Algebra II Analyzing and Working with Functions Part 1 2015-04-21 - - PDF document
Slide 1 / 166 Slide 2 / 166 Algebra II Analyzing and Working with Functions Part 1 2015-04-21 www.njctl.org Slide 3 / 166 Table of Contents click on the topic to go Part 1 to that section Function Basics Operations with Functions
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2015-04-21
The 12 Basic Functions (Parent Functions) Transforming Functions Operations with Functions Composite Functions Inverse Functions Piecewise Functions Function Basics
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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 Algebra 1.
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A relation is an association between sets of information.
The Vertical Line Test can 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 ANY time on the graph, then it is NOT a function.
Move the black vertical line to test!
The Vertical Line Test can 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 ANY time on the graph, then it is NOT a function.
Move the black vertical line to test! [This object is a pull tab]
Teacher Notes Function Not a Function
An equation is a function only if a number substituted in for x produces only 1 output or y-value. Function Reason y = 3x + 4
For each input for x, there is only
y = 5
All y values are 5.
Not a Function Reason x = 5
There are multiple values for y.
x = y2
For each x, there are two values for y.
2 3 4 7
8 x y
2 5 8 9 x y
3
4 x y Determine if each of the relations below is a function and provide an explanation to support your answer:
2 3 4 7
8 x y
2 5 8 9 x y
3
4 x y Determine if each of the relations below is a function and provide an explanation to support your answer:
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Answer Function
value has a unique y value Not a Function - x value of
more than one y value Function
value has a unique y value
Determine if each of the relations below is a function and provide an explanation to support your answer:
Determine if each of the relations below is a function and provide an explanation to support your answer:
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Answer 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.
Yes No
Yes No
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Answer Yes
Yes No
X Y
3 2
3 2
Answer
Yes No
Yes No
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Answer Yes
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.
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.
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Answer
Graph Interval Notation Inequality Notation Closed Interval a b Open Interval a b Half-Open Interval a b
D = Domain (possible input or x-values) R = Range (possible output or y-values) { } = set ∈ = is an element of (belongs to) = Set of Real Numbers = Set of Integers = Natural Numbers
= positive infinity = negative infinity
Why do you think parentheses are used in interval notation for a data set that includes or instead of brackets?
Why do you think parentheses are used in interval notation for a data set that includes or instead of brackets?
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Answer Infinity/Negative Infinity do not have a final value, they can always increase/
never end a parentheses is used instead of a bracket.
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Answer A
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Answer E
A B C D E F
A B C D E F
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Answer B
A B C D E F
A B C D E F
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Answer F
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).
Relation Domain Range
State the domain and range for each example below: 2 3 4 7
8 x y 1 2 5 8 9 x y
3
4 x y
State the domain and range for each example below: 2 3 4 7
8 x y 1 2 5 8 9 x y
3
4 x y
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Answer Domain 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.
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Answer Domain: -2 ≤ x < 2 and [-2, 2) Range: -2 ≤ y < 4 and [-2, 4)
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.
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Answer Domain: Range:
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Answer No Domain: -2 ≤ x ≤ 2
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Answer Yes
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: Fractions: There can be NO NEGATIVE values under a root. Set the radicand greater than or equal to zero (positive). Solve. In a fraction, the denominator CANNOT BE ZERO. Set the denominator equal to zero and solve.
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: Fractions: There can be NO NEGATIVE values under a root. Set the radicand greater than or equal to zero (positive). Solve. In a fraction, the denominator CANNOT BE ZERO. Set the denominator equal to zero and solve.
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Teacher Notes Note: Another restriction for domains will be logarithms of negative numbers or zero; however, students have not learned this concept yet.
Again, start with All Real Numbers . Then look for roots or fractions in your function. Find the domain of the following functions. Write your answers in interval notation. 1. 2. 3.
Again, start with All Real Numbers . Then look for roots or fractions in your function. Find the domain of the following functions. Write your answers in interval notation. 1. 2. 3.
