Algebra II. Analyzing and Working with Functions Part 1. Slide 1 / 166. Slide 2 / 166. Slide 3 / 166. Table of Contents
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1 Slide 1 / 166 Slide 2 / 166 lgebra II nalyzing and Working with Functions Part Table of ontents Slide 3 / 166 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
2 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 Relations relation is an association between sets of information. Slide 5 / 166 Slide 6 / 166
3 Graphs of Functions Slide 7 / 166 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! Graphs of Functions Slide 7 () / 166 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! Equations as Functions Slide 8 / 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.
4 Slide 9 / 166 Slide 9 () / 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 Slide 10 / 166
5 etermine if each of the relations below is a function and provide an explanation to support your answer: x -1 2 y 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 10 () / 166 Function Slide 11 / 166 etermine if each of the relations below is a function and provide an explanation to support your answer: Function etermine if each of the relations below is a function and provide an explanation to support your answer: Slide 11 () / 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.
6 Slide 12 / 166 Slide 12 () / Is the following relation a function? Slide 13 / 166 {(3,1), (2,-1), (1,1)} Yes No
7 1 Is the following relation a function? Slide 13 () / 166 {(3,1), (2,-1), (1,1)} Yes No Yes 2 Is the following relation a function? Slide 14 / 166 X Y Yes No 3 Is the following relation a function? Slide 15 / 166 Yes No
8 3 Is the following relation a function? Slide 15 () / 166 Yes No Yes Slide 16 / 166 Slide 16 () / 166
9 Slide 17 / 166 Slide 17 () / 166 Slide 18 / 166
10 Function Notation Slide 19 / 166 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. Evaluating a Function Slide 20 / 166 To Evaluate in y = Form: 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 To Evaluate in Function Notation: 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. Slide 21 / 166
11 Slide 21 () / 166 Slide 22 / 166 Slide 22 () / 166
12 Slide 23 / 166 Slide 23 () / Given and Slide 24 / 166 Find the value of.
13 Slide 24 () / 166 Slide 25 / 166 Slide 25 () / 166
14 8 Given and Slide 26 / 166 Find the value of. 8 Given and Slide 26 () / 166 Find the value of. Slide 27 / 166
15 Slide 27 () / 166 Slide 28 / 166 Slide 28 () / 166
16 Slide 29 / 166 Slide 29 () / 166 Slide 30 / 166
17 Slide 30 () / 166 Slide 31 / 166 Graph Interval Notation Inequality Notation Slide 32 / 166 a b losed Interval a Open Interval b a b Half-Open Interval
18 Slide 33 / 166 Summary Slide 34 / 166 { } = set = 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 Slide 35 / 166 Infinity Why do you think parentheses are used in interval notation for a data set that includes or instead of brackets?
19 Slide 35 () / 166 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. 13 What is the interval notation for the given graph? Slide 36 / 166 E F 13 What is the interval notation for the given graph? Slide 36 () / 166 E F
20 14 What is the inequality notation for the given graph? E F Slide 37 / What is the inequality notation for the given graph? E F E Slide 37 () / What is the interval notation for the given graph? Slide 38 / 166 E F
21 15 What is the interval notation for the given graph? Slide 38 () / 166 E F 16 What is the inequality notation for the given graph? Slide 39 / 166 E F 16 What is the inequality notation for the given graph? Slide 39 () / 166 E F F
22 omain and Range Slide 40 / 166 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). omain and Range Slide 41 / 166 Relation omain Range omain and Range State the domain and range for each example below: Slide 42 / 166 x x x 1 2 y y 4 y 5 8 9
23 omain and Range State the domain and range for each example below: Slide 42 () / 166 x x y y omain Range x y omain and Range Slide 43 / 166 State the domain and range for the function below. Write your answers in inequality and interval notation. omain and Range Slide 43 () / 166 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)
