Part I: Polynomial Functions when a = 1 Directions: Polynomial Functions Graphing Investigation Unit 3 Part B Day 1 1. For each set of factors, graph the zeros first, then use your calculator to determine how the graph moves through the zeros(make sure you get a good window).. For #1 - #3, expand the polynomial(multiply the factors) to get the equation in standard form: y = 3. Determine the degree of the function based on the number of x s. 4. Answer the questions at the end based on your graphs Graph 1: y = (x 1) Graph : y = (x 1)(x + ) Graph 3: y =(x 1)(x + )(x 3) Equation 1: Equation : Equation 3: Degree: Degree: Degree: Graph 4: Graph 5: Graph 6: y = (x 1)(x + )(x 3)(x + 4) y =(x 1)(x + )(x 3)(x + 4)(x 5) y =(x 1)(x + )(x 3)(x + 4)(x 5)(x + 6) Degree: Degree: Degree: 1
Part 1 Questions: 1. What is the relationship between the number of factors and the number of real zeros?. What is the relationship between the number of factors and the degree of the polynomial? 3. What is the relationship between the number of real zeros and the number of turning points(humps)? 4. For the odd-degree graphs, describe the end behavior in words and with a quick sketch that is, which direction to the arrows go on the left side of the graph and on the right side of the graph. 5. For the even-degree graphs, describe the end behavior in words and with a quick sketch. 6. How can you tell what the y-intercept will be from the factors? 7. What function represents the simplest odd-degree function? the simplest even-degree function? For the following set of factored equations, determine the degree, the y-intercept and the end behavior(you may sketch it) WITHOUT using the calculator? Then check it with the calculator. 7. y = ( x +.5)(x 7.5)( x 3.) 8. y = x(x + )(x 5)(x 1) For the following equations, determine the degree, the y-intercept, leading coefficient, and the end behavior(you may sketch it) WITHOUT using the calculator? Then check it with the calculator. 9. y = x 4 + 16x 3 + 34x 4x 48 10. y = x 3 5x x + 5
Part II: Graphing Negative a Graph from the factors, graph the y-intercept(no scale needed, just place it above or below the x-axis and put the value), then check your graph with the calculator to make sure your shape(end behavior, etc.) is the same. Then answer the question below the graphs. Graph 1: y = ( x + 1)(x )(x 3) y-int: Graph : y = -(x + 1)(x )(x 3) y-int: Graph 3: y = (x + 1)(x )(x 3)(x 6) Graph 4: y = -(x + 1)(x )(x 3)(x 6) Compare and contrast positive and negative a graphs. (What remains the same and what changes) 3
Part III. Graphing with double roots and triple roots. Graph the following functions form the factors. Graph 1: Graph : Graph 3: y=(x + 1)(x + 1)=(x+1) y=(x + 1)(x + 1)(x 3) y=(x + 1)(x + 1)(x 3)(x 5) y=(x+1) (x 3) y=(x+1) (x 3)(x 5) Number of real zeroes: Number of real zeroes: Number of real zeroes: Number of local min or max: Number of local min or max: Number of local min or max: Describe how the graph of a polynomial with of the same linear factors differs from graphs with distinct linear factors. Graph the following functions from the factors as you have been doing in the other parts of this investigation. Graph 4: Graph 5: y = (x + 1)(x + 1)(x + 1) y = (x + 1)(x + 1)(x + 1)(x ) y = (x + 1) 3 y = (x + 1) 3 (x ) Number of real zeroes: Number of local minima or maxima: Number of real zeroes: Number of local minima or maxima: Describe how a graph with 3 of the same linear factors differs from graphs with distinct linear factors. 4
Summary of what we have learned: 1. A polynomial of n degree has roots and factors.. The most local maxima and minima a polynomial function can have is. 3. End behavior odd-degree functions, a is positive Are modeled by function the simplest odd-degree function Arrows are Function notation: f(x) as x f(x) as x 4. End behavior odd-degree functions, a is negative Are modeled by function the simplest odd-degree function Arrows are Function notation: f(x) as x f(x) as x 5. End behavior even-degree functions, a is positive Are modeled by function the simplest odd-degree function Arrows are Function notation: f(x) as x f(x) as x 6. End behavior even-degree functions, a is negative Are modeled by function the simplest odd-degree function Arrows are Function notation: f(x) as x f(x) as x 7. A polynomial with repeated roots is a. What does the graph look like? 8. A polynomial with 3 repeated roots is b. What does the graph look like? 9. Changing the polynomial from f(x) to f(x) the real roots. Will a function of nth degree always have n roots? Will all of them be real? How many roots will the function f(x) = x 3 6x + 1x 3 have? How many REAL roots does f(x) have? How can we find the rest of the roots? 5
