Algebra II. Function Basics. Analyzing and Working with Functions Part 1. Slide 1 / 166 Slide 2 / 166. Slide 4 / 166. Slide 3 / 166.

Transcription

1 Slide 1 / 166 Slide 2 / 166 lgebra II nalyzing and Working with Functions Part Slide 3 / 166 Table of ontents Part 1 Function asics Operations with Functions omposite Functions The 12 asic Functions (Parent Functions) Part 2 Transforming Functions Inverse Functions Piecewise Functions click on the topic to go to that section Slide 4 / 166 Function asics In this section, we will review functions and relations, function notation, domain, range, along with discrete and continuous functions. These topics were also covered in 8th grade and lgebra 1. Return to Table of ontents Slide 5 / 166 Relations relation is an association between sets of information. Slide 6 / 166

2 Slide 7 / 166 Graphs of Functions The Vertical Line Testcan determine if a graph represents a function. If the vertical line intersects only one point at a time on the ENTIRE graph, then it represents a function. If the vertical line intersects more than one point at NY time on the graph, then it is NOT a function. Move the black vertical line to test! Slide 7 () / 166 Graphs of Functions The Vertical Line Testcan determine if a graph represents a function. If the vertical line intersects only Function one point at a time Not on a the Function ENTIRE graph, then it represents a function. If the vertical line intersects more than one point at NY time on the graph, then it is NOT a function. Teacher Notes Move the black vertical line to test! Slide 8 / 166 Equations as Functions Slide 9 / 166 n equation is a function only if a number substituted in for x produces only 1 output or y-value. Function Reason Not a Function Reason y = 3x + 4 For each input for x, there is only one output of y. x = 5 There are multiple values for y. y = 5 ll y values are 5. x = y 2 For each x, there are two values for y. Slide 9 () / 166 Slide 10 / 166 x -1 2 y Function etermine if each of the relations below is a function and provide an explanation to support your answer: x y 4 x y 7-3 8

3 x -1 2 y Slide 10 () / 166 etermine if each of the relations below is a function and provide an explanation to support your answer: x -2 Not a 3-5 Function - x value of -1 yields more than one y value Function y 4 Function - each x value has a unique y value x Function - each x value has a unique y value y Slide 11 / 166 Function etermine if each of the relations below is a function and provide an explanation to support your answer: Slide 11 () / 166 Function etermine if each of the relations below is a function and provide an explanation to support your answer: Slide 12 / 166 Not a Function - x value of 2 yields two different y values; does not pass vertical line test. Not a Function - multiple x values produce more than one y value; does not pass vertical line test. Function - No x values repeat. Slide 12 () / 166 Slide 13 / Is the following relation a function? {(3,1), (2,-1), (1,1)} Yes No

4 Slide 13 () / Is the following relation a function? Slide 14 / Is the following relation a function? {(3,1), (2,-1), (1,1)} X Y Yes No Yes Yes No Slide 15 / Is the following relation a function? Slide 15 () / Is the following relation a function? Yes No Yes No Yes Slide 16 / 166 Slide 16 () / 166

5 Slide 17 / 166 Slide 17 () / 166 Slide 18 / 166 Slide 19 / 166 Function Notation So why change the notation? 1) It lets the mathematician know the relation is a function. 2) If a second function is used, such as g( x) = 4x, the reader will be able to distinguish between the different functions. 3) The notation makes evaluating at a value of x easier to read. Slide 20 / 166 Evaluating a Function To Evaluate in y = Form: To Evaluate in Function Notation: Slide 21 / 166 Find the value of y = 2x + 1 when x = 3 y = 2x + 1 y = 2(3) + 1 y = 7 When x is 3, y = 7 Given f(x) = 2x + 1 find f(3) f(3) = 2(3) + 1 f(3) = 7 "f of 3 is 7" Similar methods are used to solve but function notation makes asking and answering questions more concise.

6 Slide 21 () / 166 Slide 22 / 166 Slide 22 () / 166 Slide 23 / 166 Slide 23 () / 166 Slide 24 / Given and Find the value of.

7 Slide 24 () / 166 Slide 25 / 166 Slide 25 () / 166 Slide 26 / Given and Find the value of. Slide 26 () / 166 Slide 27 / Given and Find the value of.