24 omain and Range Slide 44 / 166 State the domain and range for the function below. Write your answers in inequality and interval notation. omain and Range Slide 44 () / 166 State the domain and range for the function below. Write your answers in inequality and interval notation. omain: Range: Slide 45 / 166
25 Slide 45 () / Is -2 < x < 2 the domain of the relation? Slide 46 / 166 Yes No 18 Is -2 < x < 2 the domain of the relation? Slide 46 () / 166 Yes No No omain: -2 x 2
26 19 Is [0, 1] the range of the relation? Slide 47 / 166 Yes No 19 Is [0, 1] the range of the relation? Slide 47 () / 166 Yes No Yes Slide 48 / 166
27 Slide 48 () / 166 Slide 49 / 166 Slide 49 () / 166
28 Slide 50 / 166 Slide 50 () / 166 Slide 51 / 166
29 Slide 51 () / 166 omain 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: 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. omain 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: 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
30 omain Slide 53 / 166 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 omain Slide 53 () / 166 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 omain Slide 54 / 166 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
31 omain Slide 54 () / 166 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 Find the domain of: Slide 55 / Find the domain of: Slide 55 () / 166
32 Slide 56 / 166 Slide 56 () / Find the domain of: Slide 57 / 166
33 26 Find the domain of: Slide 57 () / Find the domain of: Slide 58 / Find the domain of: Slide 58 () / 166
34 28 Find the domain of: Slide 59 / Find the domain of: Slide 59 () / 166 Range Slide 60 / 166 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.
35 Range Slide 60 () / 166 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: Range Slide 61 / 166 Find the range of the following functions. Write your answers in interval notation Range Slide 61 () / 166 Find the range of the following functions. Write your answers in interval notation
36 29 Find the range of the following: Slide 62 / Find the range of the following: Slide 62 () / 166 Slide 63 / 166
37 Slide 63 () / 166 Slide 64 / 166 Slide 64 () / 166
38 Slide 65 / 166 Slide 65 () / 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... Slide 66 / 166 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?
39 iscrete vs ontinuous Slide 67 / 166 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. What are some continuous events? iscrete vs ontinuous Slide 68 / 166 re the following relations discrete or continuous? If continuous, state the interval of continuity. x y X Y iscrete vs ontinuous Slide 69 / 166 re the following relations discrete or continuous? If continuous, state the interval of continuity.
40 iscrete vs ontinuous Slide 69 () / 166 re the following relations discrete or continuous? If continuous, state the interval of continuity. ontinuous iscrete ontinuous iscrete vs ontinuous Slide 70 / 166 re the following relations discrete or continuous? If continuous, state the interval of continuity. Slide 70 () / 166
41 33 Is the given relation discrete or continuous? {(3,1), (2,-1), (1,1)} Slide 71 / 166 iscrete ontinuous 33 Is the given relation discrete or continuous? {(3,1), (2,-1), (1,1)} Slide 71 () / 166 iscrete ontinuous 34 Is the given relation discrete or continuous? Slide 72 / 166 iscrete ontinuous
42 34 Is the given relation discrete or continuous? Slide 72 () / 166 iscrete ontinuous 35 Is the given relation discrete or continuous? Slide 73 / 166 iscrete ontinuous 35 Is the given relation discrete or continuous? Slide 73 () / 166 iscrete ontinuous
43 36 Is the given relation discrete or continuous? Slide 74 / 166 iscrete ontinuous 36 Is the given relation discrete or continuous? Slide 74 () / 166 iscrete ontinuous 37 Is the given relation discrete or continuous? Slide 75 / 166 iscrete ontinuous
44 37 Is the given relation discrete or continuous? Slide 75 () / 166 iscrete ontinuous Slide 76 / 166 Operations with Functions Return to Table of ontents Slide 77 / 166 Goals and Objectives Students will be able to manipulate multiple functions algebraically and simplify resulting functions.
