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Algebra II
www.njctl.org
Analyzing and Working with Functions
Part 1
2015-04-21
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Table of Contents
The 12 Basic Functions (Parent Functions)
Transforming Functions
Operations with FunctionsComposite Functions
Inverse FunctionsPiecewise Functions
Function Basics
click on the topic to go to that sectionPart 1
Part 2
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Function Basics
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.
Return to Table of Contents
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RelationsA relation is an association between sets of information.
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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.
Graphs of Functions
Move the black vertical line to test!
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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.
Graphs of Functions
Move the black vertical line to test!
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Teac
her N
otes
Function Not a Function
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Equations as Functions
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 one output of y.
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.
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234
7-3 8
x y
-1 2
589
x y-2 3-5
4
x y
Determine if each of the relations below is a function and provide an explanation to support your answer:
Function
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234
7-3 8
x y
-1 2
589
x y-2 3-5
4
x y
Determine if each of the relations below is a function and provide an explanation to support your answer:
Function
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wer
Function - each x value has a unique y value
Not a Function - x value of -1 yields more than one y value
Function - each x value has a unique y value
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Determine if each of the relations below is a function and provide an explanation to support your answer:
Function
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Determine if each of the relations below is a function and provide an explanation to support your answer:
Function
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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.
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1 Is the following relation a function?
Yes
No
{(3,1), (2,-1), (1,1)}
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1 Is the following relation a function?
Yes
No
{(3,1), (2,-1), (1,1)}
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Yes
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2 Is the following relation a function?
Yes
No
X Y
-2 3
0 2
-1 -1
3 2
-2 0
Ans
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3 Is the following relation a function?
Yes
No
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3 Is the following relation a function?
Yes
No
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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.
Function Notation
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Evaluating a Function To Evaluate in y = Form:
Find the value of y = 2x + 1when x = 3
y = 2x + 1y = 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) + 1f(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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6 Given and Find the value of .
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8 Given and Find the value of .
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8 Given and Find the value of .
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Graph Interval Notation Inequality Notation
Closed Intervala b
Open Intervala b
Half-Open Intervala b
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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
Summary
= positive infinity= negative infinity
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InfinityWhy do you think parentheses are used in interval notation for a data set that includes or instead of brackets?
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InfinityWhy do you think parentheses are used in interval notation for a data set that includes or instead of brackets?
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Infinity/Negative Infinity do not have a final value, they can always increase/decrease. Since they never end a parentheses is used instead of a bracket.
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13 What is the interval notation for the given graph?
A
B
C
D
EF
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13 What is the interval notation for the given graph?
A
B
C
D
EF
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A
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14 What is the inequality notation for the given graph?
A
B
C
D
E
F
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14 What is the inequality notation for the given graph?
A
B
C
D
E
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15 What is the interval notation for the given graph?
A
B
C
D
E
F
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15 What is the interval notation for the given graph?
A
B
C
D
E
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B
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16 What is the inequality notation for the given graph?
A
B
C
D
E
F
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16 What is the inequality notation for the given graph?
A
B
C
D
E
F
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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).
Domain and Range
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Relation Domain Range
Domain and Range
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State the domain and range for each example below:
234
7-3 8
x y
12
589
x y
-2 3-5
4
x y
Domain and Range
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State the domain and range for each example below:
234
7-3 8
x y
12
589
x y
-2 3-5
4
x y
Domain and Range
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Domain Range
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State the domain and range for the function below. Write your answers in inequality and interval notation.
Domain and Range
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State the domain and range for the function below. Write your answers in inequality and interval notation.
Domain and Range
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Domain: -2 ≤ x < 2 and [-2, 2)
Range: -2 ≤ y < 4 and [-2, 4)
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State the domain and range for the function below. Write your answers in inequality and interval notation.
Domain and Range
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State the domain and range for the function below. Write your answers in inequality and interval notation.
Domain and Range
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Range:
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18 Is -2 < x < 2 the domain of the relation?
