Grade Band 912 Functions K12 Mathematics Institutes Fall 2010 Placemat Consensus Functions Common ideas are written here Individual ideas are written here Individual ideas are written here ID: 674560
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VDOE Mathematics InstituteGrade Band 9-12FunctionsK-12 Mathematics InstitutesFall 2010Slide2
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
2Slide3
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)3Slide4
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, includinga) 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.
4Slide5
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 includea) 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.
5Slide6
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.
6Slide7
Overview of Vertical ProgressionAlgebra 2 (Function Analysis)AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts includea) 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; and
h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions.7Slide8
VocabularyThe new 2009 SOL mathematics standards focus on the use of appropriate and accurate mathematics vocabulary.8Slide9
“Function” Vocabulary Across Grade LevelsRelation
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)
9Slide10
Vocabulary Across Grade LevelsEvaluate
Solve
Simplify
Apply
Analyze
Construct
Compare/contrast
Calculate
Graph
Transform
Factor
Identify
10Slide11
Wordle – Algebra I 2009 VA SOLswww.wordle.net 11Slide12
Wordle – Algebra, Functions and Data Analysis 2009 VA SOLs
12Slide13
Wordle – Algebra II 2009 VA SOLs
13Slide14
Wordle – Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry
14Slide15
Reasoning with FunctionsKey elements of reasoning and sensemaking with functions include:Using multiple representations of functionsModeling by using families of functionsAnalyzing the effects of different parameters
Adapted from
Focus in High School Mathematics:
Reasoning and Sense Making,
NCTM, 2009
15Slide16
Using Multiple Representations of FunctionsTablesGraphs or diagramsSymbolic representationsVerbal descriptions16Slide17
17Algebra Tiles ~ Adding
Add the polynomials.
(x – 2) + (x + 1)
= 2x - 1Slide18
18Algebra Tiles ~ Multiplying
x + 2 x + 3
(x + 2)(x + 3)Slide19
19Multiply the polynomials using tiles.
Create an array of the polynomials
(x + 2)
(x + 3)
x
2
+ 5x + 6Slide20
20Algebra Tiles ~ Factoring
Work backwards from the array.
(x – 1)
(x – 2)
x
2
- 3x + 2Slide21
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.
21Slide22
Polynomial DivisionDivide (x2 + 5x + 6) by (x + 3)Common factors only will be used……no long division!Let’s look at division using Algebra Tiles
22Slide23
Represent the polynomials using tiles.
x + 3
x
2
+ 5x + 6
23Slide24
Factor the numerator and denominator.
(x + 2)
(x + 3)
x
2
+ 5x + 6
(x + 2)(x + 3)
24Slide25
Represent the polynomials using tiles.
(x + 3)
x
2
+ 5x + 6
(x + 2)(x + 3)
Reduce fraction
by simplifying
like factors to
equal 1.
x + 2 is the answer
25Slide26
Points of Interest for A.2 from the Curriculum FrameworkOperations with polynomials can be represented concretely, pictorially, and symbolically.VDOE Algeblocks Training Videohttp://www.vdoe.whro.org/A_Blocks05/index.html
26Slide27
(2x + 5) + (x – 4) = 3x + 1
Algeblocks Example
27Slide28
Modeling by Using Families of FunctionsRecognize the characteristics of different families of functionsRecognize the common features of each function familyRecognize how different data patterns can be modeled using each family28Slide29
Analyzing the Effects of ParametersDifferent, but equivalent algebraic expressions can be used to define the same functionWriting functions in different forms helps identify features of the functionGraphical transformations can be observed by changes in parameters29Slide30
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).30Slide31
Time vs. Distance Graphs31Slide32
32Slide33
Slope and Linear FunctionsStudents can begin to conceptualize slope and look at multiple representations of the same relationship given real world data, tables and graphs.33Slide34
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.
