Fall 2010 VDOE Mathematics Institute Grade Band 9-12 Functions K-12 Mathematics Institutes Fall 2010...

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Fall 2010

VDOE Mathematics Institute

Grade Band 9-12Functions

K-12 Mathematics Institutes

Fall 2010

VDOE Mathematics Institute

Grade Band 9-12Functions

K-12 Mathematics Institutes

Fall 2010

Placemat ConsensusFunctions

Common ideas are written

Individual ideas are written here

Fall 2010

Overview of Vertical ProgressionOverview of Vertical ProgressionMiddle School (Function Analysis)

7.12 … represent relationships with tables, graphs, rules and words

8.14 … make connections between any two representations (tables, graphs, words, rules)

Middle School (Function Analysis)

7.12 … represent relationships with tables, graphs, rules and words

8.14 … make connections between any two representations (tables, graphs, words, rules)

Fall 2010

Overview of Vertical ProgressionOverview of Vertical ProgressionAlgebra I (Function Analysis)

A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including

a) determining whether a relation is a function;b) domain and range;c) zeros of a function;d) x- and y-intercepts;e) finding the values of a function for elements in its

domain; andf) making connections between and among multiple

representations of functions including concrete, verbal, numeric, graphic, and algebraic.

Algebra I (Function Analysis)

A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including

a) determining whether a relation is a function;b) domain and range;c) zeros of a function;d) x- and y-intercepts;e) finding the values of a function for elements in its

domain; andf) making connections between and among multiple

representations of functions including concrete, verbal, numeric, graphic, and algebraic.

Fall 2010

Overview of Vertical ProgressionOverview of Vertical ProgressionAlgebra, Functions and Data Analysis

(Function Analysis)

AFDA.1 The student will investigate and analyze function (linear, quadratic, exponential, and logarithmic) families and their characteristics. Key concepts include

a) continuity;b) local and absolute maxima and minima;c) domain and range;d) zeros;e) intercepts;f) intervals in which the function is

increasing/decreasing;g) end behaviors; andh) asymptotes.

Algebra, Functions and Data Analysis (Function Analysis)

AFDA.1 The student will investigate and analyze function (linear, quadratic, exponential, and logarithmic) families and their characteristics. Key concepts include

a) continuity;b) local and absolute maxima and minima;c) domain and range;d) zeros;e) intercepts;f) intervals in which the function is

increasing/decreasing;g) end behaviors; andh) asymptotes.

Fall 2010

Overview of Vertical ProgressionOverview of Vertical ProgressionAlgebra, Functions and Data Analysis

(Function Analysis)

AFDA.4 The student will transfer between and analyze multiple representations of functions, including algebraic formulas, graphs, tables, and words. Students will select and use appropriate representations for analysis, interpretation, and prediction.

Algebra, Functions and Data Analysis (Function Analysis)

AFDA.4 The student will transfer between and analyze multiple representations of functions, including algebraic formulas, graphs, tables, and words. Students will select and use appropriate representations for analysis, interpretation, and prediction.

Fall 2010

Overview of Vertical ProgressionOverview of Vertical ProgressionAlgebra 2 (Function Analysis)

AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include

a) domain and range, including limited and discontinuous domains and ranges;

b) zeros;c) x- and y-intercepts;d) intervals in which a function is increasing or

decreasing;e) asymptotes;f) end behavior;g) inverse of a function; andh) composition of multiple functions. Graphing calculators will be used as a tool to assist

in investigation of functions.

Algebra 2 (Function Analysis)

AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include

a) domain and range, including limited and discontinuous domains and ranges;

b) zeros;c) x- and y-intercepts;d) intervals in which a function is increasing or

decreasing;e) asymptotes;f) end behavior;g) inverse of a function; andh) composition of multiple functions. Graphing calculators will be used as a tool to assist

in investigation of functions.

Fall 2010

VocabularyVocabulary

The new 2009 SOL mathematics standards focus on the use of

appropriate and accurate mathematics vocabulary.

The new 2009 SOL mathematics standards focus on the use of

appropriate and accurate mathematics vocabulary.