8 Slide 27 () / 166 Slide 28 / 166 Slide 28 () / 166 Slide 29 / 166 Slide 29 () / 166 Slide 30 / 166

9 Slide 30 () / 166 Slide 31 / 166 Slide 32 / 166 Slide 33 / 166 Graph Interval Notation Inequality Notation a b losed Interval a Open Interval b a b Half-Open Interval { } = set Slide 34 / 166 = omain (possible input or x-values) R = Range (possible output or y-values) = is an element of (belongs to) = positive infinity = negative infinity = Set of Real Numbers = Set of Integers = Natural Numbers Summary Slide 35 / 166 Infinity Why do you think parentheses are used in interval notation for a data set that includes or instead of brackets?

10 Slide 35 () / 166 Slide 36 / What is the interval notation for the given graph? Infinity Why do you think parentheses are used in interval notation for a data set that includes or Infinity/Negative instead of brackets? Infinity do not have a final value, they can always increase/ decrease. Since they never end a parentheses is used instead of a bracket. E F Slide 36 () / What is the interval notation for the given graph? E F Slide 37 / What is the inequality notation for the given graph? E F Slide 37 () / What is the inequality notation for the given graph? Slide 38 / What is the interval notation for the given graph? E F E E F

11 Slide 38 () / What is the interval notation for the given graph? Slide 39 / What is the inequality notation for the given graph? E F E F Slide 39 () / What is the inequality notation for the given graph? E F F Slide 40 / 166 omain and Range The domain of a function or a relation is the set of all possible input values ( x-values). The range of a function or a relation is the set of all possible output values (y-values). Slide 41 / 166 omain and Range Slide 42 / 166 omain and Range State the domain and range for each example below: Relation omain Range x x x 1 2 y y 4 y 5 8 9

12 Slide 42 () / 166 omain and Range State the domain and range for each example below: Slide 43 / 166 omain and Range x x y y 4 omain Range State the domain and range for the function below. Write your answers in inequality and interval notation. x y Slide 43 () / 166 omain and Range Slide 44 / 166 omain and Range State the domain and range for the function below. Write your answers in inequality and interval notation. State the domain and range for the function below. Write your answers in inequality and interval notation. omain: -2 x < 2 and [-2, 2) Range: -2 y < 4 and [-2, 4) Slide 44 () / 166 Slide 45 / 166 omain and Range State the domain and range for the function below. Write your answers in inequality and interval notation. omain: Range:

13 Slide 45 () / 166 Slide 46 / Is -2 < x < 2 the domain of the relation? Yes No Slide 46 () / Is -2 < x < 2 the domain of the relation? Slide 47 / Is [0, 1] the range of the relation? Yes Yes No No omain: -2 x 2 No Slide 47 () / 166 Slide 48 / Is [0, 1] the range of the relation? Yes No Yes

14 Slide 48 () / 166 Slide 49 / 166 Slide 49 () / 166 Slide 50 / 166 Slide 50 () / 166 Slide 51 / 166

15 Slide 51 () / 166 Slide 52 / 166 If you are finding domain without coordinates or graphs, just assume it begins with. Then, look for roots and fractions. Restrict it with values that violate the following: Roots: omain There can be NO NEGTIVE values under a root. Set the radicand greater than or equal to zero (positive). Solve. Fractions: In a fraction, the denominator NNOT E ZERO. Set the denominator equal to zero and solve. Slide 52 () / 166 omain If you are finding domain without coordinates or graphs, just assume it begins with. Then, look for roots and fractions. Restrict it with values that violate the following: Roots: There can be NO NEGTIVE values under a root. Set Note: nother restriction the radicand greater than or equal to zero (positive). for domains will be Solve. logarithms of negative Fractions: In a fraction, the denominator numbers or zero; NNOT however, E ZERO. Set the denominator students equal to have zero not and learned solve. this concept yet. Teacher Notes gain, start with ll Real Numbers fractions in your function. Slide 53 / 166. Then look for roots or Find the domain of the following functions. Write your answers in interval notation omain 3. gain, start with ll Real Numbers fractions in your function. Slide 53 () / 166 omain. Then look for roots or Find the domain of the following functions. Write your answers in interval notation. Slide 54 / 166 omain gain, start with ll Real Numbers fractions in your function.. Then look for roots or Find the domain of the following functions. Write your answers in interval notation