34Slide35
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. 35Slide36
Function Families Linear: Absolute Value:
36Slide37
Function Families Quadratic Square Root
37Slide38
Function Families
Cube Root Rational:
38Slide39
Function Families Polynomial: Exponential:
39Slide40
Function Families Logarithmic:
40Slide41
41Linear Functions
Parent Function
f(x) = x
Other Forms:
f(x) =
mx
+ b
f(x) = b + ax
y – y
1
= m(x – x
1)
Ax + By = C
Characteristics
Algebra I
Domain & Range
:
Zero:
x-intercept:
y-intercept:
Algebra II
Increasing/Decreasing:
End Behavior:
TableSlide42
42Linear Functions
Parent Function
f(x) = x
Other Forms:
f(x) =
mx
+ b
f(x) = b + ax
y – y
1
= m(x – x
1)
Ax + By = C
Characteristics
Algebra I
Domain & Range
: {all real numbers}
Zero:
x=0
x-intercept:
(0, 0)
y-intercept:
(0, 0)
Algebra II
Increasing/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 -
∞.
TableSlide43
Absolute Value FunctionsParent Function
f(x)
= |
x
|
Other Forms:
f(x) = a
|
x - h
| + k
Characteristics
Algebra II
Domain
:
Range:
Zeros:
x-intercept:
y-intercept:
Increasing/Decreasing
:
End Behavior
:
Table of Values
43Slide44
Absolute Value FunctionsParent Function
f(x)
= |
x
|
Other Forms:
f(x)
=
a
|x - h
| + k
Characteristics
Algebra II
Domain
: {all real numbers}
Range:
{f(x)| f(x)
>
0}
Zeros:
x=0
x-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
44Slide45
Function Transformationsf(x) = |x|g(x) = |x| + 2h(x) = |x| - 3Vertical
TransformationsSlide46
Function Transformationsf(x) = |x|g(x) = |x - 2|h(x) = |x + 3| Horizontal TransformationsSlide47
47Quadratic Functions
Parent Function
Other Forms:
Characteristics
Algebra I
Domain
:
Range:
Zeros:
x-intercept:
y-intercept:
Algebra II
Increasing/Decreasing:
End Behavior:
Table
47Slide48
48Quadratic Functions
Parent Function
Other Forms:
Characteristics
Algebra I
Domain
: {all real numbers}
Range:
{f(x)| f(x)
>
0}
Zeros:
x=0
x-intercept:
(0, 0),
y-intercept:
(0, 0)
Algebra II
Increasing/Decreasing:
Dec: {x| -
∞ <
x < 0} Inc: {x| 0 < x <
∞
}
End Behavior:
As x approaches -
∞
, f(x) approaches +
∞.
As x approaches +
∞
, f(x) approaches +
∞.
TableSlide49
Exploring Quadratic Relationships through data tables and graphs49Slide50
TAKE a BREAKSlide51
51Square Root Functions
Parent Function
Other Forms:
Characteristics
Algebra II
Domain
:
Range:
Zeros:
x-intercept: y-intercept:
Increasing/Decreasing
:
End Behavior
:
Table
51Slide52
52Square Root Functions
Parent Function
Other Forms:
Characteristics
Algebra II
Domain
: {x| x
>
0 }
Range:
{f(x)| f(x)
>
0}
Zeros:
x=0
x-intercept:
(0, 0)
y-intercept:
(0, 0)
Increasing/Decreasing
:
Increasing on {x
|
0 < x <
∞
}
End Behavior
:
As x approaches +
∞
, f(x) approaches +
∞
.
Table
52Slide53
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/s
2
)
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.
53Slide54
54Cube Root Functions
Parent Function
Other Forms:
Characteristics
Algebra II
Domain
:
Range:
Zeros:
x-intercept:
y-intercept:
Increasing Interval
:
End Behavior
:
Table
54Slide55
55Cube Root Functions
Parent Function
Other Forms:
Characteristics
Algebra II
Domain
: {all real numbers }
Range:
{all real numbers }
Zeros:
x=0
x-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 +
∞
.