Fall 2010

“Function” Vocabulary Across Grade Levels

Relation Domain – limited/ discontinuous Range Continuity Zeros Intercepts Elements (values) Multiple

Representations

Local & Absolute Maxima & Minima (turning points) Increasing/ Decreasing Intervals End Behavior Inverses Asymptotes (and holes)

Fall 2010

Vocabulary Across Grade LevelsVocabulary Across Grade Levels

EvaluateSolve

SimplifyApply

AnalyzeConstruct

Compare/contrastCalculate

GraphTransform

FactorIdentify

Fall 2010

Wordle – Algebra I 2009 VA SOLswww.wordle.net

Fall 2010

Wordle – Algebra, Functions and Data Analysis 2009 VA SOLs

Fall 2010

Wordle – Algebra II 2009 VA SOLs Wordle – Algebra II 2009 VA SOLs

Fall 2010

Wordle – Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry

Fall 2010

Reasoning with FunctionsReasoning with FunctionsKey elements of reasoning and sensemaking with functions include:• Using multiple representations of

functions• Modeling by using families of

functions• Analyzing the effects of different

parameters

Adapted from Focus in High School Mathematics:Reasoning and Sense Making, NCTM, 2009

Key elements of reasoning and sensemaking with functions include:• Using multiple representations of

functions• Modeling by using families of

functions• Analyzing the effects of different

parameters

Adapted from Focus in High School Mathematics:Reasoning and Sense Making, NCTM, 2009

Fall 2010

Using Multiple Representations of Functions

• Tables• Graphs or diagrams• Symbolic representations• Verbal descriptions

Fall 2010 17

Algebra Tiles ~ AddingAlgebra Tiles ~ Adding

Add the polynomials.

(x – 2) + (x + 1)

Add the polynomials.

(x – 2) + (x + 1)

= 2x - 1

Fall 2010 18

Algebra Tiles ~ MultiplyingAlgebra Tiles ~ Multiplying

x + 2 x + 3

(x + 2)(x + 3)

Fall 2010 19

Multiply the polynomials using tiles.

Create an array of the polynomials

(x + 2)(x + 3)

x2 + 5x + 6

Fall 2010 20

Algebra Tiles ~ FactoringAlgebra Tiles ~ Factoring

Work backwards from the array.

(x – 1)(x – 2)

x2 - 3x + 2

Fall 2010

Polynomial DivisionPolynomial Division

A.2 The student will perform operations on polynomials, including

a)applying the laws of exponents to perform operations on expressions;

b)adding, subtracting, multiplying, and dividing polynomials; and

c)factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations.

A.2 The student will perform operations on polynomials, including

a)applying the laws of exponents to perform operations on expressions;

b)adding, subtracting, multiplying, and dividing polynomials; and

c)factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations.

Fall 2010

Polynomial DivisionPolynomial Division

Divide (x2 + 5x + 6) by (x + 3)

Common factors only will be used……no long division!

Divide (x2 + 5x + 6) by (x + 3)

Common factors only will be used……no long division!

Let’s look at division

using Algebra Tiles22

Fall 2010

Represent the polynomials using tiles.

x2 + 5x + 6

Fall 2010

Factor the numerator and denominator.

(x + 2)(x + 3)

x2 + 5x + 6

(x + 2)(x + 3)

Fall 2010

Represent the polynomials using tiles.

(x + 3)

x2 + 5x + 6(x + 2)(x + 3)

Reduce fraction

by simplifying

like factors to

equal 1.

Reduce fraction

by simplifying

like factors to

equal 1.

x + 2 is the answer

Fall 2010

Points of Interest for A.2 from the Curriculum Framework

Operations with polynomials can be represented concretely, pictorially, and symbolically.

VDOE Algeblocks Training Videohttp://www.vdoe.whro.org/A_Blocks05/index.html

Operations with polynomials can be represented concretely, pictorially, and symbolically.

VDOE Algeblocks Training Videohttp://www.vdoe.whro.org/A_Blocks05/index.html

Fall 2010

(2x + 5) + (x – 4) = 3x + 1(2x + 5) + (x – 4) = 3x + 1

Algeblocks ExampleAlgeblocks Example

Fall 2010

Modeling by Using Families of Functions

• Recognize the characteristics of different families of functions

• Recognize the common features of each function family

• Recognize how different data patterns can be modeled using each family

• Recognize the characteristics of different families of functions

• Recognize the common features of each function family

• Recognize how different data patterns can be modeled using each family

Fall 2010

Analyzing the Effects of Parameters

• Different, but equivalent algebraic expressions can be used to define the same function

• Writing functions in different forms helps identify features of the function

• Graphical transformations can be observed by changes in parameters

• Different, but equivalent algebraic expressions can be used to define the same function

• Writing functions in different forms helps identify features of the function

• Graphical transformations can be observed by changes in parameters

Fall 2010

Overview of Functions Looking at Patterns

Time vs. Distance Graphs allow students to relate observable patterns in one real world variable (distance) in terms of another real world variable (time).