45 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 78 / 166 Operations with Functions Slide 79 / 166 Here are the properties of combining functions: dding functions: Subtracting functions: Multiplying functions: ividing functions: Operations with Functions Slide 79 () / 166 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. ividing functions:
46 Operations with Fractions Slide 80 / 166 Given: Find: and Simplify your answers as much as possible. What happens to the domain? Operations with Fractions Slide 80 () / 166 Given: Find: and Simplify your answers as much as possible. 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. Operations with Functions Slide 81 / 166 Given: Find: and
47 Operations with Functions Slide 81 () / 166 Given: Find: and 38 Given and, find Slide 82 / Given and, find Slide 82 () / 166
48 39 Given and, find h(x) if Slide 83 / Given and, find h(x) if Slide 83 () / Given and, find Slide 84 / 166
49 40 Given and, find Slide 84 () / Given and, find Slide 85 / Given and, find Slide 85 () / 166
50 omain Slide 86 / 166 Given and, find the domain of each: a) b) c) d) Slide 86 () / 166 Slide 87 / 166
51 Slide 87 () / 166 Slide 88 / 166 Slide 88 () / 166
52 44 Find the domain of if and Slide 89 / Find the domain of if and Slide 89 () / 166 ombined Functions Slide 90 / 166 You may also be asked to evaluate combined functions when given specific values for x. Given and, find: a) b) c) d)
53 ombined Functions You may also be asked to evaluate combined functions when given specific values for x. Given a) 13 and, find: Slide 90 () / 166 a) b) b) c) c) d) undefined d) 45 Given and, find Slide 91 / Given and, find Slide 91 () /
54 46 Given and, find Slide 92 / Given and, find Slide 92 () / Given and, find Slide 93 / 166 undefined
55 47 Given and, find Slide 93 () / 166 undefined Expressions may also be used to create more complex functions. Slide 94 / 166 If and, create. Leave your answer in terms of x. Expressions may also be used to create more complex functions. Slide 94 () / 166 If and, create. Leave your answer in terms of x.
56 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
57 If and, create. Leave your answer in terms of x. Slide 96 () / If and, create. Is this equivalent to? Slide 97 / 166 Yes No 48 If and, create. Is this equivalent to? Yes No No Slide 97 () / 166
58 Slide 98 / 166 Slide 98 () / If and, create. Is this equivalent to? Slide 99 / 166 Yes No
59 Slide 99 () / 166 Slide 100 / 166 omposite Functions or Return to Table of ontents Slide 101 / 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.
60 Slide 102 / 166 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. omposite Functions Slide 103 / 166 omposite functions exist when one function is "nested" in the other function. There are 2 ways of writing a composite function: or Each form is read "f of g of x" and both mean the same thing. omposite Functions 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: Find:
61 omposite Functions Slide 104 () / 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 105 () / 166
62 omposite Functions 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. Slide 106 / 166 Given: Find: Find: omposite Functions 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. Slide 106 () / 166 Given: Find: Find: 51 If and, find the value of Slide 107 / 166
63 51 If and, find the value of Slide 107 () / Find if Slide 108 / Find if Slide 108 () / 166
64 Slide 109 / 166 Slide 109 () / 166 Slide 110 / 166
65 Slide 110 () / Find if Slide 111 / Find if Slide 111 () / 166
66 56 Find if and Slide 112 / Find if and Slide 112 () / Find the value of Slide 113 /
67 57 Find the value of Slide 113 () / 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. Slide 114 / 166 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 The 12 asic Functions Slide 115 / 166 The Identity Function y = x The Squaring Function y = x 2 The ubing Function y = x 3 The Reciprocal Function y = 1/x The bsolute Value Function y = ΙxΙ The Square Root Function