16 gain, start with ll Real Numbers fractions in your function. Slide 54 () / 166. Then look for roots or Find the domain of the following functions. Write your answers in interval notation. 4. omain 24 Find the domain of: Slide 55 / Slide 55 () / 166 Slide 56 / Find the domain of: Slide 56 () / 166 Slide 57 / Find the domain of:

17 Slide 57 () / 166 Slide 58 / Find the domain of: 27 Find the domain of: Slide 58 () / 166 Slide 59 / Find the domain of: 28 Find the domain of: 28 Find the domain of: Slide 59 () / 166 Slide 60 / 166 Range The Range is the set of all possible y values. It is extremely helpful to look at a graph when determining the range. Find the range of the following functions: 1. 2.

18 Slide 60 () / 166 Range The Range is the set of all possible y values. It is extremely helpful to look at a graph when determining the range. Find the range of the following functions: Slide 61 / 166 Range Find the range of the following functions. Write your answers in interval notation Slide 61 () / 166 Range Find the range of the following functions. Write your answers in interval notation Slide 62 / Find the range of the following: Slide 62 () / 166 Slide 63 / Find the range of the following:

19 Slide 63 () / 166 Slide 64 / 166 Slide 64 () / 166 Slide 65 / 166 Slide 65 () / 166 Slide 66 / 166 iscrete vs ontinuous relation is discrete if it is made up of separate points (only specific values are relevant). For example, you go to a bakery to buy doughnuts. How many can you purchase? 0, 1, 2, 3... You would not be able to purchase 1.2, 1.375, , etc. These values do not have meaning in this situation, therefore the data is discrete. What are some other discrete events?

20 Slide 67 / 166 iscrete vs ontinuous Slide 68 / 166 iscrete vs ontinuous relation is continuous if the points are NOT separate values exist in between. For example, a repairman says he will be to your home between 1pm and 5pm. What time could he show up? 1:00pm, 2:15pm, 3:42pm, etc... The values between 1pm and 5pm are also relevant, therefore the relation is continuous. re the following relations discrete or continuous? If continuous, state the interval of continuity. x y X Y What are some continuous events? Slide 69 / 166 iscrete vs ontinuous re the following relations discrete or continuous? If continuous, state the interval of continuity. Slide 69 () / 166 iscrete vs ontinuous re the following relations discrete or continuous? If continuous, state the interval of continuity. ontinuous iscrete ontinuous Slide 70 / 166 iscrete vs ontinuous Slide 70 () / 166 re the following relations discrete or continuous? If continuous, state the interval of continuity.

21 Slide 71 / Is the given relation discrete or continuous? {(3,1), (2,-1), (1,1)} Slide 71 () / Is the given relation discrete or continuous? {(3,1), (2,-1), (1,1)} iscrete ontinuous iscrete ontinuous Slide 72 / Is the given relation discrete or continuous? Slide 72 () / Is the given relation discrete or continuous? iscrete iscrete ontinuous ontinuous Slide 73 / Is the given relation discrete or continuous? Slide 73 () / Is the given relation discrete or continuous? iscrete ontinuous iscrete ontinuous

22 Slide 74 / Is the given relation discrete or continuous? Slide 74 () / Is the given relation discrete or continuous? iscrete iscrete ontinuous ontinuous Slide 75 / Is the given relation discrete or continuous? Slide 75 () / Is the given relation discrete or continuous? iscrete ontinuous iscrete ontinuous Slide 76 / 166 Slide 77 / 166 Operations with Functions Goals and Objectives Students will be able to manipulate multiple functions algebraically and simplify resulting functions. Return to Table of ontents

23 Slide 78 / 166 Why do we need this? In this unit, we will graphically explored transformations of functions. Sometimes, data is more complex and requires more than one representative function. lgebraically, manipulating functions allows us to combine different functions together and results in many more options for real life situations. Slide 79 / 166 Operations with Functions Here are the properties of combining functions: dding functions: Subtracting functions: Multiplying functions: ividing functions: Slide 79 () / 166 Operations with Functions Slide 80 / 166 Operations with Fractions Here are the properties of combining functions: dding functions: Subtracting functions: Students may note that the properties are "common sense." The only hard part Multiplying functions: sometimes is simplifying the expressions. Given: Find: and Simplify your answers as much as possible. What happens to the domain? ividing functions: Slide 80 () / 166 Operations with Fractions Slide 81 / 166 Operations with Functions Given: Find: and Simplify your answers as much as possible. Given: Find: and What happens to the domain? *The domain of a resulting function is subject to the domain of the original functions as well as the final function.

