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1 Algebra 2 Quarter 2 Quadratic Functions Introduction to Polynomial Functions Hybrid Electric Vehicles Since 1999, there has been a growing trend in the sales of hybrid electric vehicles. These data show the number of hybrid electric vehicles sold in each of the first eight years after What does the data (and the graph) suggest about the trend in the sales of hybrid electric vehicles? 400, , , , , , ,000 50,000 If the trend continues, how many hybrid electric vehicles do you think would be sold 10 years after 1999? Did you know in 2018, approximately 65% of all jobs in Hawaii will require some postsecondary training beyond high school? just about all 4-year universities require applicants to have completed at least Algebra 2 in high school in order to be considered for admission? just about all 2-year and 4-year institutions (community colleges and universities) require students to earn at least 3 credits in mathematics in order to earn their degree (and for many programs, much more than that)? high school students who take a math course during their senior year have higher rates of success in college math courses than those who do not? So taking more math in high school (especially in your senior year) increases your career opportunities in the future

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3 Algebra 2 Table of Contents Quarter 2 Lesson Title Page Module 7: Quadratic Functions Module 8: Introduction to Polynomial Functions Q-1.1 Products of linear expressions 1 Q-1.2 Factoring Quadratic Expressions 7 Q-2.1 How do a and c Affect the Graph of a Quadratic Function? 13 Q-2.2 Concavity and y-intercepts 23 Q-2.3 Homework 27 Q-2.4 Graphing Quadratics in Factored Form 29 Q-2.5 Building a Function from its Graph 33 Q-2.6 Matching a Graph with its Symbolic Representation 37 Q-2.7 Homework 41 Q-2.8 The Vertex of the Graph of a Quadratic Function 43 Q-2.9 Representing a Quadratic Function in Vertex Form 49 Q-2.10 Summarizing the Three Forms of a Quadratic Function 55 Q-3.1 Graphing Quadratic Functions 59 Q-3.2 Square Roots 65 Q-3.3 Homework 69 Q-3.4 Quadratic Equations in Vertex Form 71 Q-3.5 The Quadratic Formula 77 Q-3.6 Complex Numbers 85 Q-3.7 Rounds Activity 91 Q-3.8 Homework 93 Q-3.9 Determining Points of Intersection 95 P-1.1 The Standard Symbolic Form 99 P-1.2 End-behavior 103 P-1.3 Homework 111 P-1.4 A Deeper Understanding of End-behavior 113 P-1.5 Homework 117 P-2.1 End-behavior, Zeros and the y-intercept 119 P-2.2 Repeated Zeros 125 P-2.3 Graphing Polynomials in Factored Form 131 P-2.4 Homework 139 P-2.5 From Graphs to Symbolic Form 141 P-2.6 Homework 145 P-2.7 Determining a Function s Symbolic Representation from its Graph 147 P-2.8 Homework 155 Appendix Vocabulary Section 157

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5 Name Q-1.1: Products of linear expressions Pd Date Part I: Warm-up 1. Work with a partner to answer the questions below pertaining to the situation and graph shown. Jane is standing near the edge of a cliff enjoying the view of the ocean. She tosses a pebble into the ocean below her. The graph represents the height of the pebble, h(t), measured in feet, as a function of time, t, measured in seconds. h(t) a. How would you describe the shape of the graph of h(t)? How does this differ from the graph of a linear function? t b. What is the value of h(0)? What does this mean in the context of the situation? c. What is the approximate value of value of h(1.5)? What does this mean in the context of the situation? d. At approximately what values of t is h(t) = 120? Using the context of the given situation, explain why there are 2 values of t when h(t) = 120. e. At approximately what value of t is h(t) = 0? What does this mean in the context of the situation? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 1

6 Name Q-1.1: Products of linear expressions Pd Date Part II: Review of the Distributive Property Multiplying algebraic expressions is very similar to multiplying numbers. Let s look at a few examples of this that involves the Distributive Property. a(x + y) = ax + ay Example A: 5(47) Example B: 3(4x + 15) 5(47) = 5(40 + 7) 3(4x + 15) = 3(4x) + 3(15) = 5(40) + 5(7) = 12x + 45 = = 235 (a + b)(x + y) = ax + ay + bx + by Example C: 23(45) Example D: (x 3)(x + 8) 23(45) = (20 + 3)(40 + 5) x + 8 x 3 x 2 3x + 8x 24 Next, combine any terms that can be combined: = = x 2 3x + 8x 24 = 1035 = x 2 + 5x 24 Verify that (x 3)(x + 8) = x 2 + 5x 24 è pick any value for x and test it: let s use x = 4 (x 3)(x + 8) = x 2 + 5x 24 (4 3)(4 + 8) (4) 24 (1)(12) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 2

7 Name Q-1.1: Products of linear expressions Pd Date 2. After reviewing the examples on the previous page, briefly summarize the Distributive Property in your own words. Use the space below to take notes (follow your teacher s instructions). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 3

8 Name Q-1.1: Products of linear expressions Pd Date 3. Write an equivalent expression, in standard form, for each of the following. A. ( x + 5) ( x 8) B. ( x 3) ( x 7) C. ( x + 7) ( x 7) D. x + 5 ( ) x 1 2 E. x x 3 ( ) F. 8x x G. ( 7 2x) ( 5 + 3x ) H. 2x 1 4 4x 7 ( ) I. ( 4x + 5) ( 4x 5) J. 3x x ( ) K. 20x 1 4 x 2 3 L. 40 x ( )( 40 x) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 4

9 Name Q-1.1: Products of linear expressions Pd Date Part III: Rewriting more complex expressions 4. Rewrite the following expression by using the distributive property and then combining any like terms. A. ( x + 5) 2 B. ( x + a) 2 C. ( x 2 + 6x 5) D. x x( 8x 40) 2 5. Re-write the following expressions in the form ax # + bx + c. A. 4(x 2 2x + 3x) B. 9x ( x x 2 ) +8( x +1) C. x ( x + 4) 2 D. ( x 10) 2 (x ) E. 3x( x + 2) + ( 5x 1) ( 4x 3) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 5

10 Name Q-1.1: Products of linear expressions Pd Date Part IV: Reflect and Summarize (follow your teacher s instructions) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 6

11 Name Q-1.2: Factoring Quadratic Expressions Pd Date Part I: Warm-up 1. Determine if the following equations are true or false for all values of x. A. x + 2 # = x # + 4 True False B. x + 2 # = (x + 2)(x + 2) C. x + 2 # = (x 2)(x 2) D. 3x + 6 # = 3 x + 2 # E. x + 2 # = x 2 # + 8x 2. Use the Distributive Property to write an equivalent expression in the form ax 2 + bx + c. A. x + 2 x + 6 B. x 3 x + 7 C. x 4 x 5 3. Each of your answers above (in question 2) should be in the form ax 2 + bx + c. A. Compare the value of c in each of your answers to the given expression. Briefly explain how the value of c resulted from the given expression. B. Compare the value of b in each of your answers to the given expression. Briefly explain how the value of b resulted from the given expression. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 7

12 Name Q-1.2: Factoring Quadratic Expressions Pd Date Part II: Factoring a quadratic expression Now, we are going to do the opposite process. Instead of multiplying to find the product, now we are going to be given the product and we want to determine its FACTORS. Following from the problems above, keep these ideas in mind: Given an expression ax 2 + bx + c o This expression resulted from finding the product of (x + s)(x + t) o c resulted from the PRODUCT of s and t o b resulted from the sum (or difference) or s and t Therefore, when factoring a quadratic expression we need to think about o what are all the pairs of numbers can I multiply to get a product of c o and then, which of those pairs add/subtract to get the value of b Quadratic Expression Pairs of Numbers that have a product of c Which pair can I add/subtract to get b Factored Form of the Quadratic Expression x # + 11x + 24 x # 11x + 24 x # + 5x 24 x # 5x Factor the quadratic expressions below. a. x # + 9x + 18 b. x # + 7x 30 c. x # 7x + 10 d. x # 12x + 64 e. x # + 19x 20 f. x # x 30 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 8

13 Name Q-1.2: Factoring Quadratic Expressions Pd Date There are also a few special cases to consider when factoring ax 2 + bx + c. Case 1: When a, b and c all have a common factor Example: 2x 2 +18x + 40 è a, b and c are all multiples of 2 so we can factor that first 2(x 2 + 9x + 20) è now we have a simpler quadratic expression to factor o What factors of 20 add up to 9? è 5 and 4 = 2(x + 5)(x + 4) Practice: Completely factor the expression 5x 2 10x 60 Case 2: When c = 0 Example: x 2 + 7x è c = 0, therefore there is no constant term in the quadratic expression è however, we just learned in Case 1 (above) that if the terms have a common factor, we should factor it out of each term x 2 + 7x è x 2 and 7x each have a common factor of x = x(x + 7) Practice: Factor the expression 3x 2 15x Case 3: When b = 0 Example: x 2 49 è b = 0, therefore there is no linear x term in the quadratic expression è therefore, we need to see this as x 2 + 0x 49 x 2 + 0x 49 è What factors of 49 add up to 0? è to get a sum of zero, I have to add a number and its opposite, so it has to be 7 and 7 = (x + 7)(x 7) Practice: Factor the expression x Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 9

14 Name Q-1.2: Factoring Quadratic Expressions Pd Date Case 4: When a 1 Example 1: 2x 2 + 9x +10 è since a is not equal to 1, I can t simply just ask myself What factors of 10 add up to 9? (2x )(2x ) 2 (2x )(2x ) 2 (2x + 4)(2x + 5) 2 è first, I m going to write the two factors so that the first term of each factor has the same coefficient as the value of a : 2 o However, (2x)(2x) = 4x 2, which means I have too many factors of 2, so I m going to divide by 2 to keep the expression equivalent to the original; then later, I ll actually divide that extra factor of 2 out of my expression it ll all make sense when we re done. (c: è Now I can ask myself, What factors of 20 add up to 9? o In the original expression, since a = 2 and c = 10, I need to consider factors of a*c that add up to b o Since 2*10 = 20, I need to ask myself, What factors of 20 add up to 9? è 4 and 5 (2x + 4)(2x + 5) 2 2(x + 2)(2x + 5) 2 è Since we multiplied an extra factor of 2 at the very beginning, we are going to have to divide out that extra factor of 2 now. So, look for any common factors within each factor. è Divide out the common factor of 2 in 2x + 4, and then the factor of 2 in the numerator divided by the 2 in the denominator has a quotient of 1. = (x + 2)(2x + 5) Practice: Factor the expression 3x 2 +14x +8 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 10

15 Name Q-1.2: Factoring Quadratic Expressions Pd Date Use the space below to take notes (follow your teacher s instructions). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 11

