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 9-12 Functions K-12 Mathematics Institutes Fall 2010
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VDOE Mathematics Institute Grade Band 9-12 Functions K-12 Mathematics Institutes Fall 2010. Common ideas are written here. Placemat Consensus Functions. Individual ideas are written here. Individual ideas are written here. Individual ideas are written here. Individual ideas - PowerPoint PPT Presentation
Transcript of VDOE Mathematics Institute Grade Band 9-12 Functions K-12 Mathematics Institutes Fall 2010
Fall 2010
VDOE Mathematics Institute
Grade Band 9-12Functions
K-12 Mathematics InstitutesFall 2010
Fall 2010
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
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Fall 2010
Overview of Vertical ProgressionMiddle School (Function Analysis) 7.12 … represent relationships with
tables, graphs, rules and words8.14 … make connections between
any two representations (tables, graphs, words, rules)
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Overview 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.
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Overview 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.
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Overview 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.
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Overview 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.
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Vocabulary
The new 2009 SOL mathematics standards focus on the use of
appropriate and accurate mathematics vocabulary.
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“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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Vocabulary Across Grade Levels
EvaluateSolve
SimplifyApply
AnalyzeConstruct
Compare/contrastCalculate
GraphTransform
FactorIdentify
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Wordle – Algebra, Functions and Data Analysis 2009 VA SOLs
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Wordle – Algebra II 2009 VA SOLs
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Wordle – Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry
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Reasoning 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
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Using Multiple Representations of Functions
• Tables• Graphs or diagrams• Symbolic representations• Verbal descriptions
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Algebra Tiles ~ AddingAdd the polynomials. (x – 2) + (x + 1)
= 2x - 1
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Algebra Tiles ~ Multiplying
x + 2 x + 3
(x + 2)(x + 3)
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Multiply the polynomials using tiles. Create an array of the polynomials
(x + 2)(x + 3)
x2 + 5x + 6
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Algebra Tiles ~ FactoringWork backwards from the array.
(x – 1)(x – 2)
x2 - 3x + 2
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Polynomial DivisionA.2 The student will perform operations on
polynomials, includinga)applying the laws of exponents to
perform operations on expressions;b)adding, subtracting, multiplying, and
dividing polynomials; andc)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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Polynomial 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
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Represent the polynomials using tiles.
x + 3x2 + 5x + 6
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Factor the numerator and denominator.
(x + 2)(x + 3)
x2 + 5x + 6
(x + 2)(x + 3)
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Represent the polynomials using tiles.
(x + 3)x2 + 5x + 6(x + 2)(x + 3)
Reduce fraction by simplifying like factors toequal 1.
x + 2 is the answer
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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
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(2x + 5) + (x – 4) = 3x + 1
Algeblocks Example
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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
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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
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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).
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Time vs. Distance Graphs
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Slope 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.
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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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Exploring 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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Function Families Linear: Absolute Value: ( )f x x ( )f x x
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Function Families Quadratic Square Root
( )f x x2( )f x x
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Function Families
3( )f x x Cube Root Rational:
1( )f xx
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Function Families Polynomial: Exponential: 3( )f x x ( ) 2xf x
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Function Families Logarithmic:
2( ) logf x x
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Linear 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
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Linear 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
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Absolute 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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Absolute 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
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Function Transformations
f(x) = |x|
g(x) = |x| + 2
h(x) = |x| - 3Vertical
Transformations
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Function Transformations
f(x) = |x|
g(x) = |x - 2|
h(x) = |x + 3| Horizontal
Transformations
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Quadratic 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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Quadratic 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
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Exploring Quadratic Relationships through data tables and graphs
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TAKE a BREAK
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Square 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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Square 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
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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.
gd
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Cube 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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Cube 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
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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
3 2t6d
3 2)687(6d
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Rational 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 xx
( )( )( )
a xf xb x
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Rational 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 xx
( )( )( )
a xf xb x
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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
Upstream24
10 + c+
Downstream
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Rational 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.
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Applying Solving Equations and Graphing Related Functions
Algebraic5c2 - 20 = 0
c = -2 or 2 zeros
x-intercepts
Related Functionf(c) = 5c2 - 20
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Solving 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.
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Exponential FunctionsParent Function
Other Forms:
Characteristics (f(x) = 2x)Algebra IIDomain: Range: Zeros: x-intercepts: y-intercepts: Asymptote: End Behavior:
Table
181412
2321
0 11 22 4
xx y
( ) xf x b
( ) xf x ab c
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Exponential 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
181412
2321
0 11 22 4
xx y
( ) xf x b
( ) xf x ab c
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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?
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0.5 = 0.8x
log 0.5 = x log 0.8x = log 0.5 ÷ log 0.8
x = 3 days
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Logarithmic 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
2181412
log321
1 02 14 2
x y x
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Logarithmic 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
181412
log321
1 02 14 2
x y x
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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
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Inverse Functions:Exponentials and Logarithms
2
: 2 ;?
2log
x
y
Given ywhat is its inverse
xx y
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Functions 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.
Function
Not a Function
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Quadratic Functions Require Restricted Domains in order to be
InvertibleFunction:
Inverse Function:
x f(x)0 01 12 43 9
x f -1(x)0 01 14 29 3
1( )f x x
2( ) , 0f x x x
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Inverse Functions
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Polynomial FunctionsEnd behavior ~ direction of the ends of the graph
Even DegreeSame directionsOdd DegreeOpposite directions
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Teachers should facilitate students’ generalizations
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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.
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s(t) = t3 + t2 + 6t
Odd DegreeEnd Behavior
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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
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Analyzing Functions3( )2
xf xx
Domain: Range: Zeros: x-intercept:Decreasing: End Behavior:Asymptotes:
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Analyzing Functions3( )2
xf xx
Domain: {x| x < 2} U {x| x > 2 }Range: {f(x)| f(x) < 1} U {f(x)| f(x) > 1}Zeros: x = -3x-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
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Asymptotes
f(x) = 3(x – 2)
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Asymptotes
3xy = 12
xy
xy
4312
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What do you know about this rational function?
2 6( )3
x xf xx
( 2)( 3)( )3
( ) 2, 3
x xf xx
f x x x
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Discontinuity (Holes)
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2 6( )3
( ) 2, 3
x xf xx
f x x x
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Function Development 9-12
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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 representationsLocal/absolute max/minIntervals of inc/decEnd behaviorsAsymptotes
Algebra 2Domain/range (includes discontinuous domains/ranges)
Zeros
x- and y-intercepts
Function values for elements of the domain
Connections among representationsLocal/absolute max/minIntervals of incr/decrEnd behaviorsAsymptotesInverse functionsComposition of functions
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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
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Is it possible?
Why/Why Not?
Fall 2010
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
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