24 Given: Find: Slide 81 () / 166 Operations with Functions and Slide 82 / Given and, find Slide 82 () / 166 Slide 83 / Given and, find 39 Given and, find h(x) if Slide 83 () / 166 Slide 84 / Given and, find h(x) if 40 Given and, find

25 Slide 84 () / Given and, find Slide 85 / Given and, find Slide 85 () / Given and, find Slide 86 / 166 omain Given and, find the domain of each: a) b) c) d) Slide 86 () / 166 Slide 87 / 166

26 Slide 87 () / 166 Slide 88 / 166 Slide 88 () / 166 Slide 89 / Find the domain of if and Slide 89 () / Find the domain of if and Slide 90 / 166 You may also be asked to evaluate combined functions when given specific values for x. Given and, find: a) ombined Functions b) c) d)

27 Slide 90 () / 166 You may also be asked to evaluate combined functions when given specific values for x. Given a) 13 and, find: a) b) ombined Functions b) c) Slide 91 / Given and, find d) undefined c) d) Slide 91 () / 166 Slide 92 / Given and, find 46 Given and, find Slide 92 () / 166 Slide 93 / Given and, find 47 Given and, find 1728 undefined

28 Slide 93 () / Given and, find undefined Slide 94 / 166 Expressions may also be used to create more complex functions. If and, create. Leave your answer in terms of x. Slide 94 () / 166 Expressions may also be used to create more complex functions. If and, create. Leave your answer in terms of x. Slide 95 / 166 If and, create. Leave your answer in terms of x. Slide 95 () / 166 If and, create. Leave your answer in terms of x. Slide 96 / 166 If and, create. Leave your answer in terms of x.

29 Slide 96 () / 166 If and, create. Leave your answer in terms of x. Slide 97 / If and, create. Is this equivalent to? Yes No Slide 97 () / 166 Slide 98 / If and, create. Is this equivalent to? Yes No No Slide 98 () / 166 Slide 99 / If and, create. Is this equivalent to? Yes No

30 Slide 99 () / 166 Slide 100 / 166 omposite Functions or Return to Table of ontents Slide 101 / 166 Slide 102 / 166 Goals and Objectives Students will be able to recognize function notation and correctly unite two or more functions together to create a new function. Why do we need this? On many occasions, multiple situations happen to something before it obtains a final result. For example, you take extra food off of your plates before you put them in the dishwasher. Or, to wrap a present you must first put it in the box, then apply the wrapping paper, and finally tie the bow. These are multiple functions that go together to obtain a desired result. Slide 103 / 166 omposite Functions omposite functions exist when one function is "nested" in the other function. There are 2 ways of writing a composite function: or Slide 104 / 166 To simplify composite functions, substitute one function into the other in place of "x" and simplify. Work from the inside out. Given: Find: omposite Functions Find: Each form is read "f of g of x" and both mean the same thing.

31 Slide 104 () / 166 omposite Functions Slide 105 / 166 To simplify composite functions, substitute one function into the other in place Stress of the "x" differences and simplify. and the Work "nesting" from the inside out. and how that affects the order. Sometimes, students struggle the most with simplifying. Given: f(g(x)) = f(g(x)) Find: = 3(g(x)) 2 + 2(g(x)) g(f(x)) = g(f(x)) Find: = 4(f(x)) = 3(4x) 2 + 2(4x) =4(3x 2 + 2x) = 3(16x 2 = 12x ) + 8x 2 + 8x = 48x 2 + 8x Slide 105 () / 166 Slide 106 / 166 To simplify composite functions with numerical values, substitute the number into the "inner" function, simplify, and then substitute that value in for the variable in the "outer" function. Given: omposite Functions Find: Find: Slide 106 () / 166 To simplify composite functions with numerical values, substitute the number into the "inner" function, simplify, and then substitute that value in for the variable in the "outer" function. Given: Find: omposite Functions Find: Slide 107 / If and, find the value of