16 Name Q-1.2: Factoring Quadratic Expressions Pd Date Part III: Mixed Practice 5. Factor the quadratic expressions below. o Remember to take a moment to analyze the given expression to determine if it is a special case. o Verify your answer by multiplying your factors to see if the product is the original expression. a. x # 2x 35 b. 3x # + 11x 4 c. x # 6x d. x # 12x + 36 e. 2x # + 12x 14 f. x # 100 g. 4x # + 5x 6 h. 5x # + 15x 60 i. 2x # 18 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 12

17 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date Part I: Warm-up 1. Re-write the following quadratic expressions in the form ax # + bx + c. A. ( 2x 1) ( x + 7) B. 2( x +11) ( x 3) C. 4( x 3) 2 2. Factor the following expressions. A. x 2 4x + 3 B. 3x 2 12x C. 5x 2 + 9x 2 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 13

18 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date Part II: Reviewing the steps for drawing the graph of the parent quadratic function f(x) = x 2 Step 1: Plot the vertex (0, 0) è because f(0) = 0 Step 2: From the vertex, move 1 unit to the right and 1 unit up (because f(1) = 1: 1 2 = 1). And then again from the vertex, move 2 units right and 4 units up (because f(2) = 4: 2 2 = 4). Step 3: Reflect points over the y-axis Step 4: Sketch the curve through the 5 points Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 14

19 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date 3. In your words, summarize the 4 steps described on the previous page. Your summary should be stated in a way that will be useful to you to help you recall how to quickly sketch the graph of a quadratic function. 4. Use your summary above to sketch the graph of f x = x # in the coordinate plane below. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 15

20 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date Part III: Variations of the parent function f(x) = x 2 Sketching the graph of f(x) = x 2 + c è f(x) = x 2 + c è how does the value of c affect the graph? c is the constant term of the function, which means we have to ADD the value of c to x 2 (or subtract it if c < 0). Working with a partner, analyze each of the three pairs of tables below, comparing the values of x 2 to the values of x 2 + c. For the first pair of tables, compare the values of f(0) to g(0), f(1) to g(1), f(2) to g(2) and f(3) to g(3). With your partner, discuss how each of the corresponding values are different. Repeat the process to compare the values of f(x) to h(x) and then f(x) to j(x). f(x) = x 2 g(x) = x x f(x) Points on the graph of f(x) = x 2 x g(x) Points on the graph of g(x) = x (0, 0) 0 4 (0, 4) 1 1 (1, 1) 1 5 (1, 5) 2 4 (2, 4) 2 8 (2, 8) 3 9 (3, 9) 3 13 (3, 13) f(x) = x 2 h(x) = x x f(x) Points on the graph of f(x) = x 2 x h(x) Points on the graph of h(x) = x (0, 0) 0 6 (0, 6) 1 1 (1, 1) 1 7 (1, 7) 2 4 (2, 4) 2 10 (2, 10) 3 9 (3, 9) 3 15 (3, 15) f(x) = x 2 j(x) = x 2 3 x f(x) Points on the graph of f(x) = x 2 x j(x) Points on the graph of j(x) = x (0, 0) 0 3 (0, 3) 1 1 (1, 1) 1 2 (1, 2) 2 4 (2, 4) 2 1 (2, 1) 3 9 (3, 9) 3 6 (3, 6) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 16

21 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date 5. The graphs of h(x), g(x), f(x) and j(x) are shown in the same coordinate plane below. Analyze the four graphs and discuss with your partner what you notice about the four graphs. h(x) = x h(x) g(x) = x g(x) f(x) = x 2 j(x) = x 2 3 f(x) j(x) 6. Based on the conversations you had with your partner (about the tables on the previous page and the graphs above), write a brief summary to describe the general effect that the value of c has on the graph of f(x) = x 2. Note: In your summary, include a statement regarding how the 4 steps for graphing f(x) = x 2 (that were discussed above in Part II) would need to be modified when graphing g(x) = x 2 + c. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 17

22 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date Part IV: Variations of the parent function f(x) = x 2 è f(x) = ax 2 a is the coefficient of x 2, which means that we have to MULTIPLY the value of a by x 2. Working with a partner, analyze each of the three pairs of tables below, comparing the values of x 2 to the values of ax 2. For the first pair of tables, compare the values of f(0) to p(0), f(1) to p(1), f(2) to p(2) and f(3) to p(3). With your partner, discuss how each of the corresponding values are different. Repeat the process to compare the values of f(x) to q(x) and then f(x) to r(x). f(x) = x 2 p(x) = 2x 2 x x 2 Points on f(x) = x 2 x 2x 2 Points on p(x) = 2x (0, 0) 0 0 (0, 0) 1 1 (1, 1) 1 2 (1, 2) 2 4 (2, 4) 2 8 (2, 8) 3 9 (3, 9) 3 18 (3, 18) f(x) = x 2 q(x) = 5x 2 x x 2 Points on f(x) = x 2 x 5x 2 Points on q(x) = 5x (0, 0) 0 0 (0, 0) 1 1 (1, 1) 1 5 (1, 5) 2 4 (2, 4) 2 20 (2, 20) 3 9 (3, 9) 3 45 (3, 45) f(x) = x 2 r(x) = x x 2 Points on f(x) = x 2 x Points on the graph of r(x) = 1 2 x x2 0 0 (0, 0) 0 0 (0, 0) (1, 1) 1 (1, 1 ) 2 4 (2, 4) 2 2 (2, 2) 3 9 (3, 9) 3 (3, ) 4 16 (4, 16) 4 8 (4, 8) x2 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 18

23 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date 7. The graphs of q(x), p(x), f(x) and r(x) are shown in the same coordinate plane below. Analyze the four graphs and discuss with your partner what you notice about the four graphs. q(x) = 5x 2 q(x) p(x) f(x) p(x) = 2x 2 r(x) f(x) = x 2 r(x) = 1 2 x2 8. Based on the conversations you had with your partner (about the tables on the previous page and the graphs above), write a brief summary to describe the general effect that the value of a has on the graph of f(x) = x 2. Note: In your summary, include a statement regarding how the 4 steps for graphing f(x) = x 2 (that were discussed above in Part II) would need to be modified when graphing h(x) = ax 2. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 19

24 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date Part V: Sketching the graph of f(x) = ax 2 è how does the value of a affect the graph when a < 0 Working with a partner, analyze each of the three pairs of tables below, comparing the values of x 2 to the values of ax 2. For the first pair of tables, compare the values of f(0) to k(0), f(1) to k(1), f(2) to k(2) and f(3) to k(3). With your partner, discuss how each of the corresponding values are different. Repeat the process to compare the values of f(x) to m(x) and then f(x) to n(x). f(x) = x 2 k(x) = x 2 x x 2 Points on f(x) = x 2 x x 2 Points on k(x) = x (0, 0) 0 0 (0, 0) 1 1 (1, 1) 1-1 (1, -1) 2 4 (2, 4) 2-4 (2, -4) 3 9 (3, 9) 3-9 (3, -9) f(x) = x 2 m(x) = 3x 2 x x 2 Points on f(x) = x 2 x 3x 2 Points on m(x) = 3x (0, 0) 0 0 (0, 0) 1 1 (1, 1) 1-3 (1, -3) 2 4 (2, 4) 2-12 (2, -12) 3 9 (3, 9) 3-27 (3, -27) f(x) = x 2 n(x) = 1 3 x2 x x 2 Points on f(x) = x 2 x 1 Points on the graph of n(x) = 3 x2 1 3 x2 0 0 (0, 0) 0 0 (0, 0) 1 1 (1, 1) (2, 4) (1, 1 ) 3 ) (2, (3, 9) 3-3 (3, -3) 4 16 (4, 16) (4, 16 ) 3 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 20

25 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date 9. The graphs of f(x), n(x), k(x) and m(x) are shown in the same coordinate plane below. Analyze the four graphs and discuss with your partner what you notice about the four graphs. f(x) = x 2 f(x) n(x) = 1 3 x2 k(x) = x 2 m(x) = 3x 2 m(x) k(x) n(x) 10. Based on the conversations you had with your partner (about the tables on the previous page and the graphs above), write a brief summary to describe the general effect that the value of a has on the graph of f(x) = x 2 when a < 0. Note: In your summary, include a statement regarding how the 4 steps for graphing f(x) = x 2 (that were discussed above in Part II) would need to be modified when graphing j(x) = ax 2 when a < 0. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 21

26 Name Q-2.1: How do a and c Affect the Graph of a Quadratic Function? Pd Date Part VI: Graphing quadratic functions of the form f(x) = ax 2 + c. 11. Sketch the graph for each quadratic function below. The graph of the parent function f(x) = x 2 is provided simply as a reference. Identify the values of a and c and briefly state how they will affect the graph of f(x) = x 2. Sketch the graph of the function. Label the coordinates of the vertex of the graph. A. r x = x # + 3 B. g x = 4x # C. r x = x # 1 D. g x = 3x # + 2 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 22

27 Name Q-2.2: Concavity and y-intercepts Pd Date Part I: Warm-up 1. Re-write the expression 3 x 4 # in the form ax # + bx + c. 2. Factor the following expressions. A. x # + 9x 10 B. 3x # 2x 8 Part II: Understanding of what the constant term, c, tells us about the graph of a quadratic function 3. For each quadratic function below, determine the value of the function at x = 0. A. f x = 2x # + 3x 1 B. g x = x # + x 1 C. h x = 9 # x# 1 4. Compare each of your answers (in question 3 above) to the y-intercept of the graph of each function. What do you notice? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 23

28 Name Q-2.2: Concavity and y-intercepts Pd Date For a quadratic function, f x = ax # + bx + c, the value of c is often referred to as the constant term of the function. Ø f(0) = c Ø Therefore, the value of c tells us the y-coordinate of the y-intercept: (0, c) 5. While Jane is standing near the edge of a cliff enjoying the view of the ocean, she tosses a pebble upward which then falls into the ocean below. The function h t = 16t # + 45t represents the height of the pebble, h(t), measured in feet, t seconds after the pebble left her hand. a. By simply analyzing the function, determine the y-intercept of the graph of h(t). (Note: you do not have to evaluate or graph the function.) b. Interpret what the y-intercept means in the context of the given situation. 6. The Lokahi Surfboard Company uses the following function to predict its monthly profit, P(x), from selling any number of surfboards, x: P x = 8x # + 300x a. By simply analyzing the function, determine the y-intercept of the graph of P(x). (Note: you do not have to evaluate or graph the function.) b. Interpret what the y-intercept means in the context of the given situation. 7. When a baseball is hit into the air, the path that it travels takes the shape of a parabolic curve. When Patrick hit a baseball, the path of the ball could be modeled by the function f x = x # + 4x where f(x)represents the height, in feet, of ball x seconds after the ball was hit. a. By simply analyzing the function, determine the y-intercept of the graph of f(x). (Note: you do not have to evaluate or graph the function.) b. Interpret what the y-intercept means in the context of the given situation. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 24