Fall 2010

Time vs. Distance GraphsTime vs. Distance Graphs

Fall 2010 32

Fall 2010

Slope and Linear FunctionsSlope and Linear Functions

• Students can begin to conceptualize slope and look at multiple representations of the same relationship given real world data, tables and graphs.

Fall 2010

Exploring Slope using Graphs & Tables

+200 +200 +200 +200+200

+15.87 +15.87+15.86 +15.86 +15.86 +15.87 +15.86 +15.87 +16.13

The cost is approximately $15.87 for every 200kWh of electricity.

Students can then determine that the cost is about $ 0.08 per kWh of electricity.

Fall 2010

Exploring FunctionsExploring Functions

As students progress through high school mathematics, the concept of a function and its characteristics become more complex. Exploring families of functions allow students to compare and contrast the attributes of various functions.

Fall 2010

Function FamiliesFunction Families

Linear: Absolute Value:

( )f x x ( )f x x

Fall 2010

Function FamiliesFunction Families Quadratic Square Root Quadratic Square Root

( )f x x2( )f x x

Fall 2010

3( )f x x Cube Root Rational:

( )f xx

Fall 2010

Polynomial: Exponential:

3( )f x x ( ) 2xf x

Fall 2010

Logarithmic: Logarithmic:

2( ) logf x x

Fall 2010 41

Linear FunctionsLinear FunctionsParent Function

f(x) = xOther Forms:f(x) = mx + bf(x) = b + ax

y – y1 = m(x – x1)

Ax + By = C

CharacteristicsAlgebra IDomain & Range: Zero: x-intercept: y-intercept: Algebra IIIncreasing/Decreasing:End Behavior:

Fall 2010 42

Linear FunctionsLinear FunctionsParent Function

f(x) = xOther Forms:f(x) = mx + bf(x) = b + ax

y – y1 = m(x – x1)

Ax + By = C

CharacteristicsAlgebra IDomain & Range: {all real numbers}Zero: x=0x-intercept: (0, 0)y-intercept: (0, 0)Algebra IIIncreasing/Decreasing: f(x) is increasing over the interval {all real numbers}End Behavior: As x approaches + ∞, f(x) approaches + ∞. As x approaches - ∞, f(x) approaches - ∞.

Fall 2010

Absolute Value FunctionsAbsolute Value FunctionsParent Function

f(x) = |x|

Other Forms:

f(x) = a|x - h| + k

CharacteristicsAlgebra IIDomain: Range: Zeros: x-intercept: y-intercept: Increasing/Decreasing:End Behavior:

Table of Values

Fall 2010

Absolute Value FunctionsAbsolute Value FunctionsParent Function

f(x) = |x|

Other Forms:

f(x) = a|x - h| + k

CharacteristicsAlgebra IIDomain: {all real numbers}Range: {f(x)| f(x) > 0}Zeros: x=0x-intercept: (0, 0), y-intercept: (0, 0)Increasing/Decreasing: Dec: {x| -∞ < x < 0} Inc: {x| 0 < x < ∞}End Behavior: As x approaches + ∞, f(x) approaches + ∞.

As x approaches - ∞, f(x) approaches + ∞.

Table of Values

Fall 2010

Function TransformationsFunction Transformations

f(x) = |x|

g(x) = |x| + 2

h(x) = |x| - 3Vertical

Transformations

f(x) = |x|

g(x) = |x| + 2

h(x) = |x| - 3Vertical

Transformations

Fall 2010

Function TransformationsFunction Transformations

f(x) = |x|

g(x) = |x - 2|

h(x) = |x + 3| Horizontal

Transformations

f(x) = |x|

g(x) = |x - 2|

h(x) = |x + 3| Horizontal

Transformations

Fall 2010

Quadratic FunctionsQuadratic FunctionsParent Function

Other Forms:

CharacteristicsAlgebra IDomain:Range:Zeros: x-intercept: y-intercept:Algebra IIIncreasing/Decreasing:End Behavior:

2( )f x x

2( )f x ax bx c 2( ) ( )f x a x h k

Fall 2010

Quadratic FunctionsQuadratic FunctionsParent Function

Other Forms:

CharacteristicsAlgebra IDomain: {all real numbers}Range: {f(x)| f(x) > 0}Zeros: x=0x-intercept: (0, 0), y-intercept: (0, 0)Algebra IIIncreasing/Decreasing: Dec: {x| -∞ < x < 0} Inc: {x| 0 < x < ∞}End Behavior: As x approaches - ∞, f(x) approaches + ∞. As x approaches + ∞, f(x) approaches + ∞.

