CMS Curriculum Guides 2011-2012
Algebra I
Unit Title: Functions and their Graphs Part II: approximately 10 days
Enduring understanding (Big Idea): Students will understand that functions are tools that can be used to interpret real-world phenomena.
Essential Questions:
1. How are graphs of different functions the same? How are they different?
2. How can graphs help you understand different real-world situations?
Students will be able to…
Students will know…
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Formulas from Unit 2 and the formulas and graphs below:
Quadratic function: f(x)=ax2 + bx +c
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Exponential function: y  ab x
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Absolute value: f ( x)  x
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Piecewise including step functions
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Square root: f ( x) 
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Intercepts
Intervals where the function is increasing, decreasing, positive, or
negative
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Cubic: f ( x)  x 3
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Distance formula
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d  ( x2  x1 ) 2  ( y2  y1 ) 2
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Midpoint formula
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x
 x  x2 y1  y2 
M  1
,
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2 
 2
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interpret key features of graphs and tables in terms of the quantities and
sketch graphs given a verbal description for a function that models a
relationship between two quantities (verbal to graph)
relate the domain and range of a function to its graph
graph quadratic functions and show intercepts, maxima, and minima (graph to
verbal)
graph absolute value, square root, cube root, and piecewise-defined functions,
including step functions and show intercepts, maxima, and minima
use factoring and completing the square in a quadratic function to show zeros,
extreme values, and symmetry, and interpret in terms of context
compare properties of two functions represented in a different way
(algebraically, graphically, in tables, or by verbal description)
write a function that describes a relationship between two quantities
identify the effect on the graph of replacing f(x) by f(x)+k, k f(x), f(kx), and
f(x+k)
solve an equation of the form f(x)=c a simple function that has an inverse and
write an inverse and write an expression for the inverse.
Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or as a
polynomial function.
Solve applications of quadratic functions
Solve problems involving distance and midpoint
Solve systems of linear and quadratic equations
CMS Curriculum Guides 2011-2012
Algebra I
Mathematical Practices:
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
Common Core Standards
Textbook Alignment
Connection to 2003
Standards
Use properties of
rational and irrational
numbers.
Review definitions on Lesson 1-3 page 18.
Focus on p. 20 Lesson Check (1-8). Emphasize
numbers 4, 7, and 8. Allow for discussion.
Covered in Algebra 1
Foundations
Connect N.RN.3 to
physical situations.
e.g., finding the
perimeter of a square
of area 2.
Concept Byte (following 1-6)
Interpret functions
that arise in
applications in terms
of a context.
This is an extension of Unit 2: 4-2-p. 251(22); 4- Linear, exponential, and
3-p. 259 (40, 41); 5-3 pp. 331-312 (68-70, 75);
quadratic functions are
5-4 pp. 317-318 (31, 32, 33); 5-5 p. 326(64, 65); expected at this level.
7-6 pp. 451-452 (45, 46, 51); 7-7 pp. 460-461
(35-39, 43-45); 9-1 pp. 539-540 (44-48, 50, 51),
Enrichment 9-1; 9-2 pp. 545-546 (34, 36-38, 4042); 9-7 (Enrichment 9-7); 11-7 Review p. 697
Instructional notes
N.RN.3: Use properties of rational and
irrational numbers. Explain why the sum
or product of two rational numbers is
rational; that the sum of a rational number
and an irrational number is irrational; and
that the product of a nonzero rational
number and an irrational number is
irrational.
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F.IF.4: Interpret functions that arise in
applications in terms of the context. 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 maximum and minimums;
symmetries; end behavior; and periodicity.
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Focus on quadratic
functions; compare
with linear and
exponential functions
studied in Unit 2.
See other resources below for a deeper
understanding of the closure property.
CMS Curriculum Guides 2011-2012
Algebra I
F.IF.5: Interpret functions that arise in
applications in terms of the context.
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.
9-1 (Enrichment 9-1),
Linear, exponential, and
quadratic functions are
expected at this level.
F.IF.6: Interpret functions that arise in
applications in terms of the context.
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.
5-1 (Enrichment 5-1) ; 9-7 p. 579(25)
Linear, exponential, and
quadratic functions are
expected at this level.
F.IF.7: Analyze functions using different
representations. Graph functions
expressed symbolically and show key
features of the graph, by hand in simple
cases and using technology for more
complicated cases.
a. Graph linear and quadratic functions and
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Moved from 2003 Algebra
II NCSCOS.
Analyze functions
using different
representations.
For F.IF.7b, compare
and contrast absolute
value, step and
piecewise-defined
functions with linear,
Linear, exponential,
quadratic, absolute value,
step, and piecewisedefined functions are the
expectations at this level.
5-5 p. 325 (51-59), Concept Byte p. 554
CMS Curriculum Guides 2011-2012
Algebra I
show intercepts, maxima, and minima.
b. Graph square root, cube root, and
piecewise-defined functions, including step
functions and absolute value functions.
F.IF.8: Analyze functions using different
representations. Write a function defined
by an expression in different but equivalent
forms to reveal and explain different
properties of the function.
a. Use the process of factoring and
completing the square in a quadratic
function to show zeros, extreme values,
and symmetry of the graph, and interpret
these in terms of a context.
b. Use the properties of exponents to
interpret expressions for exponential
functions. For example, identify percent
rate of change in functions such as y =
(1.02)t, y = (1.01)12t, y = (1.2)t/10, and classify
them as representing exponential growth
or decay.
quadratic, and
exponential functions.
Highlight issues of
domain, range, and
usefulness when
examining piecewisedefined functions.