29 Name Q-2.2: Concavity and y-intercepts Pd Date Part III: Concavity Of the six functions graphed below, compare the three graphs in the top row to the three graphs in the bottom row. What do you notice? A. f x = 9 # x# + x 2 B. g x = 3x # 2x + 9 # C. h x = 8x # + 9x 1 D. p x = 9 > x# + x + 2 E. q x = 2x # + 9x 5 F. r x = 7x # 4x For each function, place a Ö in the appropriate columns. Each row should have two Ö. Function f(x) g(x) h(x) p(x) q(x) r(x) Graph Opens Upwards Graph Opens Downwards Coefficient of x 2 : a > 0 Coefficient of x 2 : a < 0 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 25

30 Name Q-2.2: Concavity and y-intercepts Pd Date Consider any quadratic function, f x = ax # + bx + c, with a 0. Ø The quadratic coefficient a determines the concavity of the graph of f(x): o If a > 0, then the graph of f is concave up (opens upwards). o If a < 0, then the graph of f is concave down (opens downwards). The closer the value of a is to 0, the wider the graph will be. The farther the value of a is from 0, the skinnier the graph will be. Ø The value of the constant coefficient c tells us the y-coordinate of the y-intercept: (0, c) 9. Work with a partner to label each graph with its appropriate function name: p(x), q(x), r(x), or s(x). p x = x # + 3x + 1 q x = x # + 3x 1 r x = x # + 3x + 1 s x = x # + 3x A quadratic function f x = ax # + bx + c will have two x-intercepts if the graph crosses the x- axis at two points. A. For each of the functions above in question 9, place two points on each graph to show the locations of the x-intercepts. B. However, some quadratic functions do not have any x-intercepts: their graphs will never cross the x-axis. Consider the four cases shown below. Working with a partner, circle the two cases that are guaranteed to have x-intercepts, and place an asterisk, * next to the two cases that will NOT have x-intercepts. Case 1: a > 0 and c > 0 Case 2: a > 0 and c < 0 Case 3: a < 0 and c < 0 Case 4: a < 0 and c > 0 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 26

31 Name Q-2.3: Homework Pd Date 1. For the following quadratic functions, first, indicate if its graph will be concave up or concave down; then, determine the y-intercept of the graph. Note: for some of the functions, it may be helpful to re-write the function so that it is in standard form: f x = ax # + bx + c. a. f x = 3x # 4x 5 b. f x = 9 # x# 4x E F c. f x = 3 4x # 2x d. f x = x 2 9 # x# e. f x = (x + 3)(x 2) f. f x = x 5 # g. f x = 2(3x 4)(x 1) h. f x = 2 x + 3 # 4 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 27

32 Name Q-2.3: Homework Pd Date 2. Which of the following functions will have a skinnier graph? Briefly explain why you selected that function. f x = 9 # (x# + 3x 1) or g x = 9 G (4x# + 3x 1) 3. Which of the following functions will have a wider graph? Briefly explain why you selected that function. f x = 9 # (4x 2x# ) or g x = 9 > (4x 3x# ) 4. Fill in the blank with a number so that the graph of the resulting quadratic will be concave down. There is more than one possible correct answer. a) f x = x + 3 x 2 b) f x = (3x 5x # + 2) c) f x = x + 4 # + 2 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 28

33 Name Q-2.4: Graphing Quadratics in Factored Form Pd Date Part I: Warm-up 1. A company that sells televisions uses the following function to predict its monthly profit, P(x), from selling any number of televisions, x: P x = 25x # x a. By simply analyzing the function, determine the y-intercept of the graph of P(x). (Note: you do not have to evaluate or graph the function.) b. Interpret what the y-intercept means in the context of the given situation. 2. The graph of a quadratic function, f x = ax # + bx + c, is shown below. (The scale used on both axes is 1 unit). Circle ALL statements that must be true about f(x) AND explain briefly why the statements you selected must by true. a = 0 a > 0 a < 0 c = 5 c = 1.5 c = 5 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 29

34 Name Q-2.4: Graphing Quadratics in Factored Form Pd Date Part II: Use this space to take notes following your teacher s instructions. f(x) = 1 2 x# x 5 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 30

35 Name Q-2.4: Graphing Quadratics in Factored Form Pd Date Part III: Review: determining the x-intercept of a linear function 3. Determine the x-intercept of each of the following linear functions. Show your work. A. f x = 2x + 1 B. g x = 3x + 5 C. h x = I x 3 > Part IV: The x-intercepts of a quadratic function 4. Determine the x-intercepts of each of the following quadratic functions. Show your work. A. f x = (x 3)(x + 1) B. g x = 2(x 4)(x + 3) C. h x = x # + 10x + 9 D. p x = x # 5x 14 E. k x = 9 # (x + 3)(x + E # ) F. s x = x# 400 G. r x = 2x # + 7x + 6 H. m x = x(x 5) I. f x = 18 2x # J. f x = 9x # 25 K. g x = 5x # 2x 16 L. h x = x 4 # Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 31

36 Name Q-2.4: Graphing Quadratics in Factored Form Pd Date Part V: Reflect and Summarize (follow your teacher s instructions). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 32

37 Algebra 2 Module 5: Symbolic Representation of an Exponential Function Name Q-2.5: Building a Function from its Graph Per Date Part I: Warm-up Each table of values and graph below represent an exponential function of the form f x = c a M. A. Determine the symbolic representation of each function. B. Express your answer in the form f x = c a M C. Use your function to determine the additional values of your function x k(x) x L(x) k (x) = k (0) = k (1) = k (5) = L (x) = L (0) = L (-1) = L (-4) = 3. r(x) ( 2, 400 ) r (x) = r (0) = r (4) = (0, 25) ( 1, 100 ) r (-1) = Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 33

38 Name Q-2.5: Building a Function from its Graph Pd Date Part II: Using the x-intercepts to determine the symbolic form of a quadratic function Following your teacher s instructions, use this space to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 34

39 Name Q-2.5: Building a Function from its Graph Pd Date 4. A quadratic function has x-intercepts at x = 3 and x = -5 and has a y-intercept of (0, 60). Determine the symbolic form of the function (you can express your answer in factored form). 5. The graphs of two quadratic functions are shown below. Determine the symbolic form of the quadratic functions that are graphed below. Write your answers in both factored and standard form. A. Note: The scale used on each axis of the graph is 1 unit. Factored form of f(x): B. Note: The scale used on each axis of the graph is 2 units. Factored form of g(x): Standard form of f(x): Standard form of g(x): Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 35

40 Name Q-2.5: Building a Function from its Graph Pd Date Part III: Reflect and Summarize (follow your teacher s instructions). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 36

41 Name Q-2.6: Matching a Graph with its Symbolic Representation Pd Date This page is intentionally left blank. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 37

42 Name Q-2.6: Matching a Graph with its Symbolic Representation Pd Date Match the following quadratic functions with their symbolic forms found on the following page Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 38

43 Name Q-2.6: Matching a Graph with its Symbolic Representation Pd Date Match the following quadratic functions with their graphical forms found on the previous page. Not all algebraic representations below will be used. P(x) = 0.5 (x - 2) (x + 1) s(x) = -2(x + 1) (x - 3) w(x) = -(x + 3) (x - 1) K(x) = - (x + 1) (x - 3) c(x) = -2 (x + 3)(x 1) v(x) =.25 (x - 2) (x + 1) d(x) = 2(x - 2) (x + 1) f(x) = (x + 2) (x - 1) m(x) = -.5(x + 3) (x - 1) R(x)= 2(x + 2) (x - 1) t(x) = -.25(x + 1) (x - 3) n(x) = (x + 3)(x 1) z(x) = -.5 (x - 2) (x + 1) J(x) =.5(x + 3)(x 1) g(x) = -(x + 2) (x - 1) L(x)= -2(x + 2) (x - 1) h(x) = (x - 2) (x + 1) q(x) = (x + 1) (x - 3) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 39

44 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 40

45 Name Q-2.7: Homework Pd Date 1. Determine the algebraic representation for the quadratic function whose graph is given below. Give your answer in both factored form and in standard form. 2. Determine the algebraic representation for the quadratic function whose graph is given below. Give your answer in both factored form and in standard form. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 41

46 Name Q-2.7: Homework Pd Date 3. The following is the graph of a quadratic function defined by y = f(x). a. Mark with a circle a point on the graph where x = 0. b. Mark with a square a point on the graph where f x > 0. c. Mark with a diamond a point on the graph where f x = 0. d. Determine the algebraic representation of f x. e. Use your algebraic representation to determine the value of f 3. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 42

47 Name Q-2.8: The Vertex of the Graph of a Quadratic Function Pd Date Part I: Warm-up 1. L(x) is an exponential function such that its initial value is 40 and it has a constant ratio of 9 #. A. Complete the table of values below for L(x). B. Determine the value of L(-2). x L(x) Part II: The vertex of the graph of a quadratic function 2. The graphs of two quadratic functions are shown below. Analyze each graph and discuss with a partner a few things that you notice. Write down a few ideas that you discussed regarding what might be true about each function and graph. Note: The scale used on each axis is 1 unit. p(x) r(x) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 43

48 Name Q-2.8: The Vertex of the Graph of a Quadratic Function Pd Date Following your teacher s instructions, use the space below to take notes. p x = x # + 4x 3 r x = 2x # + 5x 1 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 44

49 Name Q-2.8: The Vertex of the Graph of a Quadratic Function Pd Date Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 45

50 Name Q-2.8: The Vertex of the Graph of a Quadratic Function Pd Date 3. Determine the coordinates of the vertex of each of the following quadratic functions. A. f x = 3x # 6x + 1 B. f(x) = 9 # x# + x + 1 C. f x = 5x # 9 D. f(x) = 8x # E # x + 4 E. f x = x # 10x F. f(x) = x # x + 3 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 46

51 Name Q-2.8: The Vertex of the Graph of a Quadratic Function Pd Date 4. Below is a graph of a quadratic function, however, only the x-axis is numbered. Determine the value that should go in the blank so that the resulting function will accurately represent the graph. A. f x = x # + x + 4 B. f(x) = 5x # + x + 7 C. f x = x # + 12x + 4 D. f(x) = x # + 6x + 4 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 47

