missing factors of 3 x 10x 4x 48 ( x 4)(?)(?). D. Find the missing factors of 3 3x 9x 10x 144 3( x 3)(?)(?)

12 U n i t 3 P a r t P a g e 1 Math 3 Unit 3 Part Day 5 Finding Zeros I. Name Three other Names for Solutions:,, Types of Zeros: A. which. 1.. B. which. Total Number of Zeros is Double Root is. Graph the following polynomials on a graphing calculator. Then identify how many zeros each has, how many are real, and how many are imaginary, and if there are any double roots. 1. y = x 5 + 4x 4-3x 3-7x +. y = x 4 + x y = x 6 + 5x 5 - x 4 +5x y = x 3-5x +8x - 4 What are the possible solutions for polynomials with a higher degree? Degree of 3 Degree of 4 Degree of 5

13 U n i t 3 P a r t P a g e 13 II. Find all Rational Zeros. A. B. f x x x x 3 ( ) III. Find all Zeros. A. f x x x x x B. 4 3 ( ) f x x x x 4 3 ( ) 3 18 IV. Find all Solutions of the Equation. A. 4 3 x x x x B. 3 x 1

14 U n i t 3 P a r t P a g e 14 V. Open Ended Questions A. Two of the zeros of Explain. 3 f ( x) x 4x x 4 are 1 and -1. What type of zero is the third zero? B. How many solutions does the function f x x x x have? Explain. 5 3 ( ) 5 C. Use a calculator to graph function have? f x x x. How many real and/or complex solutions does the 4 ( ) 5 36 D. Using the factor theorem, prove whether is a solution to the equation 3 x x x E. Name all the combinations of possible solutions the polynomial can have. f x x x x 5 3 ( ) 5 F. Prove whether x + 3 is a factor of the polynomial 3 x x x 3 6 G. What are the two types of zeros?

15 U n i t 3 P a r t P a g e 15 Math 3 Unit 3 Part Day 6 Writing Polynomial Equations Find a polynomial function P(x) that has the indicated zeros. Simplify when directed to do so. A. 1,, and -3 as zeros and Simplify. What is the degree? B. zeros i and -3 and Simplify. What is the degree? C. Real coefficients and a zero s 3 7i and Simplify. What is the degree? D. zeros of 3, -3, and i What is the degree?

16 U n i t 3 P a r t P a g e 16 E. one zero is 4-5i and Simplify. What is the degree? F. real coefficients and two zeros of i and 7i What is the degree? G. zeros are 5, ½, and -3 What is the degree? H. real coefficients and a zeros of 8-i and Simpify. What is the degree?

17 Write the polynomial function as the product of factors and list the zeros. U n i t 3 P a r t P a g e 17 A. f x x x x B. 3 ( ) 1 f x x x x 3 ( )

18 Math 3 Unit 3 Part Day 7 - Polynomial Word Problems and the Closure Property U n i t 3 P a r t P a g e 18 I. The Closure Property Closure Property How do you know which operations polynomials are closed under? Whole Numbers - 0,1,,3. Does not include fractions or decimals or negative numbers. CLOSED Means that when you start with a whole number and perform the operation, you end with a whole number. Are whole numbers CLOSED under addition? Are whole numbers CLOSED under subtraction? Are whole numbers CLOSED under multiplication? Are whole numbers CLOSED under division? Counting Numbers - 1,,3 Does not include fractions or decimals or negative numbers CLOSED Means that when you start with a counting number and perform the operation, you end with a counting number. Are counting numbers CLOSED under addition? Are counting numbers CLOSED under subtraction? Are counting numbers CLOSED under multiplication? Are counting numbers CLOSED under division? Integers - -3,-,-1,0,1,,3 Does not include fractions or decimals CLOSED Means that when you start with integers and perform the operation, you end with integers. Are integers CLOSED under addition? Are integers CLOSED under subtraction? Are integers CLOSED under multiplication? Are integers CLOSED under division? Complex Numbers/Imaginary CLOSED Means that when you start with an imaginary number and perform the operation, you end with an imaginary number. Are imaginary numbers CLOSED under addition? Are imaginary numbers CLOSED under subtraction? Are imaginary numbers CLOSED under multiplication? Are imaginary numbers CLOSED under division? Polynomials Are polynomials CLOSED under addition? Are polynomials CLOSED under subtraction? Are polynomials CLOSED under multiplication? Are polynomials CLOSED under division?

19 U n i t 3 P a r t P a g e 19 Word Problems 1. The function f x x x describes the United States trade balance with Mexico in ( ) billions of dollars for the period (1994 corresponds to x = 4). A. According to the function when was the U.S. deficit greatest (largest negative value)? B. When was trade balanced (y = 0) for the period ? C. If the function continues to model U.S.-Mexican trade, will trade be balanced again? When?. The function 4 3 G( x) x x x 0.959x 1.47 describes the annual average price of a gallon of gasoline during the period where x = 0 represents A. Describe the price of gasoline according to this function including highs, lows, recent changes in the price, increasing,,decreasing etc B. Within the domain given, how is the price of gasoline best characterized? C. If this function continues to model the price of gasoline for the next three years, what kind of change in prices should we expect? D. According to this function will gasoline reach a price of $.00 per gallon? If so, when? If not, why?

20 U n i t 3 P a r t P a g e 0 3. Stephen has a set of plans to build a wooden box. He wants to reduce the volume of the box to 105 cubic meters. He would like to reduce the length of each dimension by the same amount. The plans call for the box to be 10 inches by 8 inches by 6 inches. Write and solve a polynomial to find out how much Stephen should take from each dimension? 4. The height of a box that Joan is shipping is 3 inches less than the width of the box. The length is more than twice the width. The volume of the box is 1540 cubic inches. What are the dimensions of the box? 5. From 1985 through 1995, the actual and projected number, C (in millions), of home computers sold in 3 the United Stated cam be modeled by C 0.009( t 8t 40t 400) where t = 0 represents During which year are 8.51 million computers projected to be sold?