Slide 1 / 166content.njctl.org/courses/math/algebra-ii/analyzing-and-working-with... · 21/04/2015 · Slide 8 / 166 Equations as Functions An equation is a function only if a number

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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!

[This object is a pull tab]

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


234

7-3 8

x y

-1 2

589

x y-2 3-5

4

x y


Function


Ans

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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Function



Function


Ans

wer

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)}



Yes

No

{(3,1), (2,-1), (1,1)}


Ans

wer

Yes

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Yes

No

X Y

-2 3

0 2

-1 -1

3 2

-2 0

Ans

wer

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Yes

No



Yes

No


Ans

wer

Yes

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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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Ans

wer

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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?


InfinityWhy do you think parentheses are used in interval notation for a data set that includes or instead of brackets?


Ans

wer

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



A

B

C

D

EF


Ans

wer

A

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14 What is the inequality notation for the given graph?

A

B

C

D

E

F



A

B

C

D

E

F[This object is a pull tab]

Ans

wer

E

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A

B

C

D

E

F



A

B

C

D

E

F[This object is a pull tab]

Ans

wer

B

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A

B

C

D

E

F



A

B

C

D

E

F


Ans

wer

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


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


Ans

wer

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



Domain and Range


Ans

wer

Domain: -2 ≤ x < 2 and [-2, 2)

Range: -2 ≤ y < 4 and [-2, 4)

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Domain and Range



Domain and Range


Ans

wer Domain:

Range:

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18 Is -2 < x < 2 the domain of the relation?

Yes

No


18 Is -2 < x < 2 the domain of the relation?

Yes

No


Ans

wer No

Domain: -2 ≤ x ≤ 2

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19 Is [0, 1] the range of the relation?

Yes

No


19 Is [0, 1] the range of the relation?

Yes

No


Ans

wer

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


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


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




1.

2.

3.

Domain


Ans

wer

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4.

5.

6.

Domain




4.

5.

6.

Domain


Ans

wer

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24 Find the domain of:

A

B

C

D



A

B

C

D


Ans

wer B

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A

B

C

D



A

B

C

D


Ans

wer A

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A

B

C

D



A

B

C

D


Ans

wer D

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A

B

C

D



A

B

C

D


Ans

wer

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


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


Ans

wer

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3. 4.

Find the range of the following functions. Write your answers in interval notation.

Range


3. 4.

Find the range of the following functions. Write your answers in interval notation.

Range


Ans

wer

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29 Find the range of the following:

A

B

C

D


29 Find the range of the following:

A

B

C

D


Ans

wer A

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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?


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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.


Ans

wer

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Ans

wer

Continuous Discrete Continuous

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33 Is the given relation discrete or continuous?

A DiscreteB Continuous

{(3,1), (2,-1), (1,1)}



A DiscreteB Continuous

{(3,1), (2,-1), (1,1)}


Ans

wer

A

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A Discrete

B Continuous



A Discrete

B Continuous


Ans

wer

B

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A Discrete

B Continuous



A Discrete

B Continuous


Ans

wer

B

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A Discrete

B Continuous



A Discrete

B Continuous


Ans

wer

A

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A Discrete

B Continuous



A Discrete

B Continuous


Ans

wer

B

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Operations with Functions

Return toTable ofContents

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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:



Here are the properties of combining functions:

Adding functions:

Subtracting functions:

Multiplying functions:

Dividing functions:



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


Given: and

Find: Simplify your

answers as much as possible.

What happens to the domain?

Operations with Fractions


Ans

wer

*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:



Given: and

Find:



Ans

wer

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38 Given and , find

A

B

C

D


38 Given and , find

A

B

C

D


Ans

wer

C

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39 Given

A

B

C

D

and , find h(x) if


39 Given

A

B

C

D

and , find h(x) if


Ans

wer

D

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40 Given and , find

A

B

C

D


40 Given and , find

A

B

C

D[This object is a pull tab]

Ans

wer

A

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41 Given and , find

A

B

C

D


41 Given and , find

A

B

C

D


Ans

wer

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


44 Find the domain of if and

A

B

C

D


Ans

wer

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


Given and , find:

a)

b)

c)

d)

You may also be asked to evaluate combined functions when given specific values for x.

Combined Functions


Ans

wer

a) 13

b)

c)

d) undefined

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45 Given and , find

A

B

C

D

-6

-4

12

10


45 Given and , find

A

B

C

D

-6

-4

12

10[This object is a pull tab]

Ans

wer

C

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46 Given and , find

A

B

C

D

1728

-864

864

1288


46 Given and , find

A

B

C

D

1728

-864

864

1288


Ans

wer

C

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47 Given and , find

A

B

C

D

undefined


47 Given and , find

A

B

C

D

undefined


Ans

wer

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.


Expressions may also be used to create more complex functions.