Note that this unit,
and in particular in
F.IF.8b, extends the
work begun in Unit 2
on exponential
functions with integer
exponents. For F.IF.9,
focus on expanding
the types of functions
considered to include,
linear, exponential,
and quadratic.
Extend work with
quadratics to include
relationship between
coefficients and roots,
and that once roots
are known, a
quadratic equation
can be factored.
(following 9-3)
5-8 (Reteaching 5-8) , 10-5 (Practice 10-5),
Concept Byte (following 5-8) 9-2 (Enrichment)
See Problem 2 on p. 17
See p. 22 number 71
Moved from 2003 Algebra
II NCSCS.
Linear, exponential,
quadratic, absolute value,
step, and piecewisedefined functions are the
expectations at this level.
9-4 and 9-5
7-7
CMS Curriculum Guides 2011-2012
Algebra I
F.IF.9: Analyze functions using different
representations. 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.
F.BF.1: Build a function that models a
relationship between two quantities.
Write a function that describes a
relationship between two quantities.
7-6, 9-2
Linear, exponential,
quadratic, absolute value,
step, and piecewisedefined functions are the
expectations at this level.
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a. Determine an explicit expression, a
recursive process, or steps for calculation
from a context.
b. Combine standard function types
using arithmetic operations. For example,
build a function than models the
temperature of a cooling body by adding a
constant function to a decaying
exponential, and relate these functions to
the model.
F.BF.3: Build new functions from existing
functions. Identify the effect on the graph
of replacing f(x), by f(x) + k, k f(x), f(kx), and
f(x+k) for specific values of k (both positive
and negative); find the value of k given the
graphs. Experiment with cases and
New to CCSS.
Build a function that
models a relationship
between two
quantities.
Focus on situations
that exhibit a
quadratic relationship.
Moved from 2003 Algebra
II NCSCS.
Linear, exponential, and
quadratic functions are
the expectations at this
level.
4-7, 7-8
9-7
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Build new functions
from existing
functions.
For F.BF.3, focus on
5-3, 5-4, 5-8, 7-7, 9-1, 9-2, Concept Byte (before
5-3)
Moved from 2003 Algebra
II NCSCS.
Linear, exponential,
quadratic and absolute
value functions are the
CMS Curriculum Guides 2011-2012
Algebra I
illustrate an explanation of the effects on
the graph using technology. Include
recognizing even and odd functions from
their graphs and algebraic expressions for
them.
quadratic functions,
and consider including
absolute value
functions. For F.BF.4a,
focus on linear
functions but consider
simple situations
where the domain of
the function must be
restricted in order for
the inverse to exist,
such as f(x) = x2, x>0.
F.BF.4: Build new functions from existing
functions. Find inverse functions.
a. Solve an equation of the form f(x) = c
for a simple function f that has an inverse
and write an expression for the inverse. For
example, f(x) = 2x3 or f(x) = (x+1)/(x-1) for x
≠1.
F.LE.3: Construct and compare linear,
quadratic, and exponential models and
solve problems. Observe using graphs and
tables that a quantity increasing
exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more
generally) as a polynomial function.
expectations for this level.
Moved from 2003 Algebra
II NCSCS.
Linear functions are the
expectation at this level.
Concept Byte (following 5-6)
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Construct and
compare linear,
quadratic, and
exponential models
and solve problems.
Compare linear and
exponential growth to
quadratic growth.
9-7, Concept Byte (following 9-2)
New to CCSS.
CMS Curriculum Guides 2011-2012
Algebra I
Not covered in CCSS in Algebra I
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Find the distance
between two points.
Find the midpoint of a
segment
2-4, p. 114(50) Concept Byte ( following 10-1)
p. 605
NCSCS 2.01 Find the
lengths and midpoints of
segments to solve
problems.
Prior Knowledge: Know that there are numbers that are not rational, and approximate them by rational numbers. Work with radicals and integer
exponents. Define, evaluate, and compare functions. Use functions to model relationships between quantities. Should have completed Unit 2
(functions) and Unit 4 (quadratics). Solve quadratic equations by factoring, completing the square, and using the quadratic formula
Key Vocabulary:
 Function
 Independent variable/input/domain
 Dependent variable/output/range
 Intercepts
 Roots
 Zeros
 Solution
 Intervals
 Increasing
 Decreasing
 Vertex
 Line of Symmetry
 Relative maximums (maxima)
 Relative minimums (minima)
 Inverse of a function
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Rate of change
Square root
Cube root
Linear
Piecewise-defined
Quadratic
Absolute value
Exponential growth
Exponential decay
Distance
Midpoint
CMS Curriculum Guides 2011-2012
Algebra I
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RESOURCES
For rational and irrational numbers (N.RN.3) visit:
http://mathforum.org/library/drmath/view/53048.html
http://mathforum.org/library/drmath/view/51560.html
http://www.mathsisfun.com/irrational-numbers.html
http://en.allexperts.com/q/Advanced-Math-1363/irrational-numbers.htm
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Interpreting and Analyzing functions
http://www.analyzemath.com/quadraticg/quadraticg.htm
http://www.analyzemath.com/Graphing/GraphExponentialFunction.html
http://www.analyzemath.com/Graphing/piecewise_functions.html
http://www.analyzemath.com/Graph-Basic-Functions/Graph-Basic-Functions.html
http://www.analyzemath.com/function/square_root_function.html
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MATHFORWARD/TINavigator
Activities
Assessments
Learncheck questions bank
Lessons
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INQUIRY ACTIVITIES
MARS Lesson: Forming Quadratics
Other Activities: Analyzing graphs
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PROBLEM-BASED TASKS
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PROJECTS
CMS Curriculum Guides 2011-2012
Algebra I