52 Name Q-2.8: The Vertex of the Graph of a Quadratic Function Pd Date 5. Determine the maximum or minimum value of the following quadratic functions. Ø Indicate at which x-value the maximum or minimum value of the function occurs. Ø And, specify if the function has a maximum or minimum. A. q x = 3x # + x 2 B. r x = x # + 2x 4 C. k x = 4x # + 5 D. L x = x # 6x + 9 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 48

53 Name Q-2.9: Representing a Quadratic Function in Vertex Form Pd Date Part I: Warm-up 1. Re-write the following quadratic functions in standard form: f x = ax # + bx + c A. f x = x 5 # B. g x = 3 x + 5 # C. h x = 2 x + 3 # + 1 D. k x = 9 > x + 10 # + 6 Part II: Three different, yet equivalent, ways to symbolically represent a quadratic function. So far we ve studied two forms of a quadratic function: i. Standard Form: f x = ax # + bx + c ii. Factored Form: f x = a(x r)(x s) Now we will explore another form of a quadratic function: iii. Vertex Form: f (x) = a(x h) 2 + k o The values of h and k tell us the vertex of the graph: (h, k) o Just as before, the value of a tells us if the parabola opens up or down, and tells us if the graph is wider/skinnier than the parent function. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 49

54 Name Q-2.9: Representing a Quadratic Function in Vertex Form Pd Date The quadratic function is said to be in vertex form if it is written as f(x) = a(x h) # + k. The vertex of the graph of the quadratic function is at the point (h, k). The graph of the function f x = 2 x 1 # + 3 is shown to the right. Analyzing the graph, we see that its vertex has coordinates (1, 3). f(x) Notice that in the symbolic representation of the function, h = 1 and k = 3. These values correspond exactly to the x- and y-coordinates of the vertex. 2. Re-write the function above, f x = 2 x 1 # + 3, so that it is written in standard form. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 50

55 Name Q-2.9: Representing a Quadratic Function in Vertex Form Pd Date 3. Using your answer in question 2 (above), determine the coordinates of the vertex by computing the values of Q Q and f. #R #R 4. Compare your answer for question 3 to the vertex of the graph shown on the previous page. What do you notice? Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 51

56 Name Q-2.9: Representing a Quadratic Function in Vertex Form Pd Date Part III: Re-writing a quadratic function from standard form into vertex form. 5. Rewrite the function f x = x # + 3x 9 in the form f x = a x h # + k, where a, h, and k are constants. 6. Rewrite the function f x = 5x # 20x + 64 in the form f x = a x h # + k, where a, h, and k are constants. 7. Rewrite the function f defined by f x = 4(x + 3)(x 5) in the form f x = a x h # + k, where a, h, and k are constants. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 52

57 Name Q-2.9: Representing a Quadratic Function in Vertex Form Pd Date Part IV: Graphing quadratic functions given in vertex form. 8. Consider the function g(x) = (x 2) 2-1 a. Identify the vertex and sketch the graph of the function. b. Describe how the parent function f(x) = x 2 is transformed. 9. Consider the function g(x) = (x - 5) 2 8 a. Identify the vertex and sketch the graph of the function. b. The parent function f(x) = x 2 is shown; describe how it has been transformed. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 53

58 Name Q-2.9: Representing a Quadratic Function in Vertex Form Pd Date 10. Consider the function h(x) = 2(x + 8) 2 2 a. Identify the vertex and sketch the graph of the function. b. The parent function f(x) = x 2 is shown; describe how it has been transformed. 11. Consider the function p(x) = 2 1 (x + 5) 2 16 a. Identify the vertex and sketch the graph of the function. b. The parent function f(x) = x 2 is shown; describe how it has been transformed. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 54

59 Name Q-2.10: Summarizing the Three Forms of a Quadratic Function Pd Date Part I: Warm-up 1. Fill in each blank with one of the following statements to show the three different forms for representing a quadratic function: f x = a x h # + k f x = a(x r)(x s) f x = ax # + bx + c Standard Form: Factored Form: Vertex Form: 2. Below are three equivalent forms of the same quadratic function. Standard Form: f x = 3x # + 12x 9 Vertex Form: f x = 3 x 2 # + 3 Factored Form: f x = 3(x 1)(x 3) A. Which form reveals the y-intercept without changing its form, and what is the y-intercept? B. Which form reveals the x-intercept(s) without changing its form, and what is/are they? C. Which form reveals the maximum value for f x without changing its form, and what is its value? At which x does it occur? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 55

60 Name Q-2.10: Summarizing the Three Forms of a Quadratic Function Pd Date 3. Determine whether each function in the table represents the graph of the quadratic function shown. Select Yes or No for each function. Function Yes No 4. Given the function f x = x # + 2x + 3, Place a point on the coordinate grid to show each x-intercept of the function. Place a point on the coordinate grid to show the maximum value of the function. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 56

61 Name Q-2.10: Summarizing the Three Forms of a Quadratic Function Pd Date 5. Given the function f x = 2 x + 2 # 18, Place a point on the coordinate grid to show the maximum value of the function. Place a point on the coordinate grid to show the y-intercept of the function. Ø Note: the scale used on the x-axis is 2 units and the scale used on the y-axis is 10 units. 6. Compare the quadratic function f (whose graph is shown below) to g x = (x 1)(x 4). Select whether each statement is True or False. Statement True False f The minimum value for f(x) is greater than the minimum value for g(x). The value of x when f(x) is at its minimum is greater than the value of x when g(x) is at its minimum. Both x-intercepts of g(x) occur when x is less than zero. The line of symmetry of f(x) is x = -2. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 57

62 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 58

63 Q 3.1: Graphing Quadratic Functions Name Per Date Part I: Reading an equation as a question. An equation that contains a variable is essentially asking you a question. For example: The equation 5x + 13 = 43 is asking you to answer the question, What number, when I multiply it by 5 and then add 13 is equal to 43? Translate each of the following equations into a complete sentence/question. (Do NOT solve the equations; just translate them into words.) x a. 3 7 = 2 b. 100 = 2(x + 5) c. x 2 +1= 50 d. ( x + 3) ( x 2) = 0 Part II: Solving quadratic equations written in factored form There are 4 main strategies for solving a quadratic equation, and we re going to review the 4 strategies one at a time, starting with a quadratic equation written in factored form: (x s)(x t) = 0 In order to solve a quadratic equation written in factored form, (x s)(x t) = 0, we need to review an important algebraic property. Consider the meaning of this equation: a*b = 0 Ø If we translate this equation into words, we would get a question like, What two numbers, when I multiply them, have a product of zero? Ø But we learned in 3 rd grade that any time you multiply by zero, the product is zero. Ø Therefore, in the equation a*b = 0 what must be true about the value of a or b? The Zero-Product Property: if a*b = 0, then either a = 0 or b = 0 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 59

64 Q 3.1: Graphing Quadratic Functions Name Per Date Now, let s use the Zero-Product Property to help us make sense of the solution to a quadratic equation written in factored form, (x s)(x t) = 0. Let s try to relate the equation (x s)(x t) = 0 to the equation a*b = 0. Ø The equation (x s)(x t) = 0 has 2 factors: a b Ø So, (x s) is the first factor let s refer to that quantity (the entire factor) as a Ø And, (x t) is the second factor let s refer to that quantity (the entire factor) as b Ø If we are given the fact that those 2 factors have a product of zero, then, by the Zero-Product Property, one of them MUST be equal to zero. In other words, if we know that (x s)(x t) = 0 then, either (x s) = 0 or (x t) = 0. Now we have 2 rather simple linear equations to solve: x s = 0 or x t = 0 adding s to both sides of the equation gives us x = s or x = t adding t to both sides of the equation gives us Example 1: Solve (x + 3)(x 4) = 0. Since the given equation is equal to zero, we can apply the Zero-Product Property: (x + 3)(x 4) = 0 x + 3 = 0 or x 4 = 0 x = -3 or x = 4 We ended up with 2 solutions: x = -3 or x = 4. Let s verify that these solutions make the original equation a true statement: Check the solution x = -3 Check the solution x = 4 (x + 3)(x 4) = 0 (x + 3)(x 4) = 0 (-3 + 3)(-3 4) (4 + 3)(4 4) (0)(-7) (7)(0) 0 0 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 60

65 Q 3.1: Graphing Quadratic Functions Name Per Date Part III: Solve the following equations. Show your work. 1. (x 4)(x + 10) = 0 2. (x + ¼ )(x + 9) = (x 4)(x + 10) = 0 4. x(x + 10) = 0 5. (x 40)(x + 100) = x(x - 12) = 0 7. Explain why you cannot use the Zero-Product Property to solve (x 1)(x + 2) = 5. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 61

66 Q 3.1: Graphing Quadratic Functions Name Per Date Part IV: Using factoring to solve quadratic equations. Previously we ve solved quadratic functions that were given to us in factored form. Now we want to be able to solve quadratic equations given to us in the form ax 2 + bx + c = 0. o First we factor o Then we apply the Zero-Product Property o BUT, we can apply this property only if the quadratic expression is equal to zero. Let s try a few. Solve the following equations. 8. x 2 x 30 = 0 9. x 2 + 6x = x 2 25 = x 2 = 6x 12. 2x 2 5x 3 = 0 Part V: Recall that one of the main reasons why we want to solve a quadratic equation is to find the zeros of a quadratic function. o Finding the zeros gives us information about the x-intercepts of the graph of the function o Once we know the x-intercepts, we can figure out the vertex and sketch a graph of the function. Example 1: Graph f (x) = x 2 2x 3 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 62

67 Q 3.1: Graphing Quadratic Functions Name Per Date Example 2: Graph f (x) = x 2 2x 15 Example 3: Graph f (x) = 2x 2 8x Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 63

68 Q 3.1: Graphing Quadratic Functions Name Per Date Part VI: Practice 13. Graph f(x) = x 2 6x 16. Indicate all intercepts and the vertex. 14. Graph f(x) = x 2 7x. Indicate all intercepts and the vertex. 15. Solve: x 2 + 2x 144 = 2x Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 64

69 Name Q-3.2: Square Roots Pd Date Part I: Warm-up 1. Working with a partner, discuss the 3 related vocabulary terms concerning quadratic functions, equations and graphs. Answer the questions below using the graph of g(x) as a reference. a. a solution to a quadratic equation solve 2(x 1)(x 5) = 0 o solutions: x = 1 and x = 5 g(x) = 2(x 1)(x 5) Ø How do these solutions relate to the graph of g(x)? b. a zero of a quadratic function a zero is the value of x that makes g(x) = 0 g(1) = 2(1 1)(1 5) = 2(0)(-4) = 0 g(5) = 2(5 1)(5 5) = 2(4)(0) = 0 therefore, the zeros of g(x) are 1 and 5 Ø How do the zeros relate to the solutions of the equation and the graph of the function? c. an x-intercept of the graph of a quadratic function An x-intercept is the coordinates of the point where the graph of g(x) intersects the x-axis o x-intercepts of g(x): (-1, 0) and (5, 0) Ø How do the x-intercepts relate to the solutions of the equation and the zeros of the function? 2. Briefly explain how to determine the coordinates of the VERTEX of the graph of a quadratic function if you know the zeros of the function. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 65