If and , create .



Ans

wer

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If and , create .Leave your answer in terms of x.


If and , create .Leave your answer in terms of x.


Ans

wer

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If and , create .



If and , create .



Ans

wer

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48 If and , create .

Is this equivalent to ?

Yes

No




Yes

No


Ans

wer

No

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Yes

No


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Composite Functions

Return toTable ofContents

or

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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


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


Ans

wer

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:


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:


Ans

wer

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51 If and , find the value of

AB

CD


51 If and , find the value of

AB

CD


Ans

wer

C

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52 Find if

A B C D


52 Find if

A B C D


Ans

wer

B

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55 Find if

A

B

C

D


55 Find if

A

B

C

D


Ans

wer

A

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56 Find if and

A

B

C

D

62

-88

82

19


56 Find if and

A

B

C

D

62

-88

82

19


Ans

wer

A

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57 Find the value of

A

B

C

D

0

1

2

-1


57 Find the value of

A

B

C

D

0

1

2

-1


Ans

wer

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


Example: Evaluate the x-intercept & y-intercept for the following equations:



Ans

wer

x-intercept = (5, 0)

y-intercept = (0, 10)


y-intercept = (0, -4)

x-intercept = (-2, 0)


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Example: Determine the x-intercept and y-intercept for the following graphs:



Example: Determine the x-intercept and y-intercept for the following graphs:



Ans

wer



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


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


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


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


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


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


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


Ans

wer

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


68 Is the following an odd-function, an even-function, or neither?

A Odd

B Even

C Neither


Ans

wer

A

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Even-Degree Polynomials

What do you observe about the end behavior of an even function?


Even-Degree Polynomials

What do you observe about the end behavior of an even function?


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?


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?


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?


Odd-Degree Polynomials

What do you observe about the end behavior of an odd function?


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?


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?


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.





Ans

wer

D

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A Odd and Positive

B Odd and Negative

C Even and Positive

D Even and Negative



A Odd and Positive

B Odd and Negative

C Even and Positive

D Even and Negative


Ans

wer

A

Slide 150 / 166







Ans

wer

C

Slide 151 / 166


A Odd and Positive

B Odd and Negative

C Even and Positive

D Even and Negative



A Odd and Positive

B Odd and Negative

C Even and Positive

D Even and Negative


Ans

wer

B

Slide 152 / 166

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



A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function


F Negative Lead Coefficient[This object is a pull tab]

Ans

wer

A, B, E

Slide 153 / 166


A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function





A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function




Ans

wer

C, D, E

Slide 154 / 166


A Odd- DegreeB Odd- Function

C Even- DegreeD Even- Function





A Odd- DegreeB Odd- Function

C Even- DegreeD Even- Function




Ans

wer

A, B, F

Slide 155 / 166


A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function





A Odd- Degree

B Odd- Function

C Even- Degree

D Even- Function




Ans

wer

A, E

Slide 156 / 166


A Odd- Degree

B Odd- FunctionC Even- Degree

D Even- Function





A Odd- Degree

B Odd- FunctionC Even- Degree

D Even- Function




Ans

wer

C, D, F

Slide 157 / 166

Another characteristic of odd functions is that they have rotational symmetry about the origin.

Rotational Symmetry

In other words...

Odd Functions

Slide 158 / 166

Line of Symmetry

Another characteristic of even functions is that they have symmetry about the y-axis.

In other words...

Even Functions

Slide 159 / 166

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

Slide 160 / 166

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

Slide 161 / 166

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

Slide 162 / 166

81 Identify all lines of symmetry for the equation

A x-axis

B y-axis

C diagonal (y=x)

D origin

E none



A x-axis

B y-axis

C diagonal (y=x)

D origin

E none


Ans

wer

E

Slide 163 / 166


A x-axis

B y-axis

C diagonal (y=x)

D origin

E none



A x-axis

B y-axis

C diagonal (y=x)

D origin

E none


Ans

wer

B

Slide 164 / 166

83 Identify all lines of symmetry for the graph

A x-axis

B y-axis

C diagonal (y=x)

D origin

E none



A x-axis

B y-axis

C diagonal (y=x)

D origin

E none


Ans

wer

A

Slide 165 / 166


A x-axis

B y-axis

C diagonal (y=x)

D origin

E none



A x-axis

B y-axis

C diagonal (y=x)

D origin

E none


Ans

wer

C, D

Slide 166 / 166


A x-axis

B y-axis

C diagonal (y=x)

D origin

E none



A x-axis

B y-axis

C diagonal (y=x)

D origin

E none


Ans

wer

D

Slide 1 / 166content.njctl.org/courses/math/algebra-ii/analyzing-and-working-with... · 21/04/2015 · Slide 8 / 166 Equations as Functions An equation is a function only if a number

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