2( )f x x

2( )f x ax bx c 2( ) ( )f x x h k

Fall 2010

Exploring Quadratic Relationships through data tables and graphs

Fall 2010

TAKE a BREAKTAKE a BREAK

Fall 2010

Square Root FunctionsSquare Root FunctionsParent Function

Other Forms:

CharacteristicsAlgebra IIDomain: Range: Zeros: x-intercept: y-intercept: Increasing/Decreasing: End Behavior:

( )f x x

( )f x a x h k

Fall 2010

Square Root FunctionsSquare Root FunctionsParent Function

Other Forms:

CharacteristicsAlgebra IIDomain: {x| x > 0 }Range: {f(x)| f(x) > 0}Zeros: x=0x-intercept: (0, 0) y-intercept: (0, 0)Increasing/Decreasing: Increasing on {x| 0 < x < ∞}End Behavior:

As x approaches + ∞, f(x) approaches + ∞.

( )f x x

( )f x a x h k

Fall 2010

Square Root Function Real World Application Square Root Function Real World Application

The speed of a tsunami is a function of ocean depth:

SPEED =

g = acceleration due to gravity (9.81 m/s2) d = depth of the ocean in meters

Understanding the speed of tsunamis is useful in issuing warnings to coastal regions. Knowing the speed can help predict when the tsunami will arrive at a particular location.

The speed of a tsunami is a function of ocean depth:

SPEED =

g = acceleration due to gravity (9.81 m/s2) d = depth of the ocean in meters

Understanding the speed of tsunamis is useful in issuing warnings to coastal regions. Knowing the speed can help predict when the tsunami will arrive at a particular location.

Fall 2010

Cube Root FunctionsCube Root FunctionsParent Function

Other Forms:

CharacteristicsAlgebra IIDomain:Range: Zeros: x-intercept: y-intercept: Increasing Interval:End Behavior:

3( )f x x

3( )f x a x h k

Fall 2010

Cube Root FunctionsCube Root FunctionsParent Function

Other Forms:

CharacteristicsAlgebra IIDomain: {all real numbers }Range: {all real numbers }Zeros: x=0x-intercept: (0, 0)y-intercept: (0, 0)Increasing Interval: {all real numbers}End Behavior: As x approaches - ∞, f(x) approaches - ∞; As x approaches + ∞, f(x) approaches + ∞.

3( )f x x

3( )f x a x h k

Fall 2010

Cube Root Function Real World Application

Kepler’s Law of Planetary Motion:The distance, d, of a planet from the Sun in millions of miles is equal to the cube root of 6 times the number of Earth days it takes for the planet to orbit the sun, squared. For example, the length of a year on Mars is 687 Earth-days. Thus,

d = 141.478 million miles from the Sun

Kepler’s Law of Planetary Motion:The distance, d, of a planet from the Sun in millions of miles is equal to the cube root of 6 times the number of Earth days it takes for the planet to orbit the sun, squared. For example, the length of a year on Mars is 687 Earth-days. Thus,

d = 141.478 million miles from the Sun

3 2t6d

3 2)687(6d

Fall 2010 57

Rational FunctionsRational FunctionsParent Function

Other Forms:

where a(x) and b(x) are polynomial functions

CharacteristicsAlgebra IIDomain:Range:Zeros: x-intercept & y-intercept: Increasing/Decreasing: End Behavior:Asymptotes:

1( )f x

( )( )

a xf x

Fall 2010

Rational FunctionsRational FunctionsParent Function

Other Forms:

where a(x) and b(x) are polynomial functions

CharacteristicsAlgebra IIDomain: {x| x<0} U {x| x>0}Range: {f(x)| f(x) < 0} U {f(x)| f(x) > 0}Zeros: nonex-intercept & y-intercept: noneDecreasing: {x| -∞ < x < 0} U {x| 0 < x < ∞}End Behavior: As x approaches - ∞, f(x) approaches 0; as x approaches + ∞, f(x) approaches 0.Asymptotes: x = 0, y = 0

1( )f x

( )( )

a xf x

Fall 2010 59

Rational Expressions Real World Application

A James River tugboat goes 10 mph in still water. It travels 24 mi upstream and 24 mi back in a total time of 5 hr. What is the speed of the current?

