HCPSS Curriculum Framework
Algebra 2, Unit 1
Introduction to Families of Functions
Overview (Big Ideas):
Building on their work with linear, quadratic, and exponential functions, students extend
their repertoire of functions to begin to explore other functions families. Students review key
features (domain, range, relative extrema, intervals increasing/decreasing, end behavior,
continuity) of function graphs and describe features when provided with a function or a graph.
Students begin to fit real-world data and scenarios to potential function families and begin to
explore inverses graphically.
The Mathematical Practice Standards apply throughout each course and, together with the
content standards, prescribe that students experience mathematics as a coherent, useful, and
logical subject that makes use of their ability to make sense of problem situations.
This unit serves as an introduction to function families and modeling and lays the
groundwork for deeper study of individual function families.
Enduring Understandings:
1. The concept of a function is intentionally broad and flexible, allowing it to apply to a
wide range of situations. The notion of function encompasses many types of
mathematical entities in addition to “classical” functions that describe quantities that vary
continuously.
2. Functions provide a means to describe how related quantities vary together. We can
classify, predict, and characterize various kinds of relationships by attending to the rate at
which one quantity varies with respect to the other.
3. Functions can be classified into different families of functions, each with its own unique
characteristics. Different families can be used to model different real-world phenomena.
4. Functions can be combined by adding, subtracting, multiplying, dividing, and/or
composing them. Functions sometimes have inverses. Functions can also be analyzed by
viewing them as made from other functions.
5. Functions can be represented in multiple ways, including algebraic (symbolic), graphical,
verbal, and tabular representations. Links among these different representations are
important to studying relationships and change.
Source:
Cooney, T.J., Beckmann, S., & Lloyd, G.M. (2010). Developing essential understanding of
functions for teaching mathematics in grades 9-12. Reston, VA: The National Council of
Teachers of Mathematics, Inc.
The Modeling Cycle:
Source: http://commoncoretools.me/wpcontent/uploads/2013/07/ccss_progression_modeling_2013_07_04.pdf
Analytic modeling seeks to explain data on the basis of deeper theoretical ideas (albeit with
parameters that are empirically based). Functions are an important tool for analyzing such
problems. Modeling is best interpreted not as a collection of isolated topics but rather in relation
to other standards.
Essential Questions:
o What are the key features of a graph of a function?
o What commonalities do the functions within the same family have? What makes a
particular function family unique from other types of functions?
o What types of data may be represented with a given function family?
o How can we determine if a given function family is an appropriate model for a set of
data? How can we determine if a regression equation best models a set of data?
o What is the inverse of a function? How can an inverse be displayed graphically?
Common Misconceptions:
o Students may enter this course with gaps in background knowledge, particularly in the
area of quadratic functions. Use this unit to pre-assess understanding of linear,
exponential, and quadratic units so that you can appropriately remediate in later units.
o Matching data with an appropriate function family may be difficult, particularly for
functions that students have not studied. Students are not expected to master this skill in
unit one, rather they should be able to reason why some function families are not
appropriate and begin to narrow the possibilities given the shape of the data.
o Students may have difficulty with modeling functions. Modeling is embedded in every
unit, so it is critical to review how to find a model to fit a set of data and how to check the
reasonableness of the model.
Unit 1 Content Standards:
Interpret functions that arise in applications in terms of a context.
F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs
and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* (Cross-cutting)
(Emphasize the selection of a model function based on behavior of data and context.)
Summarize, represent, and interpret data on quantitative variables.
S.ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are
related. (Cross-cutting)
a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use
given functions or choose a function suggested by the context. Emphasize linear and exponential
models.
b Informally assess the fit of a linear function by plotting and analyzing residuals.
c Fit a linear function for a scatter plot that suggests a linear association.
Analyze functions using different representations.
F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum. (Cross-cutting)
Overarching Standards (will be embedded in each unit):
Reason quantitatively and use units to solve problems.
N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. (Cross-cutting)
Interpret the structure of expressions.
A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. (Cross-cutting)
Create equations that describe numbers or relationships.
A.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. (Cross-cutting)
Interpret functions that arise in applications in terms of a context.
F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a
factory, then the positive integers would be an appropriate domain for the function.* (Cross-cutting)
Interpret linear and exponential functions that arise in applications in terms of a context.
F.IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table)
over a specified interval. Estimate the rate of change from a graph. (Cross-cutting)
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed
this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
License.