70 Name Q-3.2: Square Roots Pd Date Part II: Rewriting a radical expression in an equivalent form 3. Complete the table of values below. n n Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 66

71 Name Q-3.2: Square Roots Pd Date Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 67

72 Name Q-3.2: Square Roots Pd Date 4. Solve the following equations. If your answer involves a square root, express your answer such that the number under the radical doesn t include any perfect square factors. A. 4x # 36 = 0 B. 5x # = 0 C. x # 50 = 0 D. 9 # x# 6 = 0 E. 3(x 4) # 50 = 250 F. 2(x + 1) # 5 = 85 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 68

73 Name Q-3.3: Homework Pd Date Solve each equation. If your answer involves a square root, express your answer such that the number under the radical doesn t include any perfect square factors. 1. (x 5) = (x 7) = x 2 = (x 12) 2 5 = x = (x 10) = 0 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 69

74 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 70

75 Name Q-3.4: Quadratic Equations in Vertex Form Pd Date Part I: Following your teacher s instructions, use this space to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 71

76 Name Q-3.4: Quadratic Equations in Vertex Form Pd Date Part II: Quadratic equations in vertex form Although solving quadratic equations is the focus of this unit, we did have to do some of this in the previous unit when we were finding the x-intercepts of quadratic functions. Recall that an x-intercept of a quadratic function f is one that makes the equation f(x) = 0 true. Example: Determine the x-intercepts of f x = 9 # x 2 # 3 whose graph shown below. Of course, we can easily spot other graphical features of this quadratic from the algebraic representation: The graph is concave up this makes sense since in the function a = 9 is positive. # The vertex is at (2, -3) this makes sense since these are the h and k values in the general vertex form of a quadratic function: f x = a x h # + k. The minimum value of the function is -3 this makes sense since the y-coordinate of the vertex is -3. The y-intercept can be determined by evaluating the function at x = 0: f 0 = # 3 f 0 = f 0 = 1 Therefore, the y-intercept is (0, 1). However, we are unable to determine the exact locations of the x-intercepts by simply looking at the graph. In the graph we can see that the x-intercepts are not integers: The x-intercept on the left lies between -1 and 0. The right x-intercept on the right lies between 4 and 5. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 72

77 Name Q-3.4: Quadratic Equations in Vertex Form Pd Date To determine the exact values of the x-coordinate of the x-intercepts, we must solve f x = x 2 # 3 = x 2 # = 3 x 2 # = 6 x 2 # = 6 x 2 = ± 6 x 2 = ± 6 x = 2 ± 6 x = or x = 2 6 x = or x = Looking back at the graph of f(x), do these solutions seem to be accurate regarding the x- coordinate of the x-intercepts? Part III: Following your teacher s instructions, use the space below to take notes (using the examples provided. 1. To the right is the graph of the quadratic function f x = x 1 # 3. a. Place a square around the y-intercept on the graph. Use the algebraic representation to find the y- intercept. b. Place circles around the x-intercepts on the graph. Determine the exact value of the x-coordinate of the x-intercepts. Then, use a calculator to write your x-intercepts as decimals (accurate to 3 decimal places). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 73

78 Name Q-3.4: Quadratic Equations in Vertex Form Pd Date 2. To the right is a graph of the quadratic function f x = 2 x 2 # + 6 and the line y = 3. a. Place circles around the two points where the graph of f(x) and the line y = 3 intersect. b. Find the x-coordinates for these points of intersection. To do this, set-up the equation f x = 3, and then solve for x: 2 x 2 # + 6 = Given that k x = 3 x 1 # 12, 4. Given that L x = x + 4 # 3 determine the solutions to k x = 0 determine the solutions to L x = 2 and explain what your solutions represent and explain what your solutions represent regarding the graph of k x. regarding the graph of L x. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 74

79 Name Q-3.4: Quadratic Equations in Vertex Form Pd Date Part IV: Rewriting a quadratic function into vertex form Previously we learned how to determine the coordinates of the vertex of a quadratic function given in standard form. Let s review that process. The coordinates of the vertex of a quadratic function, f x = ax # + bx + c, will be b 2a, f b 2a Ø This means that the x-coordinate of the vertex will be the value of Q #R. Ø Then, we evaluate the function at that value to determine the y-coordinate of the vertex. Let s practice. 5. Determine the vertex of the function 6. Determine the vertex of the function f x = 3x # + 12x 7 g x = x # 8x 5 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 75

80 Name Q-3.4: Quadratic Equations in Vertex Form Pd Date If a quadratic function is given in standard form, f x = ax # + bx + c, we can rewrite the function into vertex form, f x = a(x h) # + k: Ø The value of a will be the same in both forms of the function. Ø h = Q #R Ø k = f Q #R 7. Express f x = x # 10x Express g x = 2x # + 4x 8 in vertex form: f x = a(x h) # + k in vertex form: g x = a(x h) # + k 9. Express p x = 5x # 30x Express r x = x # + 4x 8 in vertex form: p x = a(x h) # + k in vertex form: r x = a(x h) # + k Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 76

81 Name Q-3.5: The Quadratic Formula Pd Date Part I: Warm-up Solve each equation using the graph provided. The scale used on each axis is 1 unit. 1. x # 4 = # x# + 2 = 0 g(x) = 1 2 x# + 2 f(x) = x # 4 Part II: Solving a quadratic equation using the Quadratic Formula So far we ve learned 3 different methods to solve a quadratic equation: GRAPH the expressions on each side of the equation and identify the x-coordinate of the points of intersection. Set the equation equal to zero and FACTOR the quadratic expression. Rewrite the equation so that the x 2 term is isolated on one side of the equation and then take the SQUARE ROOT of each side of the equation. However, consider the equation x 2 5x 2 = 0 The x-intercepts of the graph of f (x) = x 2 5x 2 are difficult to determine because the graph crosses the x-axis at values that are not integers (i.e., decimal values that we cannot precisely determine by simply looking at the graph). The quadratic expression x 2 5x 2 is not factorable. Isolating the x 2 term to use the square root simply results in something really ugly that we don t know how to solve. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 77

82 Name Q-3.5: The Quadratic Formula Pd Date Therefore, we need another method that will help us to solve any quadratic function (particularly the more challenging quadratic equations that cannot be factored): the Quadratic Formula. The Quadratic Formula If ax 2 + bx + c = 0, where a 0, then x = b ± b2 4ac 2a This means that I can determine the solutions for any quadratic equation by setting it equal to zero and substituting the values of a, b and c into the formula. Note: The ± in the formula has two implications: First, it allows us to rewrite the quadratic formula in a way that helps us to relate the solutions to the graph: x = b 2a ± b 2 4ac 2a b o The part of the solution tells us the axis of symmetry (and thus, the x-coordinate 2a of the vertex). Second, we can separate the expression even further to show the 2 possible solutions of the equation (and these solutions are the x-coordinates of the x-intercepts): x = b 2a + b 2 4ac 2a or x = b 2a - b 2 4ac 2a Axis of Symmetry: (, 0 ) (, 0 ) Vertex at Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 78

83 Name Q-3.5: The Quadratic Formula Pd Date Example 1: Solve 2x 2 + 3x 9 = 0 First, we must identify the values of a, b and c. The standard form of a quadratic equation is ax 2 + bx + c = 0. Thus, a = 2 b = 3 c = -9 Now, substituting these values into the quadratic formula: x = b ± b2 4ac 2a x = ( 3 ) ± ( 3) 2 4( 2) ( 9) 2( 2) x = 3± 9 ( 72) 4 (-3, 0) (3/2, 0) x = 3± 81 4 x = 3± 9 4 y-intercept at (0, c): (0, -9) x = or x = x = 6 4 or x = 12 4 vertex at : ( -3/4, -81/8 ) x = 3 2 or x = 3 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 79

84 Name Q-3.5: The Quadratic Formula Pd Date 3. Let s practice using the quadratic formula to solve each of the equations below. When necessary, round solutions to the nearest hundredth. A. x 2 +8x 1= 0 B. 2x 2 11x +12 = 0 C. t = 0 a = a = a = b = b = b = c = c = c = x = b ± b2 4ac 2a x = b ± b2 4ac 2a t = b ± b2 4ac 2a Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 80

85 Name Q-3.5: The Quadratic Formula Pd Date D. x # 2x = 3 E. 3x # 17x = 15 x # F. 16t # + 25 = 0 x = b ± b2 4ac 2a x = b ± b2 4ac 2a t = b ± b2 4ac 2a Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 81

86 Name Q-3.5: The Quadratic Formula Pd Date 4. Work with a partner to discuss the following problems and work together to answer the question. A. Max was working with a quadratic function, f(x), and made the following conclusion: The x-intercepts of the graph of f(x) occur at x = -1 and x = 13, and the vertex is at (3,-5). Even though you don t know what the actual function is, explain why Max s conclusion is not true. B. The graph of some quadratic function, g(x), has x-intercepts at (4, 0) and (7, 0). What is the x-coordinate of the vertex of g(x)? C. The graph of some quadratic function, p(x), has its vertex located at I #, 7. Maya claimed that the each of the following pairs of values could possibly be the x-coordinates of the x-intercepts of the graph of p(x): x = -3 and x = -2 x = 0 and x = 5 x = -5 and x = 5 x = -5 and x = 0 Is Maya s claim correct? Explain why or why not? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 82

87 Name Q-3.5: The Quadratic Formula Pd Date 5. A ball is hit by a baseball player, and the height of the ball is given by the function h t = 16t # + 58t + 3 where h(t) is the height of the ball (in feet), t seconds after the ball is hit. a. What is the value of h(0). What does this value represent in the context of the given situation? b. What is the value of h(1). What does this value represent in the context of the given situation? c. If you graphed the function, h(t), the vertex will represent the maximum height that the ball reaches after being hit. Determine the coordinates of the vertex of h(t) and interpret what these values mean in the context of the given situation. d. Recall that an equation is actually asking you a question. Translate the equation h(t) = 0 into a question, and then explain what the answer to this question represents in the context of the given situation. e. Set up an equation and solve it using the quadratic formula, to determine the time, t, when the ball will hit the ground. Round your answer to the thousandths place. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 83