Distance Speed Time

Upstream 24 10 – c t1

Downstream 24 10 + c t2

Distance Speed Time

Upstream 24 10 – c 24/(10 – c )

Downstream 24 10 + c 24/(10 + c )

10 - c

Upstream

10 + c+

Downstream

Fall 2010 60

Rational Expressions Real World ApplicationRational Expressions

Real World Application

10 - c

10 + c+(10 – c) (10 + c) (10 – c) (10 + c)

24(10 + c) + 24 (10 – c) = 5 (100 – c2)

480 = 500 - 5c2 5c2 - 20 = 0

c = 2 or -2 5(c + 2)(c – 2) = 0

The speed of the current is 2 mph.

Fall 2010 61

Applying Solving Equations and Graphing Related Functions

Algebraic5c2 - 20 = 0

c = -2 or 2 zeros

x-intercepts

Related Functionf(c) = 5c2 - 20

Fall 2010

Solving Equations & FunctionsSolving Equations & FunctionsA.4 The student will solve

multistep linear and quadratic equations in two variables…..

FrameworkIdentify the root(s) or zero(s) of a …..

function over the real number system as the solution(s) to the ….. equation that is formed by setting the given …… expression equal to zero.

A.4 The student will solve multistep linear and quadratic equations in two variables…..

FrameworkIdentify the root(s) or zero(s) of a …..

function over the real number system as the solution(s) to the ….. equation that is formed by setting the given …… expression equal to zero.

Fall 2010

Exponential FunctionsExponential FunctionsParent Function

Other Forms:

Characteristics (f(x) = 2x)Algebra IIDomain: Range: Zeros: x-intercepts: y-intercepts: Asymptote: End Behavior:

( ) xf x b

( ) xf x ab c

Fall 2010

Exponential FunctionsExponential FunctionsParent Function

Other Forms:

Characteristics (f(x) = 2x)Algebra IIDomain: {all real numbers} Range: {f(x)| f(x) > 0}Zeros: none x-intercepts: none y-intercepts: (0, 1)Asymptote: y = 0End Behavior: As x approaches ∞, f(x) approaches + ∞. As x approaches - ∞, f(x) approaches 0.

( ) xf x b

( ) xf x ab c

Fall 2010

Exponential Function Real World Application Exponential Function

Real World Application Homemade chocolate chip cookies can lose their freshness over time. When the cookies are fresh, the taste quality is 1. The taste quality decreases according to the function:

f(x) = 0.8x, where x represents the number of days since the cookies were baked and f(x) measures the taste quality.

When will the cookiestaste half as good aswhen they were fresh?

Homemade chocolate chip cookies can lose their freshness over time. When the cookies are fresh, the taste quality is 1. The taste quality decreases according to the function:

f(x) = 0.8x, where x represents the number of days since the cookies were baked and f(x) measures the taste quality.

When will the cookiestaste half as good aswhen they were fresh?

0.5 = 0.8x

log 0.5 = x log 0.8x = log 0.5 ÷ log 0.8

x = 3 days

Fall 2010

Logarithmic FunctionsLogarithmic FunctionsParent Function

f(x) = logb x, b > 0, b 1

Characteristics (f(x) = log x)Algebra IIDomain: Range:Zeros: x-intercepts: y-intercepts: Asymptotes: End Behavior:

Fall 2010

Logarithmic FunctionsLogarithmic FunctionsParent Function

f(x) = logb x, b > 0, b 1

Characteristics (f(x) = log x)Algebra IIDomain: {x| x > 0} Range: {all real numbers} Zeros: x=1 x-intercepts: (1, 0) y-intercepts: noneAsymptotes: x = 0End Behavior: As x approaches ∞, y approaches + ∞.

Fall 2010

Logarithmic Function Real World Application Logarithmic Function

Real World Application The wind speed, s (in miles per hour), near the

center of a tornado can be modeled by s = 93 log d + 65

Where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles

through three states. Estimate the wind speed near the tornado’s center.

s = 93 log d + 65s = 93 log 220 + 65s = 93(2.342) + 65

s = 282.806 miles/hour

The wind speed, s (in miles per hour), near the center of a tornado can be modeled by

s = 93 log d + 65 Where d is the distance (in miles) that the tornado

travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed

near the tornado’s center.

s = 93 log d + 65s = 93 log 220 + 65s = 93(2.342) + 65

s = 282.806 miles/hour

Fall 2010

Inverse Functions:Exponentials and Logarithms

Given y

what is its inverse

Fall 2010

Functions and InversesFunctions and InversesEvery function has an inverse

relation, but not every inverse relation is a function.