Yes
No
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18 Is -2 < x < 2 the domain of the relation?
Yes
No
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Domain: -2 ≤ x ≤ 2
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19 Is [0, 1] the range of the relation?
Yes
No
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19 Is [0, 1] the range of the relation?
Yes
No
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Yes
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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.
Domain
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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.
Domain
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Teac
her N
otes Note: Another restriction
for domains will be logarithms of negative
numbers or zero; however, students have not learned
this concept yet.
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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.
Domain
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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.
Domain
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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.
Domain
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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.
Domain
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24 Find the domain of:
A
B
C
D
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24 Find the domain of:
A
B
C
D
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26 Find the domain of:
A
B
C
D
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26 Find the domain of:
A
B
C
D
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27 Find the domain of:
A
B
C
D
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27 Find the domain of:
A
B
C
D
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28 Find the domain of:
A
B
C
D
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28 Find the domain of:
A
B
C
D
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B
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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.
Range
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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.
Range
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3. 4.
Find the range of the following functions. Write your answers in interval notation.
Range
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3. 4.
Find the range of the following functions. Write your answers in interval notation.
Range
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29 Find the range of the following:
A
B
C
D
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29 Find the range of the following:
A
B
C
D
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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?
Discrete vs Continuous
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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?
Discrete vs Continuous
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234
7-3 8
x y X Y
1 3
2 4
5 -5
3 9
4 7
Are the following relations discrete or continuous? If continuous, state the interval of continuity.
Discrete vs Continuous
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Are the following relations discrete or continuous? If continuous, state the interval of continuity.
Discrete vs Continuous
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Are the following relations discrete or continuous? If continuous, state the interval of continuity.
Discrete vs Continuous
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Continuous Discrete Continuous
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Are the following relations discrete or continuous? If continuous, state the interval of continuity.
Discrete vs Continuous
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33 Is the given relation discrete or continuous?
A DiscreteB Continuous
{(3,1), (2,-1), (1,1)}
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33 Is the given relation discrete or continuous?
A DiscreteB Continuous
{(3,1), (2,-1), (1,1)}
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A
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34 Is the given relation discrete or continuous?
A Discrete
B Continuous
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34 Is the given relation discrete or continuous?
A Discrete
B Continuous
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B
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35 Is the given relation discrete or continuous?
A Discrete
B Continuous
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35 Is the given relation discrete or continuous?
A Discrete
B Continuous
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B
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36 Is the given relation discrete or continuous?
A Discrete
B Continuous
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36 Is the given relation discrete or continuous?
A Discrete
B Continuous
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A
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37 Is the given relation discrete or continuous?
A Discrete
B Continuous
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37 Is the given relation discrete or continuous?
A Discrete
B Continuous
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B
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Goals and ObjectivesStudents will be able to manipulate multiple functions algebraically and simplify resulting
functions.
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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.
Algebraically, manipulating functions allows us to combine different functions together and results in
many more options for real life situations.
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Here are the properties of combining functions:
Adding functions:
Subtracting functions:
Multiplying functions:
Dividing functions:
Operations with Functions
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Here are the properties of combining functions:
Adding functions:
Subtracting functions:
Multiplying functions:
Dividing functions:
Operations with Functions
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Ans
wer Students may note that the
properties are "common sense." The only hard part sometimes is simplifying the expressions.
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Given: and
Find: Simplify your
answers as much as possible.
What happens to the domain?
Operations with Fractions
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Given: and
Find: Simplify your
answers as much as possible.
What happens to the domain?
Operations with Fractions
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*The domain of a resulting function is subject to the domain of the original functions as well as the final function.
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Given: and
Find:
Operations with Functions
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Given: and
Find:
Operations with Functions
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38 Given and , find
A
B
C
D
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38 Given and , find
A
B
C
D
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C
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39 Given
A
B
C
D
and , find h(x) if
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39 Given
A
B
C
D
and , find h(x) if
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D
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40 Given and , find
A
B
C
D
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40 Given and , find
A
B
C
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41 Given and , find
A
B
C
D
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41 Given and , find
A
B
C
D
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D
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Given and , find the domain of each:
a) b) c) d)
Domain
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44 Find the domain of if and
A
B
C
D
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44 Find the domain of if and
A
B
C
D
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B
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Given and , find:
a)
b)
c)
d)
You may also be asked to evaluate combined functions when given specific values for x.
