# UNIT ALIGNMENT

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```					                                         APS MATHEMATICS UNIT ALIGNMENT
Integrated Algebra/Geometry/Trigonometry 3 UNIT 6: Families of Functions

CDE Benchmarks:
2.A* Use rational, polynomial, trigonometric, and inverse functions to model real-world phenomena

2.1 Model real-world phenomena (for example, distance-versus-time relationships, compound interest, amortization tables, mortality rates) using
functions, equations, inequalities, and matrices (2.1)

2.2 Represent functional relationships using written explanations, tables, equations, and graphs, and describe the connections among these
representations (2.2)

2.3 Solve problems involving functional relationships using graphing calculators and/or computers as well as appropriate paper-and-pencil
techniques (2.3)

2.4 Analyze and explain the behaviors, transformation and general properties of types of equations and functions (for example, linear, quadratic,
exponential) (2.4)

2.5 Interpret algebraic equations and inequalities geometrically and describe geometric relationships algebraically (2.5)

4.1 Find and analyze relationships among geometric figures using transformations (for example, reflections, translations, rotations, dilations) in
coordinate systems (4.1)

The * denotes additional knowledge “Beyond the CSAP” measured standards

Big Ideas:
 Represent functional relationships of power and trigonometric functions using tables, graphs, and equations and describe the connections among them
 Recognize how patterns in graphs, tables, and rules of functions relate to the functions’ transformed graphs, tables, and rules
 Write function rules which are reflections, translations, or stretches of basic functions

Prior Learning Experiences
Core 2 Unit 2 Patterns of Location, Shape, and Size: transformations (flags)
Core 2 Unit 4 Power Models: inverse models (travel time, penlight intensity), quadratic models, solve quadratic equations (platform diver, concert
n
promoter), radicals (drawing a spiral), cube roots, other power models in form y = ax
Core 2 Unit 6 Geometric Form and Its Function: sketch graphs and find period and amplitude of functions y = AsinBx and y = AcosBx (Ferris wheel)

Integrated Algebra/Geometry 3: Unit 6 Families of Functions                                                                                             July 2009
Investigation Number      BENCHMARK(S)                                      KEY LEARNING                                 NUMERACY               CONNECTIONS
BEING                                         QUESTION                                                            TO OTHER
DEVELOPED
and/or                                                                                                           STANDARDS
MAINTAINED
Lesson 1: Function Models Revisited
Investigation 1: Modeling   2.2                               For the function families studied in this investigation   Percent growth and       Perimeter, area, volume
Atmospheric Change          2.1                               (linear models, exponential models, power models,         decay                    in terms of radius
2.4                               and quadratic models), what general patterns do           Order of operations
2.3                               you see in graphs, tables, and equations?                 Inversely proportional
2.5                                                                                         Directly proportional
What conditions or data patterns in problem
situations provide clues about appropriateness of
using each of the function families?
Investigation 2: Modeling               2.A*                  How are the amplitude and period of trigonometric         Order of operations
Periodic Change                         2.2                   functions reflected in tables, graphs, and
2.1                   equations?
2.3
What clues in problem situations suggest using the
cosine as the basic building block? What clues
suggest using the sine function?

Suppose you are modeling a periodic phenomenon
using the function f(x) = a cos (bx). How would you
determine the values of a and b?
Investigation 3: It’s All in the        2.4                   How do the parameters of each of the function rules       Order of operations      Recursive processes
Family                                  2.2                   of each of the function families (linear, exponential,                             (O3)
power, and trigonometric) relate to patterns in                                    Depreciation, percent
tables and graphs?                                                                 decrease (M1)
Cylinder volume (M2)
How does an integer exponent in direct power                                       Step graphs (M3)
models relate to patterns in corresponding tables                                  Percent increase (M5)
and graphs? How does an inverse power model                                        Asymptotes (E1)
relate to patterns in corresponding tables and
graphs?

Group Process: What actions helped the group to
work productively? What actions could be added to
make the group even more productive?

Integrated Algebra/Geometry 3: Unit 6 Families of Functions                                                                                                    July 2009
Lesson 2: Customized Models 1: Reflections and Vertical Transformations
Investigation 1: Vertical   2.4               Suppose f(x) and g(x) are any two functions. What                                                Absolute value function
Translation                 4.1               connection between the rules for these functions
would you expect when the graph of f(x) is a vertical
translation of the graph of g(x)?
What similarities in table patterns correspond to
functions with graphs that can be translated
vertically onto each other?
Investigation 2: Reflection 4.1               How are the rules of f(x) and g(x) connected if the
across the                  2.2               graph of f(x) is a vertical translation of the graph of
x-axis                      2.4               g(x)?

How are the rules of f(x) and g(x) connected if the
graph of g(x) is a reflection across the x-axis of the
graph of f(x)?
Investigation 3: Vertical               4.1                   What connections between rules for functions h(x)        Percent increase (E4)   Fit exponential model
Stretching and Shrinking                2.2                   and k(x) would you expect when the graph of one is                               to data (M2)
2.1                   found by vertically stretching or shrinking the other?                           Fit linear model to data
2.4                                                                                                    (M3)
What table patterns correspond to vertically                                     Now-Next equations
stretching and shrinking graphs?                                                 (E2)

What connections between rules for functions f(x)
and g(x) would you expect when the graph of one is
vertically stretched and then translated vertically?
Lesson 3: Customizing Models 2: Horizontal Transformations
Investigation 1: Horizontal 4.1                                                       2
For the graphs f(x) = x or f(x) = x , explain how                                               Probability, sum of dice
Shifts                      2.4                                                                                                                rolls
f(x – a) for a > 0 will transform the function.

How will the graph of g(x) = a cos (x – b) + c, where
a, b, and c are fixed, positive numbers, be related to
the graph of c(x) = cos x?

Group Process: What actions helped the group to
work productively? What actions could be added to
make the group even more productive?
Investigation 2: Horizontal             4.1                   How is the graph of a function with rule                                         Quadratic model (M5)
Stretching and Compression              2.A*                   f(x) = sin (kx) related to the graph of s(x) = sin x                            Circle equations (R5)
2.4                   when k > 1? When 0<k<1? How is the situation                                     Equivalent expressions
similar for g(x) = cos (kx) and c(x) = cos x?                                    (M5, E2)
equations (E4)

Integrated Algebra/Geometry 3: Unit 6 Families of Functions                                                                                                   July 2009

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