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Answer
Again, start with All Real Numbers . Then look for roots or fractions in your function. Find the domain of the following functions. Write your answers in interval notation. 4. 5. 6.
Again, start with All Real Numbers . Then look for roots or fractions in your function. Find the domain of the following functions. Write your answers in interval notation. 4. 5. 6.
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Answer
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Answer
B
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Answer
A
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Answer
D
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Answer
B
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.
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.
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Answer
Find the range of the following functions. Write your answers in interval notation.
Find the range of the following functions. Write your answers in interval notation.
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Answer
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Answer
A
A 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, 3.5899, etc. These values do not have meaning in this situation, therefore the data is discrete. What are some other discrete events?
A 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?
2 3 4 7
8 x y
X Y 1 3 2 4 5
3 9 4 7
Are the following relations discrete or continuous? If continuous, state the interval of continuity.
Answer
Are the following relations discrete or continuous? If continuous, state the interval of continuity.
Are the following relations discrete or continuous? If continuous, state the interval of continuity.
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Answer Continuous Discrete Continuous
Are the following relations discrete or continuous? If continuous, state the interval of continuity.
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Answer A
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Answer B
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Answer B
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Answer A
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Answer B
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Students will be able to manipulate multiple functions algebraically and simplify resulting functions.
In this unit, we will graphically explored transformations
requires more than one representative function. Algebraically, manipulating functions allows us to combine different functions together and results in many more options for real life situations.
Here are the properties of combining functions: Adding functions: Subtracting functions: Multiplying functions: Dividing functions:
Here are the properties of combining functions: Adding functions: Subtracting functions: Multiplying functions: Dividing functions:
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Answer Students may note that the properties are "common sense." The only hard part sometimes is simplifying the expressions.
Given: and Find:
Simplify your answers as much as possible. What happens to the domain?
Given: and Find:
Simplify your answers as much as possible. What happens to the domain?
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Answer
*The domain of a resulting function is subject to the domain of the
Given: and Find:
Given: and Find:
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Answer
38 Given and , find A B C D
38 Given and , find A B C D
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Answer C
39 Given A B C D and , find h(x) if
39 Given A B C D and , find h(x) if
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Answer D
40 Given and , find A B C D
40 Given and , find A B C D
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Answer A
41 Given and , find A B C D
41 Given and , find A B C D
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Answer D
Given and , find the domain of each: a) b) c) d)
44 Find the domain of if and A B C D
44 Find the domain of if and A B C D
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Answer B
a) b) c) d) You may also be asked to evaluate combined functions when given specific values for x.
a) b) c) d) You may also be asked to evaluate combined functions when given specific values for x.
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Answer a) 13 b) c) d) undefined
45 Given and , find A B C D
45 Given and , find A B C D
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Answer C
46 Given and , find A B C D 1728
864 1288
46 Given and , find A B C D 1728
864 1288
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Answer C
47 Given and , find A B C D undefined
47 Given and , find A B C D undefined
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Answer D
Expressions may also be used to create more complex functions. If and , create . Leave your answer in terms of x.
Expressions may also be used to create more complex functions. If and , create . Leave your answer in terms of x.
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Answer
If and , create . Leave your answer in terms of x.
If and , create . Leave your answer in terms of x.
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Answer
If and , create . Leave your answer in terms of x.
If and , create . Leave your answer in terms of x.
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Answer
48 If and , create . Is this equivalent to ?
48 If and , create . Is this equivalent to ?
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Answer
No
50 If and , create . Is this equivalent to ? Yes No
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Students will be able to recognize function notation and correctly unite two or more functions together to create a new function.
On many occasions, multiple situations happen to something before it obtains a final result. For example, you take extra food
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.
Composite functions exist when one function is "nested" in the other function. There are 2 ways of writing a composite function: Each form is read "f of g of x" and both mean the same thing.