TableSlide56
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
56Slide57
57Rational Functions
Parent Function
Other Forms:
where a(x) and b(x) are
polynomial functions
Characteristics
Algebra II
Domain:
Range:
Zeros:
x-intercept & y-intercept:
Increasing/Decreasing:
End Behavior:
Asymptotes:
TableSlide58
58Rational Functions
Parent Function
Other Forms:
where a(x) and b(x) are
polynomial functions
Characteristics
Algebra II
Domain:
{x| x<0} U {x| x>0}
Range:
{f(x)| f(x) < 0} U {f(x)| f(x) > 0}
Zeros:
none
x-intercept & y-intercept:
none
Decreasing:
{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
TableSlide59
59Rational 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
t1Downstream
2410 + ct2
Distance
SpeedTimeUpstream24
10 – c 24/(10 – c )Downstream24
10 + c24/(10 + c )
= 5
24
10 - c
Upstream
24
10 + c
+
DownstreamSlide60
60Rational Expressions Real World Application
= 5
24
10 - c
24
10 + c
+
(10 – c) (10 + c)
(10 – c) (10 + c)
24(10 + c) + 24 (10 – c) = 5 (100 – c
2
)
480 = 500 - 5c
2
5c
2
- 20 = 0
c
= 2 or -2
5(c + 2)(c – 2) = 0
The speed of the current is 2 mph.Slide61
61Applying Solving Equations and Graphing Related Functions
Algebraic
5c
2
- 20 = 0
c
= -2 or 2
zerosx-intercepts
Related Functionf(c) = 5c2 - 20Slide62
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.62Slide63
Exponential FunctionsParent Function
Other Forms:
Characteristics (
f(x)
= 2
x
)
Algebra II
Domain:
Range:
Zeros:
x-intercepts:
y-intercepts:
Asymptote:
End Behavior:
Table Slide64
Exponential FunctionsParent Function
Other Forms:
Characteristics (
f(x)
= 2
x
)
Algebra II
Domain:
{all real numbers}
Range:
{f(x)| f(x) > 0}
Zeros:
none
x-intercepts:
none
y-intercepts:
(0, 1)
Asymptote:
y = 0
End Behavior:
As x approaches
∞
, f(x) approaches +
∞.
As x approaches -
∞
, f(x) approaches 0.
Table Slide65
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 cookies
taste half as good as
when they were fresh?
65
0.5 = 0.8xlog 0.5 = x log 0.8x = log 0.5 ÷ log 0.8x = 3 days Slide66
Logarithmic FunctionsParent Function
f(x) =
log
b
x
, b > 0, b 1
Characteristics (
f(x)
= log x)
Algebra II
Domain:
Range:
Zeros:
x-intercepts:
y-intercepts:
Asymptotes:
End Behavior:
Table
66Slide67
Logarithmic FunctionsParent Function
f(x) =
log
b
x
, b > 0, b 1
Characteristics (
f(x)
= log x)
Algebra II
Domain: {x| x > 0}
Range:
{all real numbers} Zeros:
x=1
x-intercepts:
(1, 0)
y-intercepts:
none
Asymptotes:
x
= 0
End Behavior
: As x approaches
∞
, y approaches +
∞.
Table
67Slide68
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 + 65
s = 93 log 220 + 65
s = 93(2.342) + 65
s = 282.806 miles/hour
68Slide69
Inverse Functions:Exponentials and Logarithms69Slide70
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
70Slide71
Quadratic Functions Require Restricted Domains in order to be InvertibleFunction: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
71Slide72
Inverse Functions72Slide73
Polynomial FunctionsEnd behavior ~ direction of the ends of the graph
Even Degree
Same directions
Odd Degree
Opposite directions
73
Teachers should facilitate students’ generalizationsSlide74
Real World ApplicationPolynomial FunctionSuppose 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.
74Slide75
s(t) = t3 + t2 + 6t
Odd Degree
End Behavior
75Slide76
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 seconds76Slide77
Analyzing Functions
Domain
:
Range:
Zeros:
x-intercept:
Decreasing: End
Behavior:Asymptotes:
77Slide78
Analyzing Functions
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
78Slide79
Asymptotesf(x) = 3(x – 2)
79Slide80
Asymptotes3xy = 12
80Slide81
What do you know about this rational function?81Slide82
Discontinuity (Holes)82
3Slide83
Function Development 9-1283
Algebra I
Relation or function?
Domain/range
Zeros
x- and y-intercepts
Function values for elements of the domain
Connections among representations
AFDA
Continuity
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 2
Domain/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
83Slide84
Draw a function that has the following characteristicsDomain: {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?Slide85
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