88 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 84

89 Name Q-3.6: Complex Numbers Pd Date Part I: Warm-up 1. Determine the x-intercepts and the vertex of the following quadratic functions. Then, use your answers to make a sketch of the graph. A. f x = x # + 2x 8 B. f x = x # Rewrite the following radical expressions so that the number under the radical doesn t include any perfect square factors. A. 18 B. 75 C D. 200 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 85

90 Name Q-3.6: Complex Numbers Pd Date Part II: Following your teacher s instructions, use the space below to take notes. The quadratic function f x = x # + 1 is graphed below. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 86

91 Name Q-3.6: Complex Numbers Pd Date Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 87

92 Name Q-3.6: Complex Numbers Pd Date Part III: 3. Write an equivalent expression for each of the following. a b. EWW # c. X>I E d. X>Y G e f g h Solve each equation. Express your solutions so that the number under the radical has no perfect square factors and is not negative. A. x # + 2x + 10 = 0 B. x # + 7 = 0 C. x # + 5 = 3x Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 88

93 Name Q-3.6: Complex Numbers Pd Date D. 2x # 3x + 4 = 0 E. 9 # x# + 5x = 14 F. x # = x 2 Part IV: Reflect and Summarize Each equation above in question 4, can be expressed in the form f x = 0. Some of the equations are given in this form, but for three of the equations (C, E, and F) we had to rewrite them to make the f x equal to zero. Using a graphing calculator graph each function, f x, that is embedded in each equation. 5. What do you notice about the x-intercepts of all six graphs? 6. Review your solutions to all six equations that you solved above (in question 4) and consider your solutions in light of the x-intercepts of all six graphs. What conclusion can you make? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 89

94 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 90

95 Algebra 2 Module 2: Algebra Review Name Q-3.7: Rounds Activity Per Date Follow your teacher s instructions for this activity. Round 1 Round 2 Given: f x x x 2 ( ) = Given: f x x x 2 ( ) = Factor completely: 1. Rewrite f(x) so it is in vertex form: 2. y-intercept: 2. y-intercept: 3. x-intercepts: 3. x-intercepts: 4. Vertex: 4. Vertex: 5. Graph f x x x 2 ( ) = Graph f x x x 2 ( ) = Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 91

96 Name Q-3.7: Rounds Activity Pd Date Round 3 Round 4 Given f x x x 2 ( ) = Given f x 2 ( ) = 2x Use the quadratic formula to solve 1. y-intercept: = 2 2x 3x y-intercept: 2. x-intercepts: 3. x-intercepts: 3. Vertex: 4. Vertex: 2 4. Graph f( x) = 2x Graph f x x x 2 ( ) = Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 92

97 Name Q-3.8: Homework Pd Date 1. Consider the three functions given below: f x = 2x # + 6x 55 g x = 3(x 2)(x + 4) h x = 2(x + 1) # 4 a. For which function above is it easiest to determine the vertex? What is the vertex of that function? b. For which function above is it easiest to find the x intercepts? What are the x-intercepts for that function? c. For which function is it easiest to find the y intercept? What is the y-intercept for that function? 2. Graph the function defined by f x = (x 2)(x + 4) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 93

98 Name Q-3.8: Homework Pd Date 3. Rewrite f(x) = 2x 2 12x + 14 to express it in vertex form and interpret what your result tells you about the graph of f(x). Note: you do NOT have to sketch the graph. 4. Use the quadratic formula to help you determine the x-intercepts for the graph of each function below. Your answer should be stated as coordinates. A. f(x) = 2x 2 12x + 14 B. g(x) = 2x 2 + 9x 14 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 94

99 Algebra 2 Module 8: Introduction to Polynomial Functions Name Q-3.9: Determining Points of Intersection Pd Date Part I: Warm-up 1. Consider the equation 5x # + 6x + 2 = 0 Solve the equation. The equation above is written in the form f x = 0. Write a statement to explain what your solution tells us about the graph of f x. 2. Consider the function p x = 5x # + 6x + 2. Rewrite p x so that it is expressed in vertex form. Then, write a statement to explain what your solution represents regarding the graph of p x. 3. The same function is written below three times, but expressed in a different form each time: k x = 3(x 2)(x + 4) k x = 3x # 6x + 24 k x = 3 x + 1 # + 27 Determine the following characteristics regarding the graph of k x. Ø Note: you do not need calculate or graph anything; all the information that you need is provided among the three forms of the function provided above. A. y-intercept: B. vertex: C. x-intercepts: D. concavity: Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 95

100 Algebra 2 Module 8: Introduction to Polynomial Functions Name Q-3.9: Determining Points of Intersection Pd Date Part II: What does it mean when we set two functions equal to each other? Consider the functions f x = x # 2 and g x = 2x Set-up the equation f x = g(x) and then solve it. 5. The graphs of f x = x # 2 and g x = 2x + 1 are shown in the coordinate plane below. A. What are the coordinates of the points where f x and g x intersect? B. Compare your answer to question 4 (above) to the coordinates of the points of intersection. What do you notice? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 96

101 Algebra 2 Module 8: Introduction to Polynomial Functions Name Q-3.9: Determining Points of Intersection Pd Date Part III: Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 97

102 Algebra 2 Module 8: Introduction to Polynomial Functions Name Q-3.9: Determining Points of Intersection Pd Date Part IV: For each pair of functions given, determine the coordinates of the points where their graphs will intersect. 6. f x = x # 5x + 7 and g x = 2x f x = x # 2x + 7 and g x = 5x f x = x # + 3x 5 and g x = x # 4x f x = 2x # 5x + 3 and g x = x # 4x + 2 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 98

103 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.1: The Standard Symbolic Form Pd Date Part I: Warm-up 1. Determine the following sums or differences. Write your final answer such that the exponents on the variables are in descending order (from largest to smallest): for example, ax 2 + bx + c. A. 21x # + 15x x + 16x # B. 5m # m 6m + m # 1 2. Determine the following products. Write your final answer such that the exponents on the variables are in descending order (from largest to smallest): e.g., ax 2 + bx + c. A. 5 9x 25 B. 3x + 5 4x 9 C. 4n 7 n # 8 D. 3p # + 1 p # + 5p 2 3. Working with a partner, compare your answers to question 2 above. Then, discuss the statements below and circle the appropriate phrase that should be used to complete the sentence. A. In question 2A above, when we multiplied a constant by a linear expression, the product that resulted was. a LINEAR expression a QUADRATIC expression some other kind of expression we haven t learned yet B. In question 2B above, when we multiplied a linear expression by a linear expression, the product that resulted was. a LINEAR expression a QUADRATIC expression some other kind of expression we haven t learned yet C. In question 2C above, when we multiplied a linear expression by a quadratic expression, the product that resulted was. a LINEAR expression a QUADRATIC expression some other kind of expression we haven t learned yet Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 99

104 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.1: The Standard Symbolic Form Pd Date Part II: Polynomial Functions Polynomial Functions: The standard symbolic form of an n th degree polynomial is given by P(x) = c \ x \ + c \X9 x \X9 + c \X# x \X# + + c # x # + c 9 x + c W While this general definition may at first seem complicated, the fact is that a polynomial is nothing more k than the sum of simple terms, each of the form c x. Ø A polynomial expressed in standard form has its terms written in order from the highest power of x to the lowest power of x. Ø The value of c is called the coefficient of the term. Ø The term that shows only a coefficient (i.e., a factor of x k is not shown) is called the constant term. Ø The largest power, k, that appears in the polynomial tells us the DEGREE of the polynomial. Ø The coefficient of the term with the largest power is called the leading coefficient. 4. Express each polynomial in standard form, then identify the degree, the leading coefficient and the constant term of each polynomial. Degree A. p x = x # 9 + [ E x> I x + 7xE E Leading Coefficient Constant Term B. q x = 9 # x + 5 2x# + 6x Degree Leading Coefficient Constant Term Degree C. r x = 1 4x E 1 + 4x E Leading Coefficient Constant Term Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 100

105 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.1: The Standard Symbolic Form Pd Date Work with a partner to answer the following questions. 5. A polynomial function P(x) is created by finding the product of f(x) and g(x). If f x = 3x # + 2, determine a function, g(x), such that P(x) is a 5 th -degree polynomial with leading coefficient 12 and a constant term of A polynomial function Q(x) is created by finding the product of j(x) and k(x). If j x = 2x # + 5x 10, determine a function, k(x), such that Q(x) is a 12 th -degree polynomial with leading coefficient 1 and a constant term of R(x) = v(x)* w(x). If R(x) is a 10 th -degree polynomial and v x = Y > xe + F x, What must be true > about the degree of w(x)? 8 f(x) = g(x)* h(x), such that f(x) is a second-degree polynomial. Select all of the following statements that must be true about g(x) and h(x). A. Both g(x) and h(x) are quadratic functions. B. Both g(x) and h(x) are linear functions. C. g(x) could be a linear function and h(x) could be a quadratic function. D. g(x) could be a linear function and h(x) could contain only a constant term. E. g(x) could be a quadratic function and h(x) could contain only a constant term Determine whether each expression is equivalent to 8x Select Yes or No for each expression Yes No Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 101

106 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.1: The Standard Symbolic Form Pd Date Determine whether each expression is equivalent to 1000x Select Yes or No for each expression. Yes No 11. Determine to which family each function below belongs by placing an X in each row of the table. Function Linear Quadratic Polynomial Exponential None Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 102

107 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Part I: Warm-up 1. Determine the x-intercepts and the vertex of the following quadratic functions. Then, use your answers to make a sketch of the graph. A. f x = x # 4x 5 B. f x = 16x # + 96x 2. If f x = x #, g x = x E + x, and h x = x > 1, determine the functions that are created as a result of multiplying the indicated functions. A. p x = g x h(x) B. q x = f x h(x) C. r x = f x g(x) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 103

108 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Part II: Exploring higher-degree polynomials 3. Use a graphing calculator (e.g., a handheld unit or the graphing calculator at to sketch the graph of each of the following functions. For each graph, show and label the coordinates of the points that represent the values of f (-2), f (-1), f (0), f (1) and f (2). A. f x = x E B. f x = x > C. f x = x I D. f x = x G Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 104

109 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date 4. Work with a partner to answer the following questions regarding the 4 graphs you just created. A. Which functions have similar shapes? B. Compare the shapes of f x = x E and f x = x I in contrast to the shapes of the graphs of f x = x > and f x = x G. Describe the difference in the shapes between each pair of graphs. C. What conclusion can you draw about what the degree of the polynomial tells us about the shape of the graph? D. Without using a graphing calculator, quickly sketch the general shape of the graph of each of the following functions. g x = x 9> h x = x 9E j x = x F k x = x 9W 5. Return to the 4 graphs you created on the previous page. In each coordinate plane, use a dashed curve to show the graph of the g x = f(x). In other words, in the same coordinate plane as the graph of f x = x E, add the graph of g x = x E (using a dashed curve). Then, repeat to show the graphs of g x = x >, etc. Use a graphing calculator (e.g., a handheld unit or the graphing calculator at to help you. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 105