Combined Functions
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Given and , find:
a)
b)
c)
d)
You may also be asked to evaluate combined functions when given specific values for x.
Combined Functions
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a) 13
b)
c)
d) undefined
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45 Given and , find
A
B
C
D
-6
-4
12
10
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45 Given and , find
A
B
C
D
-6
-4
12
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C
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46 Given and , find
A
B
C
D
1728
-864
864
1288
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46 Given and , find
A
B
C
D
1728
-864
864
1288
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C
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47 Given and , find
A
B
C
D
undefined
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47 Given and , find
A
B
C
D
undefined
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D
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Expressions may also be used to create more complex functions.
If and , create .
Leave your answer in terms of x.
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Expressions may also be used to create more complex functions.
If and , create .
Leave your answer in terms of x.
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If and , create .Leave your answer in terms of x.
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If and , create .Leave your answer in terms of x.
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If and , create .
Leave your answer in terms of x.
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If and , create .
Leave your answer in terms of x.
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48 If and , create .
Is this equivalent to ?
Yes
No
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48 If and , create .
Is this equivalent to ?
Yes
No
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No
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50 If and , create .
Is this equivalent to ?
Yes
No
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Goals and ObjectivesStudents will be able to recognize
function notation and correctly unite two or more functions together to
create a new function.
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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.
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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.
or
Composite Functions
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To simplify composite functions, substitute one function into the other in place of "x" and simplify. Work from the inside out.
Given:
Find: Find:
Composite Functions
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To simplify composite functions, substitute one function into the other in place of "x" and simplify. Work from the inside out.
Given:
Find: Find:
Composite Functions
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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(16x2) + 8x
= 48x2 + 8x
g(f(x)) = g(f(x))
= 4(f(x))
=4(3x2 + 2x)
= 12x2 + 8x
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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:
Composite Functions
Find:
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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:
Composite Functions
Find:
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51 If and , find the value of
AB
CD
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51 If and , find the value of
AB
CD
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C
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52 Find if
A B C D
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52 Find if
A B C D
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B
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55 Find if
A
B
C
D
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55 Find if
A
B
C
D
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56 Find if and
A
B
C
D
62
-88
82
19
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56 Find if and
A
B
C
D
62
-88
82
19
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A
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57 Find the value of
A
B
C
D
0
1
2
-1
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57 Find the value of
A
B
C
D
0
1
2
-1
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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.
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The 12 Basic Functions (Parent Functions)
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
match. Some people get paid a lot of money to do this!
Return to Table of Contents
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The 12 Basic FunctionsThe Identity Function
y = xThe Squaring Function
y = x2The Cubing Function
y = x3
The Reciprocal Functiony = 1/x
The Absolute Value Functiony = ΙxΙ
The Square Root Function
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The 12 Basic Functionscontinued
The Exponential Functiony = ex
The Natural Log Functiony = lnx
The Logistic Function
The Sine Functiony = sinx
The Cosine Functiony = cosx
The Greatest Integer Functiony = [x]
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Recall from Algebra 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.
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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:
Function Basics
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Graph of Intercepts
x-intercept
y-intercept
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Example: Evaluate the x-intercept & y-intercept for the following equations:
x-intercept & y-intercept
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Example: Evaluate the x-intercept & y-intercept for the following equations:
x-intercept & y-intercept
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Ans
wer
x-intercept = (5, 0)
y-intercept = (0, 10)
x-intercept = (2, 0)
y-intercept = (0, -4)
x-intercept = (-2, 0)
y-intercept = (0, 4)
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Example: Determine the x-intercept and y-intercept for the following graphs:
x-intercept & y-intercept
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Example: Determine the x-intercept and y-intercept for the following graphs:
x-intercept & y-intercept
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Ans
wer
x-intercept = (2, 0)
y-intercept = (0, 4)
x-intercept = Does Not Exist
intercept = (0, 3)
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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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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 [This object is a pull tab]
Ans
wer
A
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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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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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Ans
wer
B
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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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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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Ans
wer
D
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Increasing and Decreasing Functions
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
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Maxima and Minima
maxima
A maxima occurs at the HIGHEST point of a graph.
minima
A minima occurs at the LOWEST point of a graph
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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.