32 Slide 107 () / 166 Slide 108 / If and, find the value of 52 Find if Slide 108 () / 166 Slide 109 / Find if Slide 109 () / 166 Slide 110 / 166

33 Slide 110 () / 166 Slide 111 / Find if 55 Find if Slide 111 () / 166 Slide 112 / Find if and Slide 112 () / 166 Slide 113 / Find if and 57 Find the value of

34 Slide 113 () / 166 Slide 114 / Find the value of Note: If students are struggling with the notation using, have them rewrite as f(h(g(x))). It is sometimes easier to see the nesting. The 12 asic Functions (Parent Functions) Many situations in the world that people study and collect data on follow one of the following 12 patterns. y recognizing a general pattern, or what we call the Parent Function, and then algebraically manipulating the function, you can almost come up with an exact match. Some people get paid a lot of money to do this! Return to Table of ontents Slide 115 / 166 Slide 116 / 166 The 12 asic Functions The 12 asic Functions continued The Identity Function y = x The Squaring Function y = x 2 The ubing Function y = x 3 The Exponential Function y = e x The Natural Log Function y = lnx The Logistic Function The Reciprocal Function y = 1/x The bsolute Value Function y = ΙxΙ The Square Root Function The Sine Function y = sinx The osine Function y = cosx The Greatest Integer Function y = [x] Slide 117 / 166 Recall from lgebra I The x-intercept of a graph is the point where the graph crosses the x- axis and has the ordered pair (x, 0). To find the x-intercept using an equation, substitute 0 for y and solve for x. The x-intercept is also referred to as the root or zero. The y-intercept of a graph is the point where the graph crosses the y- axis and has the ordered pair (0, y). To find the y-intercept using an equation, substitute 0 for x and solve for y. Slide 118 / 166 When studying the graphs of functions, scientists like to analyze many different aspects of the graph. omain: Range: Minimum (Min): Maximum (Max): Intercepts: x-intercepts: y-intercepts: Function asics Increasing intervals: ecreasing intervals: Odd/Even/Neither: End ehavior:

35 Slide 119 / 166 Graph of Intercepts y-intercept Slide 120 / 166 x-intercept & y-intercept Example: Evaluate the x-intercept & y-intercept for the following equations: x-intercept Slide 120 () / 166 x-intercept & y-intercept x-intercept = (5, 0) Example: Evaluate the x-intercept y-intercept & y-intercept = (0, 10) for the following equations: Slide 121 / 166 x-intercept & y-intercept Example: etermine the x-intercept and y-intercept for the following graphs: x-intercept = (2, 0) y-intercept = (0, -4) x-intercept = (-2, 0) y-intercept = (0, 4) Slide 121 () / 166 x-intercept & y-intercept Example: etermine the x-intercept and y-intercept for the following graphs: Slide 122 / Evaluate the y-intercept of the following equation: (0, 6) x-intercept = (2, 0) y-intercept = (0, 4) x-intercept = oes Not Exist intercept = (0, 3) (6, 0) (2,0) (0,2) E oes not exist

36 Slide 122 () / Evaluate the y-intercept of the following equation: Slide 123 / What is the x-intercept of the graph below? (0, 6) (6, 0) (2,0) (0,2) E oes not exist (0, 7) (7, 0) (0,21) annot be determined by graph E oes Not Exist Slide 123 () / What is the x-intercept of the graph below? Slide 124 / Why does the x-intercept for the graph below NOT exist? (0, 7) (7, 0) (0,21) annot be determined by graph E oes Not Exist The x-intercept does exist The graph is misleading There is not enough of the graph shown to determine the reason The graph does not pass through the x-axis Slide 124 () / 166 Slide 125 / Why does the x-intercept for the graph below NOT exist? The x-intercept does exist The graph is misleading There is not enough of the graph shown to determine the reason The graph does not pass through the x-axis