110 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Summary: There are only two basic shapes that occur for the parent polynomial functions of the form n f( x) = c x. n - If n is even then the graph of f( x) = c x resembles that of a parabola. n - If n is odd then the graph of f( x) = c x resembles that of a cubic. The end-behavior of a function refers to what happens at the ends of each graph. As the x-values keep getting larger and larger (e.g., x = 100, x = 1,000, x = 100,000, etc.), what happens to the value of f(x)? What direction does the graph of f(x) go toward? As the x-values keep getting smaller and smaller (e.g., x = 100, x = 1,000, x = 100,000, etc.), what happens to the value of f(x)? What direction does the graph of f(x) go toward? The end-behavior of a polynomial function of the form f( x) = c x is determined by two characteristics: whether or not the degree of the polynomial, n, is even or odd; whether or not the leading coefficient, c, is positive or negative; n 6. n Each of the following graphs represent a polynomial function of the form f( x) = c x. For each graph, indicate the following: if the degree of f(x) is even or odd, and if the leading coefficient (LC) of f(x) is positive or negative. A. Degree: B. Degree: C. Degree: D. Degree: LC: LC: LC: LC: Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 106

111 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Part III: End-behavior for generalized polynomial functions n We've seen that the end behavior for functions of the form f( x) = c x is easily identified. But what about more complicated cubic polynomials, or polynomials of higher degree? Luckily, as the following examples indicate, the end behavior of more complicated polynomial functions mimics that of the parent functions. Example A: Cubic (3 rd -degree) Polynomials f( x) = x 3 3 gx ( ) =- x+ 3x hx x x x 3 2 ( ) = Analyze the graph of h(x) to help you complete the following statements. The leading coefficient of h(x) is (pos/neg). For large positive values of x, h(x) is (pos/neg). For large negative values of x, h(x) is (pos/neg). Briefly describe how the end-behavior of the graph of h(x) is similar to the end-behavior of the graph of f x = x E. Analyze the graph of g(x) to help you complete the following statements. The leading coefficient of g(x) is (pos/neg). For large positive values of x, g(x) is (pos/neg). For large negative values of x, g(x) is (pos/neg). Briefly describe how the end-behavior of the graph of g(x) is different from the end-behavior of the graph of f x = x E. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 107

112 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Example B: Quartic (4 th -degree) Polynomials f x = x > g x = 2x > + 6x # + x + 1 h x = 0.5x > 2x # + 2 Analyze the graph of h(x) to help you complete the following statements. The leading coefficient of h(x) is (pos/neg). For large positive values of x, h(x) is (pos/neg). For large negative values of x, h(x) is (pos/neg). Briefly describe how the end-behavior of the graph of h(x) is similar to the end-behavior of the graph of f x = x >. Analyze the graph of g(x) to help you complete the following statements. The leading coefficient of g(x) is (pos/neg). For large positive values of x, g(x) is (pos/neg). For large negative values of x, g(x) is (pos/neg). Briefly describe how the end-behavior of the graph of g(x) is different from the end-behavior of the graph of f x = x >. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 108

113 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Example C: Quintic (5 th -degree) Polynomials f x = x I g x = x I 5x E + 4x h x = x > + 2x Analyze the graph of g(x) to help you complete the following statements. The leading coefficient of g(x) is (pos/neg). For large positive values of x, g(x) is (pos/neg). For large negative values of x, g(x) is (pos/neg). Briefly describe how the end-behavior of the graph of g(x) is similar to the end-behavior of the graph of f x = x I. Analyze the graph of h(x) to help you complete the following statements. The leading coefficient of h(x) is (pos/neg). For large positive values of x, h(x) is (pos/neg). For large negative values of x, h(x) is (pos/neg). Briefly describe how the end-behavior of the graph of h(x) is different from the end-behavior of the graph of f x = x I. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 109

114 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.2: End-behavior Pd Date Part IV: Reflect and Summarize a. What can you conclude about the left and right end behavior (i.e. which way do the two ends of the function point relative to each other) of the graphs for your polynomials of degree 3 and 5? b. What can you conclude about the left and right end behavior of the graphs for your polynomials of degree 2 and 4? c. Why do you think that all cubic polynomials (3 rd degree) have similar end behavior regardless of the terms in the polynomial other than the leading 3 rd degree term? d. Why is it that the end behavior of even degree polynomials is different from that of odd degree polynomials? (Hint: the end behavior is completely determined by output values for large input values.) e. The number at the end of the function s symbolic representation (the constant term) identifies what part of the graph? Look back at your graphs to see for yourself. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 110

115 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.3: Homework Pd Date Given the degree and leading coefficient (LC) of a polynomial function, sketch only its end-behavior: 1) Degree = 4 2) Degree = 3 3) Degree = 5 4) Degree = 6 LC = 3 LC = -2 LC = 2 LC = -2 For each of the following functions, state the degree, the leading coefficient (LC), and the y-intercept. Then, draw a sketch showing only the end-behavior of the graph of the function. 5) 2 y x x = Degree: 6) y = x 3 4x 2 + 7x 5 Degree: LC: y-intercept: LC: y-intercept: 7) y = 8+ x 2 Degree: LC: y-intercept: 8) y = 2x 4 + 3x 2 7 Degree: LC: y-intercept: 5 3 9) y=-x - x+ 2x + 4 Degree: LC: y-intercept: 10) y =1 2x + x 3 +5x 4 Degree: LC: y-intercept: Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 111

116 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.3: Homework Pd Date For each of the following polynomial graphs, circle whether the polynomial is of even or odd degree, whether its leading coefficient is positive or negative, and then state a possible parent function. 13) 14) 15) Degree: even odd Degree: even odd Degree: even odd Lead Coefficient: + - Lead Coefficient: + - Lead Coefficient: + - Parent f( x ) = Parent f( x ) = Parent f( x ) = 16) 17) 18) Degree: even odd Degree: even odd Degree: even odd Lead Coefficient: + - Lead Coefficient: + - Lead Coefficient: + - Parent f( x ) = Parent f( x ) = Parent f( x ) = Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 112

117 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.4: A Deeper Understanding of End-behavior Pd Date Part I: Warm-up Question: How can we determine the degree, leading coefficient, and y-intercept of a polynomial function written in factored form? An answer (the long one): Use multiplication to first rewrite the polynomial in standard form. Practice: determine the degree, leading coefficient and y-intercept of the following functions by first rewriting the polynomial in standard form. 1. f (x) = 3x(x +1)(x 4)(x 2) 2. f (x) = 2(2x +5)(x 3) 2 Degree: Leading Coefficient: y-int: Degree: Leading Coefficient: y-int: Another answer (a more efficient strategy): The degree and leading coefficient can be determined by multiplying all terms containing x. The y-intercept can be determined by finding the product of all constant terms. Practice: determine the degree, leading coefficient and y-intercept of the following functions by using the more efficient strategy described above. 3. f( x) = ( x- 2)( x+ 5)( x+ 3) Degree: Leading Coefficient: y-int: 4. f x x x 2 ( ) =-5( - 1) ( + 2) Degree: Leading Coefficient: y-int: 5. f( x) = 5 x(3x- 2)( x+ 4) 3 6. f( x) = ( x+ 1) ( x-4)( - 2x+ 3) 2 2 Degree: Degree: Leading Coefficient: Leading Coefficient: y-int: y-int: Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 113

118 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.4: A Deeper Understanding of End-behavior Pd Date Part II: Making sense of end-behavior Question: Why, when determining end-behavior, do we only have to consider the term of largest degree? The answer (let s figure it out): Consider the function 3 f( x) = x x x 3 x value of each term x f( x) = x x ,000 10, ,000 Which term contributes the most to the value of f( x ) for small values of x ( x = 1 and x = 10)? Which term contributes the most to the value of f( x ) for large values of x (x > 1000)? Summary: For sufficiently large input values x, the leading term contributes the most to the output value f(x). This is true regardless of the magnitude of the coefficients of the other terms. àthe dominance of the term of largest degree explains end-behavior: End-behavior refers to the behavior of a function as x approaches positive or negative infinity (in other words, when x is positive and very large, or x is negative and very large in absolute value, respectively). Let s see why the previously discussed rules for end-behavior make sense by evaluating the following parent functions at relatively large x-values, say x = ±100 : Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 114

119 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.4: A Deeper Understanding of End-behavior Pd Date Parent Function f( x) = x 2 Key Features degree: even odd leading coef: + - Sketch End-behavior Evaluate the parent function at x = ±100 and describe endbehaviors Right: f (100) =. Large positive x values yield f(x) values, so the graph of f to the right will. Left: f ( 100) =. Negative x values with large absolute value yield f(x) values, so the graph of f to the left will. degree: even odd leading coef: Right: f (100) =. Large positive x values yield f(x) values, so the graph of f to the right will. f( x) =-x Left: f ( 100) =. Negative x values with large absolute value yield f(x) values, so the graph of f to the left will. degree: even odd leading coef: Right: f (100) =. Large positive x values yield f(x) values, so the graph of f to the right will. f( x) = x Left: f ( 100) =. Negative x values with large absolute value yield f(x) values, so the graph of f to the left will. f( x) =-x 3 degree: even odd leading coef: + - Right: f (100) =. Large positive x values yield f(x) values, so the graph of f to the right will. Left: f ( 100) =. Negative x values with large absolute value yield f(x)values, so the graph of f to the left will. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 115

120 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.4: A Deeper Understanding of End-behavior Pd Date Parent Function f (x) = x 4 Key Features degree: even odd leading coef: + - Sketch End-behavior Evaluate the parent function at x = ±100 and describe endbehaviors Right: f (100) =. Large positive x values yield f(x) values, so the graph of f to the right will. Left: f ( 100) =. Negative x values with large absolute value yield f(x) values, so the graph of f to the left will. f (x) = x 4 degree: even odd leading coef: + - Right: f (100) =. Large positive x values yield f(x) values, so the graph of f to the right will. Left: f ( 100) =. Negative x values with large absolute value yield f(x) values, so the graph of f to the left will. Explain (in your own words) why we only have to consider the term of largest degree when determining end-behavior. Why do the end-behavior rules make sense? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 116

121 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-1.5: Homework Pd Date Indicate for each polynomial function graphed below whether its degree is even or odd and whether its leading coefficient is positive or negative by placing two X s in each row of the table. Polynomial Even Degree Odd Degree A B C D Positive Leading Coefficient Negative Leading Coefficient Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 117