To simplify composite functions, substitute one function into the
Given: Find: Find:
To simplify composite functions, substitute one function into the
Given: Find: Find:
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Answer
Stress the differences and the "nesting" and how that affects the order. Sometimes, students struggle the most with simplifying.
f(g(x)) = f( g(x))
= 3(g(x))2 + 2(g(x)) = 3(4x)2 + 2(4x) = 3(16x
2) + 8x
= 48x
2 + 8x
g(f(x)) = g( f(x))
= 4(f(x)) =4(3x2 + 2x ) = 12x
2 + 8x
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:
Find:
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:
Find:
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Answer
51 If and , find the value of A B C D
51 If and , find the value of A B C D
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Answer C
52 Find if A B C D
52 Find if A B C D
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Answer B
55 Find if A B C D
55 Find if A B C D
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Answer A
56 Find if and A B C D 62
82 19
56 Find if and A B C D 62
82 19
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Answer A
57 Find the value of A B C D 1 2
57 Find the value of A B C D 1 2
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Answer C 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.
Many situations in the world that people study and collect data on follow one of the following 12 patterns. By 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
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The Identity Function y = x The Squaring Function y = x
2
The Cubing Function y = x
3
The Reciprocal Function y = 1/x The Absolute Value Function y = ΙxΙ The Square Root Function
continued
The Exponential Function y = e
x
The Natural Log Function y = lnx The Logistic Function The Sine Function y = sinx The Cosine Function y = cosx The Greatest Integer Function y = [x]
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.
When studying the graphs of functions, scientists like to analyze many different aspects of the graph. Domain: Range: Minimum (Min): Maximum (Max): Intercepts: x-intercepts: y-intercepts: Increasing intervals: Decreasing intervals: Odd/Even/Neither: End Behavior:
x-intercept y-intercept
Example: Evaluate the x-intercept & y-intercept for the following equations:
Example: Evaluate the x-intercept & y-intercept for the following equations:
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Answer x-intercept = (5, 0) y-intercept = (0, 10) x-intercept = (2, 0) y-intercept = (0, -4) x-intercept = (-2, 0) y-intercept = (0, 4)
Example: Determine the x-intercept and y-intercept for the following graphs:
Example: Determine the x-intercept and y-intercept for the following graphs:
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Answer x-intercept = (2, 0) y-intercept = (0, 4) x-intercept = Does Not Exist intercept = (0, 3)
58 Evaluate the y-intercept of the following equation: A (0, 6) B (6, 0) C (2,0) D (0,2) E Does not exist
58 Evaluate the y-intercept of the following equation: A (0, 6) B (6, 0) C (2,0) D (0,2) E Does not exist
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Answer A
59 What is the x-intercept of the graph below? A (0, 7) B (7, 0) C (0,21) E Does Not Exist D Cannot be determined by graph
59 What is the x-intercept of the graph below? A (0, 7) B (7, 0) C (0,21) E Does Not Exist D Cannot be determined by graph
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Answer B
60 Why does the x-intercept for the graph below NOT exist? A The x-intercept does exist B The graph is misleading C There is not enough of the graph shown to determine the reason D The graph does not pass through the x-axis
60 Why does the x-intercept for the graph below NOT exist? A The x-intercept does exist B The graph is misleading C There is not enough of the graph shown to determine the reason D The graph does not pass through the x-axis
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Answer
A function is said to be increasing when the graph is travelling in an upward direction (when traveled from left to right). A function is said to be decreasing when the graph is travelling in a downward direction (when traveled from left to right).
Increasing Decreasing
maxima
A maxima occurs at the HIGHEST point of a graph.
minima
A minima occurs at the LOWEST point of a graph
A maximum occurs when a function changes from increasing to decreasing. A minimum occurs when a function changes from decreasing to increasing. There are 2 types of maximums/minimums: –Local: Any turning point in the graph. Note: a Local max/min CANNOT occur at endpoints. –Absolute: The highest/lowest point on the graph. Note: an Absolute max/min CAN occur at an endpoint.
Local Max Local Max Local Min Local Min ABSOLUTE MAX ABSOLUTE MIN Increasing Decreasing Increasing Decreasing Increasing
The concavity of a function is the direction of the "bowl shape" of a graph. A graph is concave up if the bowl faces upward. A graph is concave down if the bowl is upside down.