Graphs of Polynomial Functions A. B. C. 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 increases decreases D. E. F. Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior increases decreases Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior increases decreases Type of Function Degree (even or odd) Number of Zeros Left Behavior Right Behavior increases decreases 6
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 4) ( x 1)( x 3) 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 4) (1 x) 5. 3 f ( x) ( x )( x 4) ( x 1)( x ) 6. 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 7
Find the end behavior of each function as x and x 4 6 3 5 1. f x x. f x x 3. f x x 4. f x x 4 7 9 5. f x x 6. f x x 7. f x x 8. f x x Find the degree and leading coefficient of each polynomial 7 9. 4x 6 10. 5x 11. 5 x 3 1. 6 3x 4x 4 13. x 3 x x 1 5 4 14. 6x x x 3 x 3 x 4 (3x 1) 3x 1 x 1 (4x 3) 15. 16. Find the end behavior of each function as x and x 4 17. x 3 x x 1 5 4 18. 6x x x 3 19. 3 x x 3 0. x x x 3 1. What is the maximum number of x-intercepts and turning points for a polynomial of degree 5? Draw a sketch to support your reasoning.. What is the maximum number of x-intercepts and turning points for a polynomial of degree 8? Draw a sketch to support your reasoning. What is the least possible degree of the polynomial function shown in each graph? 3. 4. 5. 6. 7. 8. 9. 30. Find the vertical and horizontal intercepts (x and y intercepts) of each function. f t t 1 t ( t 3) f x 3 x 1 x 4 ( x 5) 31. 3. 33. g n 3n 1 (n 1) 34. k u n n 35. 4 3 f ( x) 3 x( x 3) (5x )(x 1) (4 x ) 36. 3 4 (4 3) f ( x) x( x ) ( x 1) 3 8
Unit 3 Part B Dividing Polynomials Day I. Dividing by Monomials 1. (4x 3 y + 8xy 1x y 3 ) (4xy). ( 6w 3 z 4 3w z 5 + 4w + 5z)(w z) 1 II. Long Division Review: 579 5 Steps: 1. Divide. Multiply 3. Subtract 4. Bring down next number Examples: 1. (x + x 4) (x 4). (4x4 5x +x+4) (y 1) 9
3. (x 4 3x 3 + 5x 6) (x + ) 4. (4y 4 y 3 8y + y) (y 1) III. Synthetic Division used when divisor is linear. 1. (x 3 + 4x 6)(x + 3) 1. (x 4 3x 3 11x + 3x + 10) (x 5) 3. (4x 3 + 5x 7) (x 1) 4. (6x 4 + 15x 3 8x 6) (x + ) 10
Homework Unit 3 Part B Dividing Polynomials Use long division. 1. (3x 3 + 7x 4x + 3) (x + 3). x3 +6x+15 x+4 3. (4x 4 17x + 14x 3) (x 3) 4. (3x 3 x + 4x 3) (x + 3x + 3) Use synthetic division. 5. (x 3 + 4x 6) (x + 3) 6. (6y 4 + 15y 3 8y 6) (y + ) 7. (x 3 + 5x x 15) (x 3) 8. x3 +7 x+3 9. (6x 5x 15)(y + 3) 1 10. (6x 3 + 5x x + 1) (3x + 1) 11
Math 3 Unit 3 Part B Day 3 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. Equation Example: X F(x) - -1 0 1 Even Functions f x ( ) x 1 Odd Functions Equation Example: f ( x) x X F(x) - -1 0 1 Graph the function above: Graph the function above: ***Very Important Trick*** EVEN functions should look the same when you take your paper and. ***Very Important Trick*** ODD functions should look the same when you take your paper and. 1
Algebraically: Replace x with x and compare the result to f(x). If f ( x) f ( x) the function is EVEN. 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 Steps: 1. Substitute x with x. Simplify and compare result with given function (they should be the same). Simplify and compare result with given function (they should be opposites) 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. rx ( ) {(1,3),( 1, 3),(4,11),( 4, 11)} - -10-1 -4 x -6-6 sx ( ) 1 - -5-8 1-4 -10 13
For question 4, classify as even, odd, or neither based on the given graph. For questions 5-8, determine algebraically whether the function is even, odd, or neither. 4 5. f x 1x 3x 6. g x 6x x 4x 8 7 5 7 5 7. h x 1x 9x 8. k x x x 14