122 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 118

123 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.1: End-behavior, zeros and the y-intercept Pd Date Part I: Warm-up 1. Of the six functions listed below, circle all that are polynomial functions. a(x) = (x 5)(x + 3) d(x) = x I + 3x > 8x # + 2 b(x) = M`9 #MXF 2. Kimo just sat down at the table with his math homework when his mom asked him to feed his baby brother. Unfortunately for Kimo, poi was on the menu what a mess! Oh no, said Kimo, There is poi on my homework. My graphs are ruined. How can I match the graphs to the functions when I can t see all of the graphs? Please help Kimo match each graph to its equation. e(x) = 2x 1 c(x) = 2 3 M f(x) = 7x 5 f(x ) g(x) x x h(x) k(x) x x = -2x(x + 4)(x 1) = 3x 4 9x 3 + 2x 2 5x + 1 = - (x + 3) 3 (x 1) = x 3 100x Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 119

124 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.1: End-behavior, zeros and the y-intercept Pd Date Part II: Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 120

125 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.1: End-behavior, zeros and the y-intercept Pd Date Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 121

126 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.1: End-behavior, zeros and the y-intercept Pd Date Part III: Sketch a possible graph for each of the following polynomial functions based on the given information. 3. f is a polynomial function such that its degree is even its leading coefficient is negative its constant term is negative it has zeros only at x = 1 and x = 4 4. f is a polynomial function such that its degree is odd its leading coefficient is positive its constant term is negative it has zeros only at x = 3, x = 1 and x = 2 5. f is a polynomial function such that its degree is odd its leading coefficient is negative its constant term is negative it has zeros only at x = 5, x = 4, x = 1 x = 1 and x = 3 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 122

127 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.1: End-behavior, zeros and the y-intercept Pd Date 6. f is a polynomial function such that its degree is even its leading coefficient is positive its constant term is positive it has zeros only at x = 1, x = 2, x = 3 and x = 4 7. f is a polynomial function such that its degree is even its leading coefficient is positive its constant term is positive it has zeros only at x = 4, x = 3, x = 1 and x = 2 8. f is a polynomial function such that its degree is odd its leading coefficient is positive its constant term is negative it has zeros only at x = 4, x = 1, x = 1 x = 2 and x = 4 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 123

128 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.1: End-behavior, zeros and the y-intercept Pd Date Part IV: Reflect and Summarize (follow your teacher s instructions). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 124

129 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.2: Repeated Zeros Pd Date Part I: Warm-up 1. Rewrite the function f x = 4x # 16x + 21 in the form f x = a x h # + k, where a, h, and k are constants. Show how you determined your answer. 2. Referring to your answer above, explain the values of a, h, and k tell us about the graph of f x = 4x # 16x Note: you do not need to graph the function. 3. Consider the equation x # + 4x + 5 = 0 Solve the equation. The equation above is written in the form f x = 0. Write a statement to explain what your solution tells us about the graph of f x. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 125

130 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.2: Repeated Zeros Pd Date Part II: Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 126

131 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.2: Repeated Zeros Pd Date Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 127

132 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.2: Repeated Zeros Pd Date Part III: Sketch a possible graph for each of the following polynomial functions based on the given information. 4. f is a polynomial function such that its degree is odd its leading coefficient is negative it has zeros only at x = 1 and x = 4, where x = 4 has multiplicity f is a polynomial function such that its degree is even its leading coefficient is positive it has zeros only at x = 3, x = 1 and x = 2, where x = 1 has multiplicity f is a polynomial function such that its degree is odd its leading coefficient is positive it has zeros only at x = 4 and x = 2, where x = 4 has multiplicity 3 and x = 2 has multiplicity 2. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 128

133 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.2: Repeated Zeros Pd Date 7. f is a polynomial function such that its degree is even its leading coefficient is negative it has zeros only at x = 2 and x = 2, where x = 2 has multiplicity 2 and x = 2 has multiplicity f is a polynomial function such that its degree is even its leading coefficient is positive it has zeros only at x = 4, x = 1.5 and x = 2, where x = 4 has multiplicity 2, and x = 1.5 has multiplicity f is a polynomial function such that its degree is odd its leading coefficient is positive it has zeros only at x = 4, x = 1 and x = 2, where x = 1 has multiplicity 3. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 129

134 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.2: Repeated Zeros Pd Date Part IV: Reflect and Summarize Discuss the following questions with a partner and write down the ideas you discussed. Then, confirm or correct your understanding during the whole class discussion that your teacher will lead. 10. Make a conjecture about the sum of the multiplicities of the zeros for an n th degree polynomial. Assume each zero that is not repeated has multiplicity Is it possible for an even degree polynomial to have no real zeros? 12. Is it possible for an odd degree polynomial to have no real zeros? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 130

135 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date Part I: Warm-up 1. Without graphing, explain what you know must be true about the end-behavior of the polynomial function L x = 2 x 3 x Graph of L x = 2 x 3 x + 1 in the coordinate plane provided below. Show how you determined the x-intercepts. Show how you determined the y-intercept. Label your axes to indicate the scale you used on each axis. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 131

136 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date Part II: Following your teacher s instructions, use the space below to take notes. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 132

137 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date Part III: Sketch a graph of each polynomial function below. Use the guiding questions to help with your sketch. Your sketch should accurately represent the behavior of the function, however, you do not have to be precise regarding the points where f changes direction. 3. Sketch the graph of f(x) = (x 2)(x + 1)(x + 4) A. Describe the end behavior? Explain how you know. B. What are the zeros? Show or explain how you know. C. What is the y-intercept? Show or explain how you know. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 133

138 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date 4. Sketch the graph of f(x) = x(x + 5)(x 4) A. Describe the end behavior? Explain how you know. B. What are the zeros of the function? Show or explain how you know. C. What is the y-intercept? Show or explain how you know. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 134

139 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date 5. Sketch the graph of f(x) = (x + 4)(x + 1)(x 2) 6. Sketch the graph of f(x) = 2(x + 5)(x + 1)(x 2) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 135

140 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date 7. Sketch the graph of f(x) = (x + 2)(x 2)² 8. Sketch the graph of f(x) = 2(x 1) E (x + 3) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 136

141 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.3: Graphing Polynomials in Factored Form Pd Date Part IV: Work with a partner to discuss and answer the following questions. 9. Below is the graph of the function defined by f x x x 2 ( ) = ( - 1) ( + 3) a. What is the y-intercept for f? b. What are the x-intercepts for f? c. Suppose we wish to build a new polynomial function g, such that g has exactly the same intercepts as f, the same end-behavior, and the same behavior close to the x-intercepts, but g has a y-intercept at (0, 15). How could we alter the symbolic representation of f to achieve this? In particular, determine a possible symbolic representation for g(x). 10. Below is the graph of the function defined by kx x x x 2 ( ) =- ( + 1) ( -1)( - 2) a. What is the y-intercept for k? b. What are the x-intercepts for k? c. Suppose r is a polynomial function with the same x-intercepts as k, the same end-behavior as k, and the same behavior as k close to its x-intercepts, but with y-intercept at (0, -1). Determine a possible symbolic representation for r(x). Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 137

142 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 138

143 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.4: Homework Pd Date 1. Sketch the graph of the function defined by E(x) = x(2x + 3)(x 3)(x 6). Answer the 3 guiding questions: What is the end behavior? What are the zeros? What is the y-intercept? 2. Sketch the graph of the function defined by C(x) = (x + 4)(x + 1)(x 2)(x 4). Answer the 3 guiding questions: What is the end behavior? What are the zeros? What is the y-intercept? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 139

144 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.4: Homework Pd Date 3. Sketch the graph of the function defined by D(x) = (x + 3)(2x + 1)(x 3). Answer the 3 guiding questions: What is the end behavior? What are the zeros? What is the y-intercept? 4. Sketch the graph of the function defined by H(x) = (x + 5)(x 3)³. Answer the 3 guiding questions: What is the end behavior? What are the zeros? What is the y-intercept? Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 140

145 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.5: From Graphs to Symbolic Form Pd Date Part I: Warm-up 1. Graph the following functions. Label the axes to indicate the scale used on each axis. Label the coordinates of all x- and y-intercepts. A. f x = x + 5 B. f x = x # 25 C. h x = x + 1 x 5 x + 5 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 141

146 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.5: From Graphs to Symbolic Form Pd Date Part II: Follow your teacher s instructions for this activity Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 142

147 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.5: From Graphs to Symbolic Form Pd Date A f(x) = (x + 1) (x - 2) (x - 4) B f(x) = x (x + 1) (x - 2) C f(x) = 0.5 (x + 1) (x - 1)² (x - 3) D E F f(x) = (x + 1) (x-1)(x - 2) (x - 3) f(x) = -(x + 1) (x - 2) 2 f(x) = -(x + 1) (x - 2) (x - 4) G H I f(x) = (x + 1) (x - 1) f(x) = -0.5(x - 4) (x 2 x 2) f(x) = 8(x + 1) (x - 2) (x - 4) J K L f(x) = -0.5 (x + 1) (x - 1) (x - 3) f(x) = 0.5(x+1)(x - 1)(x - 2) (x - 3) f(x) = x (x + 1) (x - 2) 2 M N O f(x) = (x + 1) (x - 1)³ f(x) = 2(x + 1) (x - 2) f(x) = (x + 1) (x - 2) 2 P Q R f(x) = -x (x + 1) (x - 2) (x - 3) f(x) = 2 (x + 2) (x + 1) (x - 1) f(x) = -4(x + 1) (x - 2) (x - 4) Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 143

148 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 144

149 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.6: Homework Pd Date 1. Sketch the graph of E(x) = -(x + 5)²(x - 1)(x - 3)³. Use what you know about end behavior (as determined by the degree and the sign of the leading coefficient), location of zeros, location of the y-intercept, and behavior of the functions at each type of zero (repeated and non-repeated, even number of factors and odd). 2. Sketch the graph of F(x) = x(x + 3)²(x - 2)³. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 145

150 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 146

151 Algebra 2 Module 8: Introduction to Polynomial Functions Name P-2.7: Determining a Function s Symbolic Representation from its Graph Pd Date Part I: Warm-up (work with a partner to discuss and answer exercises 1 and 2 below) 1. Indicate which of the following could be the symbolic representation of the polynomial function graphed below by placing an X in the appropriate box for each row in the table. Function F(x) = (x + 1) 2 (x 3) Is a Possible Symbolic Representation Is NOT a Possible Symbolic Representation F(x) = (x + 5)(x 2) 2 F(x) = 2(x 1) 2 (x + 5) F(x) = -3(x + 5)(x 1) 2 2. Referring back to your answers above (in question 1), for all functions that you indicated could NOT be a possible symbolic representation of the graph, write a briefly explain why you came to that conclusion. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license. 147