68 The 12 asic Functions continued Slide 116 / 166 The Exponential Function y = e x The Natural Log Function y = lnx The Logistic Function The Sine Function y = sinx The osine Function y = cosx The Greatest Integer Function y = [x] Recall from lgebra I Slide 117 / 166 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. Function asics 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: Increasing intervals: ecreasing intervals: Odd/Even/Neither: End ehavior:
69 Graph of Intercepts Slide 119 / 166 y-intercept x-intercept x-intercept & y-intercept Slide 120 / 166 Example: Evaluate the x-intercept & y-intercept for the following equations: x-intercept & y-intercept x-intercept = (5, 0) Example: Evaluate the x-intercept y-intercept & y-intercept = (0, 10) for the following equations: Slide 120 () / 166 x-intercept = (2, 0) y-intercept = (0, -4) x-intercept = (-2, 0) y-intercept = (0, 4)
70 x-intercept & y-intercept Slide 121 / 166 Example: etermine the x-intercept and y-intercept for the following graphs: x-intercept & y-intercept Slide 121 () / 166 Example: etermine the x-intercept and y-intercept for the following graphs: x-intercept = (2, 0) y-intercept = (0, 4) x-intercept = oes Not Exist intercept = (0, 3) 58 Evaluate the y-intercept of the following equation: Slide 122 / 166 (0, 6) (6, 0) (2,0) (0,2) E oes not exist
71 58 Evaluate the y-intercept of the following equation: Slide 122 () / 166 (0, 6) (6, 0) (2,0) (0,2) E oes not exist 59 What is the x-intercept of the graph below? Slide 123 / 166 (0, 7) (7, 0) (0,21) annot be determined by graph E oes Not Exist 59 What is the x-intercept of the graph below? Slide 123 () / 166 (0, 7) (7, 0) (0,21) annot be determined by graph E oes Not Exist
72 60 Why does the x-intercept for the graph below NOT exist? Slide 124 / 166 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 60 Why does the x-intercept for the graph below NOT exist? Slide 124 () / 166 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 125 / 166
73 Increasing and ecreasing Functions Slide 126 / 166 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). Increasing ecreasing Maxima and Minima Slide 127 / 166 maxima occurs at the HIGHEST point of a graph. minima occurs at the LOWEST point of a graph maxima minima Maxima and Minima Slide 128 / 166 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.
74 Maxima and Minima Slide 129 / 166 SOLUTE MX Increasing Local Max Local Max Increasing SOLUTE MIN Local Min ecreasing Local Min ecreasing Increasing oncavity Slide 130 / 166 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. oncavity Slide 130 () / 166 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
75 Slide 131 / 166 Slide 131 () / 166 Slide 132 / 166
76 Slide 132 () / 166 Slide 133 / 166 Slide 133 () / 166
77 Slide 134 / 166 Slide 134 () / 166 Slide 135 / 166
78 Slide 135 () / 166 Slide 136 / 166 Slide 136 () / 166
79 Slide 137 / 166 Slide 138 / 166 Slide 138 () / 166
80 68 Is the following an odd-function, an even-function, or neither? Slide 139 / 166 Odd Even Neither 68 Is the following an odd-function, an even-function, or neither? Slide 139 () / 166 Odd Even Neither Slide 140 / 166
81 Slide 140 () / 166 Slide 141 / 166 Slide 141 () / 166
82 Slide 142 / 166 Slide 142 () / 166 Even-egree Polynomials Slide 143 / 166 What do you observe about the end behavior of an even function?
83 Even-egree Polynomials Slide 143 () / 166 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? Even-egree Polynomials Positive Lead oefficient Negative Lead oefficient Slide 144 / 166 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? Even-egree Polynomials Positive Lead oefficient Negative Lead oefficient Slide 144 () / 166 Positive: High to High Negative: Low to Low 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?
84 Odd-egree Polynomials Slide 145 / 166 What do you observe about the end behavior of an odd function? Odd-egree Polynomials Slide 145 () / 166 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? Positive Lead oefficient Odd-egree Polynomials Negative Lead oefficient Slide 146 / 166 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?