37 Slide 126 / 166 Increasing and ecreasing Functions function is said to be increasing when the graph is travelling in an upward direction (when traveled from left to right). function is said to be decreasing when the graph is travelling in a downward direction (when traveled from left to right). maxima occurs at the HIGHEST point of a graph. Slide 127 / 166 Maxima and Minima maxima minima occurs at the LOWEST point of a graph minima Increasing ecreasing Slide 128 / 166 Maxima and Minima Slide 129 / 166 Maxima and Minima maximum occurs when a function changes from increasing to decreasing. minimum occurs when a function changes from decreasing to increasing. There are 2 types of maximums/minimums: Local: ny turning point in the graph. Note: a Local max/min NNOT occur at endpoints. bsolute: The highest/lowest point on the graph. Note: an bsolute max/min N occur at an endpoint. Increasing SOLUTE MIN Increasing Local Max Local Min ecreasing Local Max SOLUTE MX Local Min ecreasing Increasing Slide 130 / 166 oncavity Slide 130 () / 166 oncavity The concavity of a function is the direction of the "bowl shape" of a graph. graph is concave down if the bowl is upside down. graph is concave up if the bowl faces upward. The concavity of a function is the direction of the "bowl shape" of a graph. Students may also find it easy to remember by graph is concave down if the graph is concave up if the bowl is upside down. using the following: bowl faces upward. oncave own = Frown oncave Up = up

38 Slide 131 / 166 Slide 131 () / 166 Slide 132 / 166 Slide 132 () / 166 Slide 133 / 166 Slide 133 () / 166

39 Slide 134 / 166 Slide 134 () / 166 Slide 135 / 166 Slide 135 () / 166 Slide 136 / 166 Slide 136 () / 166

40 Slide 137 / 166 Slide 138 / 166 Slide 138 () / 166 Slide 139 / Is the following an odd-function, an even-function, or neither? Odd Even Neither Slide 139 () / 166 Slide 140 / Is the following an odd-function, an even-function, or neither? Odd Even Neither

41 Slide 140 () / 166 Slide 141 / 166 Slide 141 () / 166 Slide 142 / 166 Slide 142 () / 166 Slide 143 / 166 Even-egree Polynomials What do you observe about the end behavior of an even function?

42 Slide 143 () / 166 Even-egree Polynomials Slide 144 / 166 Even-egree Polynomials Positive Lead oefficient Negative Lead oefficient Start and end in same place (High to High or Low to Low) What do you observe about the end behavior of an even function? What do you observe about the end behavior of an even function with a positive lead coefficient? What do you observe about the end behavior of an even function with a negative lead coefficient? Slide 144 () / 166 Even-egree Polynomials Positive Lead oefficient Negative Lead oefficient Slide 145 / 166 Odd-egree Polynomials Positive: High to High Negative: Low to Low What do you observe about the end behavior of an odd function? What do you observe about the end behavior of an even function with a positive lead coefficient? What do you observe about the end behavior of an even function with a negative lead coefficient? Slide 145 () / 166 Odd-egree Polynomials Slide 146 / 166 Odd-egree Polynomials Positive Lead oefficient Negative Lead oefficient Start and end in different places (High to Low or Low to High) What do you observe about the end behavior of an odd function? What do you observe about the end behavior of an odd function with a positive lead coefficient? What do you observe about the end behavior of an odd function with a negative lead coefficient?

43 Slide 146 () / 166 Odd-egree Polynomials Positive Lead oefficient Negative Lead oefficient Slide 147 / 166 End ehavior When describing the end behavior of a polynomial, we are describing what the y-values are approaching. Lead oefficient is Positive Lead oefficient is Negative Positive: Low to High Negative: High to Low Even- egree Polynomial Left End Right End Left End Right End What do you observe about the end behavior of an odd function with a positive lead coefficient? What do you observe about the end behavior of an odd function with a negative lead coefficient? Odd- egree Polynomial Slide 148 / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 148 () / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Odd and Positive Odd and Negative Even and Positive Even and Negative Odd and Positive Odd and Negative Even and Positive Even and Negative Slide 149 / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 149 () / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Odd and Positive Odd and Positive Odd and Negative Even and Positive Odd and Negative Even and Positive Even and Negative Even and Negative