Maxima and Minima
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LocalMax
LocalMax
LocalMin
LocalMin
ABSOLUTEMAX
ABSOLUTEMIN
Increasing
Decreasing
Increasing
Decreasing
Increasing
Maxima and Minima
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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.
Concavity
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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.
Concavity
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Students may also find it easy to remember by using the following:
Concave Down = Frown
Concave Up = Cup
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68 Is the following an odd-function, an even-function, or neither?
A Odd
B Even
C Neither
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68 Is the following an odd-function, an even-function, or neither?
A Odd
B Even
C Neither
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Ans
wer
A
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Even-Degree Polynomials
What do you observe about the end behavior of an even function?
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Even-Degree Polynomials
What do you observe about the end behavior of an even function?
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Ans
wer Start and end in same place
(High to High or Low to Low)
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Even-Degree PolynomialsPositive 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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Even-Degree PolynomialsPositive 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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Ans
wer Positive: High to High
Negative: Low to Low
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Odd-Degree Polynomials
What do you observe about the end behavior of an odd function?
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Odd-Degree Polynomials
What do you observe about the end behavior of an odd function?
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Ans
wer Start and end in different places
(High to Low or Low to High)
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Odd-Degree PolynomialsPositive 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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Odd-Degree PolynomialsPositive 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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Ans
wer Positive: Low to High
Negative: High to Low
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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 Coefficientis Negative
Left End Right End
Even- Degree Polynomial
Odd- Degree Polynomial
End Behavior
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A Odd and PositiveB Odd and NegativeC Even and PositiveD 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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A Odd and PositiveB Odd and NegativeC Even and PositiveD 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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Ans
wer
D
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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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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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Ans
wer
A
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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 PositiveB Odd and NegativeC Even and PositiveD Even and Negative
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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 PositiveB Odd and NegativeC Even and PositiveD Even and Negative
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Ans
wer
C
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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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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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Ans
wer
B
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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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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[This object is a pull tab]
Ans
wer
A, B, E
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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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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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Ans
wer
C, D, E
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78 Pick all that apply to describe the graph below:
A Odd- DegreeB Odd- Function
C Even- DegreeD Even- Function
E Positive Lead Coefficient
F Negative Lead Coefficient
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78 Pick all that apply to describe the graph below:
A Odd- DegreeB Odd- Function
C Even- DegreeD Even- Function
E Positive Lead Coefficient
F Negative Lead Coefficient
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Ans
wer
A, B, F
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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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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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wer
A, E
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80 Pick all that apply to describe the graph below:
A Odd- Degree
B Odd- FunctionC Even- Degree
D Even- Function
E Positive Lead Coefficient
F Negative Lead Coefficient
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80 Pick all that apply to describe the graph below:
A Odd- Degree
B Odd- FunctionC Even- Degree
D Even- Function
E Positive Lead Coefficient
F Negative Lead Coefficient
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Ans
wer
C, D, F
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Another characteristic of odd functions is that they have rotational symmetry about the origin.
Rotational Symmetry
In other words...
Odd Functions
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Line of Symmetry
Another characteristic of even functions is that they have symmetry about the y-axis.
In other words...
Even Functions
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Identifying SymmetryWe 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
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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
Identifying Symmetry
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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.
Symmetry
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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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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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wer
E
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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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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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wer
B
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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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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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Ans
wer
A
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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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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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wer
C, D
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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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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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Ans
wer
D