The concavity of a function is the direction of the "bowl shape" of a graph. A graph is concave up if the bowl faces upward. A graph is concave down if the bowl is upside down.
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Answer Students may also find it easy to remember by using the following: Concave Down = Frown Concave Up = Cup
68 Is the following an odd-function, an even-function,
A Odd B Even C Neither
68 Is the following an odd-function, an even-function,
A Odd B Even C Neither
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Answer A
What do you observe about the end behavior of an even function?
What do you observe about the end behavior of an even function?
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Answer Start and end in same place (High to High or Low to Low)
Positive Lead Coefficient Negative Lead Coefficient 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?
Positive Lead Coefficient Negative Lead Coefficient 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?
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Answer 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 odd function?
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Answer Start and end in different places (High to Low or Low to High)
Positive Lead Coefficient Negative Lead Coefficient 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?
Positive Lead Coefficient Negative Lead Coefficient 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?
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Answer Positive: Low to High Negative: High to Low
When describing the end behavior of a polynomial, we are describing what the y-values are approaching.
Lead Coefficient is Positive
Left End Right End
Lead Coefficient is Negative
Left End Right End
Even- Degree Polynomial Odd- Degree Polynomial
A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative 72 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative.
A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative 72 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative.
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Answer D
73 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative. A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative
73 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative. A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative
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Answer A
74 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative. A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative
74 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative. A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative
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Answer C
75 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative. A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative
75 Determine if the graph represents an odd-degree or an even-degree polynomial AND if the lead coefficient is positive or negative. A Odd and Positive B Odd and Negative C Even and Positive D Even and Negative
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Answer B
76 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
76 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
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Answer A, B, E
77 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
77 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
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Answer C, D, E
78 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
78 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
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Answer A, B, F
79 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
79 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
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Answer A, E
80 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
80 Pick all that apply to describe the graph below: A Odd- Degree B Odd- Function C Even- Degree D Even- Function E Positive Lead Coefficient F Negative Lead Coefficient
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Answer C, D, F
Another characteristic of odd functions is that they have rotational symmetry about the origin.
Rotational Symmetry
In other words...
Line of Symmetry
Another characteristic of even functions is that they have symmetry about the y-axis. In other words...
We can identify symmetry by comparing values of f(x). If f(x) has symmetry over the y-axis, then f(x)=f(-x) Notice: f(x)=f(-x), therefore f(x) is symmetrical over the y-axis, as we would expect for this even function. Example: Given the even function: 1) Plug in -x. 2) Simplify
If the function is symmetrical about the origin, then f(-x)=-f(x) Notice: f(-x)=-f(x), therefore the function is symmetrical about the origin, as we would expect for an odd function. Example: Given the odd function: 1) Plug in -x 2)Simplify
If a function has symmetry over the x-axis, then f(x)=-f(x) 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 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 A x-axis B y-axis C diagonal (y=x) D origin E none
81 Identify all lines of symmetry for the equation A x-axis B y-axis C diagonal (y=x) D origin E none
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Answer E
82 Identify all lines of symmetry for the equation A x-axis B y-axis C diagonal (y=x) D origin E none
82 Identify all lines of symmetry for the equation A x-axis B y-axis C diagonal (y=x) D origin E none
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Answer B
83 Identify all lines of symmetry for the graph A x-axis B y-axis C diagonal (y=x) D origin E none
83 Identify all lines of symmetry for the graph A x-axis B y-axis C diagonal (y=x) D origin E none
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Answer A
84 Identify all lines of symmetry for the graph A x-axis B y-axis C diagonal (y=x) D origin E none
84 Identify all lines of symmetry for the graph A x-axis B y-axis C diagonal (y=x) D origin E none
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Answer C, D
85 Identify all lines of symmetry for the equation A x-axis B y-axis C diagonal (y=x) D origin E none
85 Identify all lines of symmetry for the equation A x-axis B y-axis C diagonal (y=x) D origin E none
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Answer D