44 Slide 150 / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 150 () / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Odd and Positive Odd and Negative Even and Positive Even and Negative Odd and Positive Odd and Negative Even and Positive Even and Negative Slide 151 / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 151 () / etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Odd and Positive Odd and Positive Odd and Negative Even and Positive Even and Negative Odd and Negative Even and Positive Even and Negative Slide 152 / Pick all that apply to describe the graph below: Odd- egree Odd- Function Slide 152 () / Pick all that apply to describe the graph below: Odd- egree Odd- Function Even- egree Even- Function Even- egree,, E Even- Function E Positive Lead oefficient E Positive Lead oefficient F Negative Lead oefficient F Negative Lead oefficient

45 Slide 153 / Pick all that apply to describe the graph below: Slide 153 () / Pick all that apply to describe the graph below: Odd- egree Odd- egree Odd- Function Odd- Function E F Even- egree Even- Function Positive Lead oefficient Negative Lead oefficient E F Even- egree Even-, Function, E Positive Lead oefficient Negative Lead oefficient Slide 154 / Pick all that apply to describe the graph below: Slide 154 () / Pick all that apply to describe the graph below: E Odd- egree Odd- Function Even- egree Even- Function Positive Lead oefficient E Odd- egree Odd- Function Even- egree Even-, Function, F Positive Lead oefficient F Negative Lead oefficient F Negative Lead oefficient Slide 155 / Pick all that apply to describe the graph below: Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient Slide 155 () / Pick all that apply to describe the graph below: Odd- egree Odd- Function Even- egree Even- Function, E E Positive Lead oefficient F Negative Lead oefficient F Negative Lead oefficient

46 Slide 156 / Pick all that apply to describe the graph below: Slide 156 () / Pick all that apply to describe the graph below: E F Odd- egree Odd- Function Even- egree Even- Function Positive Lead oefficient Negative Lead oefficient E F Odd- egree Odd- Function Even- egree Even-,, Function F Positive Lead oefficient Negative Lead oefficient Slide 157 / 166 nother characteristic of odd functions is that they have rotational symmetry about the origin. In other words... Odd Functions Slide 158 / 166 nother characteristic of even functions is that they have symmetry about the y-axis. In other words... Even Functions Rotational Symmetry Line of Symmetry Slide 159 / 166 Identifying Symmetry We can identify symmetry by comparing values of f(x). If f(x) has symmetry over the y-axis, then f(x)=f(-x) Example: Given the even function: 1) Plug in -x. 2) Simplify Slide 160 / 166 Identifying Symmetry If the function is symmetrical about the origin, then f(-x)=-f(x) Example: Given the odd function: 1) Plug in -x 2)Simplify Notice: f(x)=f(-x), therefore f(x) is symmetrical over the y-axis, as we would expect for this even function. Notice: f(-x)=-f(x), therefore the function is symmetrical about the origin, as we would expect for an odd function.

47 Slide 161 / 166 Symmetry In addition to the previous two, there are other types of symmetry which a function can have including symmetry over the x-axis and diagonal symmetry. If a function has symmetry over the x-axis, then f(x)=-f(x) If the function has diagonal symmetry, then the function is the same when x and y are interchanged. Slide 162 / Identify all lines of symmetry for the equation x-axis y-axis diagonal (y=x) origin E none Slide 162 () / Identify all lines of symmetry for the equation Slide 163 / Identify all lines of symmetry for the equation x-axis y-axis diagonal (y=x) origin E none E x-axis y-axis diagonal (y=x) origin E none Slide 163 () / Identify all lines of symmetry for the equation Slide 164 / Identify all lines of symmetry for the graph x-axis y-axis diagonal (y=x) origin E none x-axis y-axis diagonal (y=x) origin E none

48 Slide 164 () / Identify all lines of symmetry for the graph Slide 165 / Identify all lines of symmetry for the graph x-axis y-axis diagonal (y=x) origin E none x-axis y-axis diagonal (y=x) origin E none Slide 165 () / Identify all lines of symmetry for the graph x-axis y-axis Slide 166 / Identify all lines of symmetry for the equation x-axis y-axis diagonal (y=x) origin E none, diagonal (y=x) origin E none Slide 166 () / Identify all lines of symmetry for the equation x-axis y-axis diagonal (y=x) origin E none