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
Fall 2010
Placemat ConsensusFunctions
Placemat ConsensusFunctions
Common ideas are written
here
Individual ideas are written here
Individual ideas are written here
Individual ideas are written here
Individual ideas are written here
2
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)
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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.
4
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.
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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.
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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.
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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.
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Fall 2010
“Function” Vocabulary Across Grade Levels
“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)
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Fall 2010
Vocabulary Across Grade LevelsVocabulary Across Grade Levels
EvaluateSolve
SimplifyApply
AnalyzeConstruct
Compare/contrastCalculate
GraphTransform
FactorIdentify
10
Fall 2010
Wordle – Algebra I 2009 VA SOLswww.wordle.net
Wordle – Algebra I 2009 VA SOLswww.wordle.net
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Fall 2010
Wordle – Algebra, Functions and Data Analysis 2009 VA SOLs
Wordle – Algebra, Functions and Data Analysis 2009 VA SOLs
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Fall 2010
Wordle – Algebra II 2009 VA SOLs Wordle – Algebra II 2009 VA SOLs
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Fall 2010
Wordle – Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry
Wordle – Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry
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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
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Fall 2010
Using Multiple Representations of Functions
Using Multiple Representations of Functions
• Tables• Graphs or diagrams• Symbolic representations• Verbal descriptions
• Tables• Graphs or diagrams• Symbolic representations• Verbal descriptions
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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.
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.
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.
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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.
Represent the polynomials using tiles.
x + 3
x2 + 5x + 6
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Fall 2010
Factor the numerator and denominator.
Factor the numerator and denominator.
(x + 2)(x + 3)
x2 + 5x + 6
(x + 2)(x + 3)
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Fall 2010
Represent the polynomials using tiles.
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
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Fall 2010
Points of Interest for A.2 from the Curriculum Framework
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
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Fall 2010
(2x + 5) + (x – 4) = 3x + 1(2x + 5) + (x – 4) = 3x + 1
Algeblocks ExampleAlgeblocks Example
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Fall 2010
Modeling by Using Families of Functions
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
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Fall 2010
Analyzing the Effects of Parameters
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
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Fall 2010
Overview of Functions Looking at Patterns
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).
Time vs. Distance Graphs allow students to relate observable patterns in one real world variable (distance) in terms of another real world variable (time).
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Fall 2010
Time vs. Distance GraphsTime vs. Distance Graphs
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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.
• Students can begin to conceptualize slope and look at multiple representations of the same relationship given real world data, tables and graphs.
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Fall 2010
Exploring Slope using Graphs & Tables
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.
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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.
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.
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Fall 2010
Function FamiliesFunction Families
Linear: Absolute Value:
Linear: Absolute Value:
( )f x x ( )f x x
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Fall 2010
Function FamiliesFunction Families Quadratic Square Root Quadratic Square Root
( )f x x2( )f x x
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Fall 2010
Function FamiliesFunction Families
3( )f x x Cube Root Rational:
1
( )f xx
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Fall 2010
Function FamiliesFunction Families
Polynomial: Exponential:
Polynomial: Exponential:
3( )f x x ( ) 2xf x
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Fall 2010
Function FamiliesFunction Families
Logarithmic: Logarithmic:
2( ) logf x x
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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:
Table
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 - ∞.
Table
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
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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
44
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
47
Quadratic FunctionsQuadratic FunctionsParent Function
Other Forms:
CharacteristicsAlgebra IDomain:Range:Zeros: x-intercept: y-intercept:Algebra IIIncreasing/Decreasing:End Behavior:
Table
2( )f x x
2( )f x ax bx c 2( ) ( )f x a x h k
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Fall 2010
48
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 + ∞.
Table
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
Exploring Quadratic Relationships through data tables and graphs
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Fall 2010
TAKE a BREAKTAKE a BREAK
Fall 2010
51
Square Root FunctionsSquare Root FunctionsParent Function
Other Forms:
CharacteristicsAlgebra IIDomain: Range: Zeros: x-intercept: y-intercept: Increasing/Decreasing: End Behavior:
Table
( )f x x
( )f x a x h k
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Fall 2010
52
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 + ∞.