When is a function invertible?

A function is invertible if its inverse relation is also a function.

Every function has an inverse relation, but not every inverse relation is a function.

When is a function invertible?

A function is invertible if its inverse relation is also a function.

Function

Not a Function

Fall 2010

Quadratic Functions Require Restricted Domains in order to be

Invertible

Quadratic Functions Require Restricted Domains in order to be

InvertibleFunction:

Inverse Function:

Function:

Inverse Function:

x f(x)

x f -1(x)

1( )f x x

2( ) , 0f x x x

Fall 2010

Inverse FunctionsInverse Functions

Fall 2010

Polynomial FunctionsPolynomial Functions

End behavior ~ direction of the ends

of the graph

End behavior ~ direction of the ends

of the graph

Even DegreeSame directionsOdd DegreeOpposite directions

Teachers should facilitate students’ generalizations

Fall 2010

Real World ApplicationPolynomial Function

Suppose an object moves in a straight line so that its distance s(t) after t seconds, is represented by s(t)= t3 + t2 + 6t feet from its starting point. Determine the distance traveled in the first 4 seconds.

Fall 2010

s(t) = t3 + t2 + 6t

Odd DegreeEnd Behavior

Fall 2010

Time is our constraint, so we are only concerned with the positive domain

s(t) = t3 + t2 + 6t

s(4) = (4)3 + (4)2 + 6(4)

s(4) = 64+ 16 + 24

s(4) = 104

Determine the distance traveled after 4 seconds.

The object traveled 104 feet in 4 seconds

Fall 2010

Analyzing FunctionsAnalyzing Functions

Domain:

Range:

Zeros:

x-intercept:

Decreasing:

End Behavior:

Asymptotes:

Fall 2010

Analyzing FunctionsAnalyzing Functions

Domain: {x| x < 2} U {x| x > 2 }

Range: {f(x)| f(x) < 1} U {f(x)| f(x) > 1}

Zeros: x = -3

x-intercept: (-3, 0)

Decreasing:

{x| x < 2} U {x| x > 2 }

End Behavior: As x approaches - ∞, f(x) approaches 1. As x approaches + ∞, f(x) approaches 1.

Asymptotes: x = 2, y = 1

Fall 2010

AsymptotesAsymptotes

f(x) = 3(x – 2)f(x) = 3(x – 2)

Fall 2010

AsymptotesAsymptotes

3xy = 123xy = 12

Fall 2010

What do you know about this rational function?

2 6( )

x xf x

( 2)( 3)( )

3( ) 2, 3

x xf x

xf x x x

Fall 2010

Discontinuity (Holes)Discontinuity (Holes)

2 6( )

3( ) 2, 3

x xf x

xf x x x

Fall 2010

Function Development 9-12Function Development 9-12

Algebra IRelation or function?

Domain/range

x- and y-intercepts

Function values for elements of the domain

Connections among representations

AFDAContinuity

Domain/range

x- and y-intercepts

Local/absolute max/min

Intervals of inc/dec

End behaviors

Asymptotes

Algebra 2Domain/range (includes discontinuous domains/ranges)

x- and y-intercepts

Local/absolute max/min

Intervals of incr/decr

End behaviors

Asymptotes

Inverse functions

Composition of functions

Fall 2010

Draw a function that has the following characteristics

Domain: {all real numbers}

Range: {f(x)| f(x)>0}

Increasing: {x| -2<x<2 U x>5}

Decreasing: {x| 2<x<5}

Relative maximum(turning point): (2, 4)

Relative minimum(turning point): (-2, 1)

End Behavior: As x approaches ∞, f(x) approaches ∞. As x approaches - ∞, f(x) approaches ∞.

Asymptotes: y=0

Domain: {all real numbers}

Range: {f(x)| f(x)>0}

Increasing: {x| -2<x<2 U x>5}

Decreasing: {x| 2<x<5}

Relative maximum(turning point): (2, 4)

Relative minimum(turning point): (-2, 1)

End Behavior: As x approaches ∞, f(x) approaches ∞. As x approaches - ∞, f(x) approaches ∞.

Asymptotes: y=0

Is it possible?

Why/Why Not?

Fall 2010

Revisit Placemat ConsensusFunctions

Common ideas are written

Individual ideas are written here

Fall 2010 VDOE Mathematics Institute Grade Band 9-12 Functions K-12 Mathematics Institutes Fall 2010...

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