85 Positive Lead oefficient Odd-egree Polynomials Negative Lead oefficient Slide 146 () / 166 Positive: Low to High Negative: High to Low 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? End ehavior Slide 147 / 166 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 Left End Right End Left End Right End Even- egree Polynomial Odd- egree Polynomial 72 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 148 / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative
86 72 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 148 () / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative 73 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 149 / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative 73 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 149 () / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative
87 74 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 150 / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative 74 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 150 () / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative 75 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 151 / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative
88 75 etermine if the graph represents an odd-degree or an even-degree polynomial N if the lead coefficient is positive or negative. Slide 151 () / 166 Odd and Positive Odd and Negative Even and Positive Even and Negative 76 Pick all that apply to describe the graph below: Odd- egree Odd- Function Slide 152 / 166 E F Even- egree Even- Function Positive Lead oefficient Negative Lead oefficient 76 Pick all that apply to describe the graph below: Odd- egree Odd- Function Slide 152 () / 166 E F Even- egree,, E Even- Function Positive Lead oefficient Negative Lead oefficient
89 77 Pick all that apply to describe the graph below: Slide 153 / 166 E F Odd- egree Odd- Function Even- egree Even- Function Positive Lead oefficient Negative Lead oefficient 77 Pick all that apply to describe the graph below: Slide 153 () / 166 E F Odd- egree Odd- Function Even- egree Even-, Function, E Positive Lead oefficient Negative Lead oefficient 78 Pick all that apply to describe the graph below: Slide 154 / 166 E F Odd- egree Odd- Function Even- egree Even- Function Positive Lead oefficient Negative Lead oefficient
90 78 Pick all that apply to describe the graph below: Slide 154 () / 166 E F Odd- egree Odd- Function Even- egree Even-, Function, F Positive Lead oefficient Negative Lead oefficient 79 Pick all that apply to describe the graph below: Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient Slide 155 / 166 F Negative Lead oefficient 79 Pick all that apply to describe the graph below: Odd- egree Odd- Function Even- egree Even- Function, E E Positive Lead oefficient Slide 155 () / 166 F Negative Lead oefficient
91 80 Pick all that apply to describe the graph below: Slide 156 / 166 E F Odd- egree Odd- Function Even- egree Even- Function Positive Lead oefficient Negative Lead oefficient 80 Pick all that apply to describe the graph below: Slide 156 () / 166 E F Odd- egree Odd- Function Even- egree Even-,, Function F Positive Lead oefficient Negative Lead oefficient Odd Functions Slide 157 / 166 nother characteristic of odd functions is that they have rotational symmetry about the origin. In other words... Rotational Symmetry
92 Even Functions Slide 158 / 166 nother characteristic of even functions is that they have symmetry about the y-axis. In other words... Line of Symmetry Identifying Symmetry Slide 159 / 166 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 Notice: f(x)=f(-x), therefore f(x) is symmetrical over the y-axis, as we would expect for this even function. Identifying Symmetry Slide 160 / 166 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 the function is symmetrical about the origin, as we would expect for an odd function.
93 Symmetry Slide 161 / 166 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. 81 Identify all lines of symmetry for the equation Slide 162 / 166 x-axis y-axis diagonal (y=x) origin E none 81 Identify all lines of symmetry for the equation Slide 162 () / 166 x-axis y-axis diagonal (y=x) origin E none E
94 82 Identify all lines of symmetry for the equation Slide 163 / 166 x-axis y-axis diagonal (y=x) origin E none 82 Identify all lines of symmetry for the equation Slide 163 () / 166 x-axis y-axis diagonal (y=x) origin E none 83 Identify all lines of symmetry for the graph Slide 164 / 166 x-axis y-axis diagonal (y=x) origin E none
95 83 Identify all lines of symmetry for the graph Slide 164 () / 166 x-axis y-axis diagonal (y=x) origin E none 84 Identify all lines of symmetry for the graph Slide 165 / 166 x-axis y-axis diagonal (y=x) origin E none 84 Identify all lines of symmetry for the graph Slide 165 () / 166 x-axis y-axis diagonal (y=x) origin E none,
96 85 Identify all lines of symmetry for the equation Slide 166 / 166 x-axis y-axis diagonal (y=x) origin E none 85 Identify all lines of symmetry for the equation Slide 166 () / 166 x-axis y-axis diagonal (y=x) origin E none
Algebra II. Function Basics. Analyzing and Working with Functions Part 1. Slide 1 / 166 Slide 2 / 166. Slide 4 / 166. Slide 3 / 166.
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