Table
( )f x x
( )f x a x h k
52
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.
gd
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Fall 2010
54
Cube Root FunctionsCube Root FunctionsParent Function
Other Forms:
CharacteristicsAlgebra IIDomain:Range: Zeros: x-intercept: y-intercept: Increasing Interval:End Behavior:
Table
3( )f x x
3( )f x a x h k
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Fall 2010
55
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 + ∞.
Table
3( )f x x
3( )f x a x h k
Fall 2010
Cube Root Function Real World Application
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
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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:
Table
1( )f x
x
( )( )
( )
a xf x
b x
Fall 2010
58
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
Table
1( )f x
x
( )( )
( )
a xf x
b x
Fall 2010 59
Rational Expressions Real World Application
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 )
= 524
10 - c
Upstream
24
10 + c+
Downstream
Fall 2010 60
Rational Expressions Real World ApplicationRational Expressions
Real World Application
= 524
10 - c
24
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
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.
62
Fall 2010
Exponential FunctionsExponential FunctionsParent Function
Other Forms:
Characteristics (f(x) = 2x)Algebra IIDomain: Range: Zeros: x-intercepts: y-intercepts: Asymptote: End Behavior:
Table
18
14
12
2
3
2
1
0 1
1 2
2 4
xx y
( ) 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.
Table
18
14
12
2
3
2
1
0 1
1 2
2 4
xx y
( ) 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?
65
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:
Table
2
18
14
12
log
3
2
1
1 0
2 1
4 2
x y x
66
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 + ∞.
Table
2
18
14
12
log
3
2
1
1 0
2 1
4 2
x y x
67
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
68
Fall 2010
Inverse Functions:Exponentials and Logarithms
Inverse Functions:Exponentials and Logarithms
2
: 2 ;
?
2
log
x
y
Given y
what is its inverse
x
x y
69
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
70
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)
0 0
1 1
2 4
3 9
x f -1(x)
0 0
1 1
4 2
9 3
1( )f x x
2( ) , 0f x x x
71
Fall 2010
Inverse FunctionsInverse Functions
72
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
73
Teachers should facilitate students’ generalizations
Fall 2010
Real World ApplicationPolynomial Function
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.
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.
74
Fall 2010
s(t) = t3 + t2 + 6t
Odd DegreeEnd Behavior
75
Fall 2010
Time is our constraint, so we are only concerned with the positive domain
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
76
Fall 2010
Analyzing FunctionsAnalyzing Functions
3( )
2
xf x
x
Domain:
Range:
Zeros:
x-intercept:
Decreasing:
End Behavior:
Asymptotes:
77
Fall 2010
Analyzing FunctionsAnalyzing Functions
3( )
2
xf x
x
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
78
Fall 2010
AsymptotesAsymptotes
f(x) = 3(x – 2)f(x) = 3(x – 2)
79
Fall 2010
AsymptotesAsymptotes
3xy = 123xy = 12
xy
xy
43
12
80
Fall 2010
What do you know about this rational function?
What do you know about this rational function?
2 6( )
3
x xf x
x
( 2)( 3)( )
3( ) 2, 3
x xf x
xf x x x
81
Fall 2010
Discontinuity (Holes)Discontinuity (Holes)
82
2 6( )
3( ) 2, 3
x xf x
xf x x x
3
Fall 2010
Function Development 9-12Function Development 9-12
83
Algebra IRelation or function?
Domain/range
Zeros
x- and y-intercepts
Function values for elements of the domain
Connections among representations
AFDAContinuity
Domain/range
Zeros
x- and y-intercepts
Function values for elements of the domain
Connections among representations
Local/absolute max/min
Intervals of inc/dec
End behaviors
Asymptotes
Algebra 2Domain/range (includes discontinuous domains/ranges)
Zeros
x- and y-intercepts
Function values for elements of the domain
Connections among representations
Local/absolute max/min
Intervals of incr/decr
End behaviors
Asymptotes
Inverse functions
Composition of functions
83
Fall 2010
Draw a function that has the following characteristics
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
84
Is it possible?
Why/Why Not?
Fall 2010
Revisit Placemat ConsensusFunctions
Revisit Placemat ConsensusFunctions
Common ideas are written
here
Individual ideas are written here
Individual ideas are written here
Individual ideas are written here
Individual ideas are written here
85