Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
Enduring understanding (Big Idea): Students will understand that a function is a relationship between variables that can be represented in a
variety of ways. Students will learn how families of functions are developed through transformations of the function and that new functions can be
made from other functions by applying arithmetic operations or applying one function to the output of another. Students will recognize that a
function can model a real-world situation and can be used to make estimates or predictions about future occurrences.
Essential Questions:
1. What are the characteristics of linear, exponential and quadratic functions?
2. How can you build a new function from an existing function?
3. How can you use functions to model real-world situations?
4. What kinds of relationships can proportions represent?
BY THE END OF THIS UNIT:
Students will know…
 Formulas from Unit 2 and the formulas and graphs
below:
 Quadratic function: f(x) = ax2 + bx + c
 Exponential function: y = abx
 Intercepts
 Intervals where the function is increasing,
decreasing, positive or negative.
Vocabulary:
Domain, Range, Dependent Variable, Independent Variable, Linear
Equation, Exponential Function, Quadratic Function, x-intercept and yintercept, Growth Factor/Decay, Factor, Parabola,
Roots, Input, Compound Interest, Axis of Symmetry, Zeros, Output,
Exponential Decay, Vertex, Intervals, Rational Function,
Maximum/Minimum, Increasing/Decreasing, Nonlinear Function,
Arithmetic Sequence, Common Difference, Asymptote, Inverse of a
function
Unit Resources
MARS Lesson: Forming Quadratics
Mathematical Practices in Focus:
4. Model with mathematics
5. Use appropriate tools strategically.
6. Attend to precision.
CCSS-M Included:
F.BF.1a-b, F.BF.2, F.IF.4-5, F.IF.7a, F.IF.8Aa-b, F.IF.9, F.LE.3,
A.SSE.3a-b
Suggested Pacing:
20 -25 Days
Released Test Questions:
4, 9, 11, 19, 24, 38
Algebra I Project Binder
Pages 159 – 172, 189 – 225, 227 – 239
Algebra I Station Activities
Seeing Structure in Expressions p. 53
Interpreting Functions p. 271
Interpreting Functions p. 285 station 1, 2, 4
Students will be able to…
F.BF.1a,b: I can determine an explicit expression from a given context. I
can determine a recursive process from a given context. I can determine
the steps for calculating a solution in a given context.
F.BF.2 I can write arithmetic sequences recursively. I can write
arithmetic sequences with an explicit formula. I can use arithmetic
sequences to model situations. I can translate the equation of an
arithmetic sequence between recursive and explicit forms. I can write
geometric sequences recursively. I can write geometric sequences with
an explicit formula. I can use geometric sequences to model situations. I
can translate the equation of a geometric sequence between recursive
and explicit forms.
F.BF.3 I can identify the effects of k (positive and negative) on a given
function: f(x) + k, f(x + k), k f(x), f(kx). I can find the value of k given a
graph. I can experiment using technology to illustrate the effects of k on
a graph. I can recognize whether functions are even or odd from their
algebraic equations (comparing f(x) to f(-x)).
F.IF.4: I can interpret key features of graphs, including intercepts,
intervals where the function is increasing – decreasing – positive – or
negative, relative maxima and minima, symmetries, end-behavior, and
periodicity. I can interpret key features of tables, including intercepts,
intervals where the function is increasing – decreasing – positive – or
negative, relative maxima and minima, symmetries, end-behavior, and
periodicity. I can interpret tables in terms of quantities. I can sketch
graphs showing graphing key features given verbal descriptions.
F.IF.5:I can relate the domain of a function to its graph. I can interpret the
relationship of the domain to the function.
F.IF.7a: I can graph linear functions using slope and y-intercept. I can
graph linear functions using x and y intercepts. I can show x and y
intercepts of a linear function from the graph. I can graph quadratic
functions using vertex form. I can graph quadratic functions using
standard form. I can show the maximum/minimum from a graph of a
quadratic function. I can show x and y intercept(s) from a graph of a
quadratic function. I can graph functions using technology (y = f(x))…i.e.
non-integer coefficients)
F.IF.8a,b:I can factor a quadratic function to show zeros, extreme values,
and symmetry of the graph. I can complete the square for a quadratic
function (put in vertex form) to show zeros, extreme values, and
symmetry of the graph. I can interpret the zeros, extreme values, and
symmetry for a context represented by the quadratic function.
I can use properties of exponents to interpret exponential functions as
growth or decay.
F.IF.9: I can compare properties of the same type of functions
represented different ways (algebraically, graphically, numerically, and
verbally).
F.LE.3: I can observe that the output of an exponential function
eventually exceeds the output of other increasing polynomial functions by
analyzing a graph and/or table.
A.SSE.3a,b: I can factor a quadratic expression. I can identify (reveal) the zeros of a
quadratic expression. I can complete the square of a quadratic expression. I can find
the maximum or minimum value of a quadratic function.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Build a function that models a relationship between two quantities
Standard: F.BF.1: Write a function that describes a relationship between two quantities.
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 that models the
temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions
to the model.
Concepts and Skills to Master:


Given a linear or exponential context, find an expression, recursive process, or steps to model a context with mathematical
representations.
Combine linear and/or exponential functions using addition, subtraction, multiplication, and division.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Simplifying expressions
Academic Vocabulary
Function, intercepts, explicit expression, recursive
Suggested Instructional Strategies:
 Toothpick patterns
 Give examples and use arithmetic operations to
linear and exponential functions to fit the data
Resources:
NCDPI Unpacking:
F.BF.1a: Recognize when a relationship exists between
two quantities and write a function to describe them. Use
steps, the recursive process, to make the calculations from
context in order to write the explicit expression that
represents the relationship.
F.BF.1b: Students should take standard function types
such as constant, linear and exponential functions and add,
subtract, multiply and divide them. Also explain how the
function is effected and how it relates to the model. At this
level, limit to addition or subtraction of a constant function
to linear, exponential, or quadratic functions or addition of
linear functions to linear or quadratic functions.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.

Textbook Correlation: 5-4, 5-5, CC-2, CC-10, CC-17

MARS Problem Solving Lesson:
Generalizing Patterns: Table Tiles

MARS Apprentice Tasks:
Patchwork
Sidewalk Patterns

MARS Expert Task:
Sidewalk Stones
Course Name: Middle School Math I
Unit # 7
Sample Assessment Tasks
Skill-based task
1. If f(x) = x + 4 and g(x) = 3x – 5, find (f + g)(x)
Unit Title: Functions and Their Graphs Part II
Problem Task
1. Find an expression, process or calculation to determine
the number of squares needed to make the next three
patterns in the series.
2. Anne is shopping and finds a $30 sweater on sale
for 20% off. When she buys the sweater, she must
also pay 6% sales tax. Write an expression for the
final price of the sweater in such a way that the
original price is still evident. (Extension: if the clerk
just adds 14% will the price be correct?)
2. Ten bacteria are placed in a test tube and each one splits in
two after one minute. After 1 minute, the resulting 10 bacteria
each split in two, creating 20 bacteria. This process continues for
one hour until test tube is full.
a. How many bacteria are in the test tube after 5 minutes? 15
minutes?
b. Describe how to take any current number of bacteria to find the
number of bacteria at the next minute (this is writing a NOW NEXT rule).
c. Write an equation that will determine the number of bacteria
after any number of minutes.
d. How many bacteria are in the test tube after one hour?
e. For further research, Dr. Bland removes 5 bacteria after each
minute from the original test tube to start a new cell culture. Write
the resulting equation and describe how this affected your rule in
parts b. and c.?
3. A single bacterium is placed in a test tube and splits in two
after one minute. After two minutes, the resulting two bacteria
split in two, creating four bacteria. This process continues for one
hour until test tube is filled up. How many bacteria are in the test
tube after 5 minutes? 15 minutes? Write a recursive rule to find
the number of bacteria in the test tube after n minutes. Convert
this rule into explicit form. How many bacteria are in the test tube
after one hour?
4. Suppose Kevin had $10,000 to invest in a CD account paying
8% interest compounded yearly. The function representing this
situation is y = 10000(1.08x). When the constant function y = 50 is
added to the function, what effect does it have on the exponential
function? What does that mean in the context of the problem?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Build a function that models a relationship between two quantities
Standard F.BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, and use them to model
situations and translate between the two forms. FOCUS IS EXPONENTIAL FUNCTIONS
Concepts and Skills to Master:



Write arithmetic sequences both recursively and with an explicit formula, use the two forms to model a situation and translate
between the two forms.
Write geometric sequences both recursively and with an explicit formula, use the two forms to model a situation and translate
between the two forms.
Understand that linear functions are the explicit form of recursively-defined arithmetic sequences and that exponential functions
are the explicit form of recursively-defined geometric sequences.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Identify arithmetic and geometric sequences
Academic Vocabulary
Arithmetic sequence, geometric sequence, recursive, explicit, NOW/NEXT EQUATIONS
Suggested Instructional Strategies:


Resources:
Use tables to elicit the difference between recursive and
explicit formulas for the same pattern
Write recursive and explicit formulas for patterns made by
adding toothpicks to existing patterns.
Match sequences expressed recursively with those
expressed explicitly
NCDPI Unpacking:
F-BF.2: Use arithmetic, common difference, and geometric,
common ratio, sequences to model real-life situations. Write both
the recursive and exact rule for each type of sequence, and
translate from one form of the sequence to the other. Write the
recursive and explicit forms of the arithmetic and geometric
sequences. Translate between the recursive and explicit forms.
Use the recursive and explicit forms of arithmetic and geometric
sequences to model real-world situations. In an arithmetic
sequence, each term is obtained from the previous term by adding
the same number each time. This number is called the common
difference. In a geometric sequence, each term is obtained from the
previous term by multiplying by a constant amount, called the
common ratio. Connect arithmetic sequences to linear functions
and geometric sequences to exponential functions. At this level,
formal recursive notation is not used. Instead, use of informal
recursive notation (such as NEXT = NOW + 5, starting at 3) is
intended.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.


Textbook Correlation: CC-2, CC-10
MARS Problem Solving Lesson:
Generalizing Patterns: Table Tiles
 MARS Apprentice Tasks:
Patchwork
Sidewalk Patterns
 MARS Expert Task:
Sidewalk Stones
Course Name: Middle School Math I
Sample Assessment Tasks
Skill-based task

Write two formulas that model the pattern:
3, 9, 27, 81…
Unit # 7
Unit Title: Functions and Their Graphs Part II
Problem Task
The diagram below represents a famous mathematical
fractal called Koch’s Curve. Each new stage is formed by
replacing the middle of each line segment with a “tent”
shape, creating additional line segments as shown below.
This process continues indefinitely!
a. Create a sketch of stage 3.
b. Record the number of resulting line segments in the table
below.
c. Write a rule that shows how the number of line segments
for the next step depends on the prior number of line
segments (Hint: think NOW- NEXT)
d. Write a rule to determine the number of line segments at
any stage.
e. Compare the rules you developed in parts (c) and (d).
Write an explanation as to what characteristics of the
sequence can be seen when using each form? Also include
the pros and cons when using each of the forms.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Build new functions from existing functions.
Standard: F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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 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.
Concepts and Skills to Master:
 Perform translations and transformations on quadratic and absolute value graphs
 Find the value of k given f(x) replaced by f(x) + k, kf(x), and f(x + k) on a graph of a quadratic or absolute value
function
 Relate the vertical translations of a quadratic or absolute value function to its y-intercept
 Relate the horizontal translations of a quadratic or absolute value function to its x-intercept
 Relate vertical stretch (vertical shrink) to the narrowing (widening) of the graph.
 Describe what will happen to a function f(x) is replaced by f(x) + k, kf(x), or f(x + k) for different values of k.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Graphing quadratic, exponential, and absolute value functions
Academic Vocabulary
Translation, transformation, y-intercept, x-intercept, vertical shift, horizontal shift, vertical stretch, vertical shrink
Suggested Instructional Strategies:
Use graphing technology to explore translations of
functions
Resources:
Textbook Correlation: 7-7, 9-1, CB5-3, CC-4, CC-6, CC-11,
CC13
NCDPI Unpacking:
F.BF.3 Know that when adding a constant, k, to a function, it moves
the graph of the function vertically. If k is positive, it translates the
graph up, and if k is negative, it translates the graph down.
If k is either added or subtracted from the x-value, it translates the
graph of the function horizontally. If we add k, the graph shifts left and if
we subtract k, the graph shifts right. The expression (x + k) shifts the
graphs k units to the left because when x + k = 0, x = -k.
Use the calculator to explore the effects these values have when
applied to a function and explain why the values affect the function the
way it does. The calculator visually displays the function and its
translation making it simple for every student to describe and
understand translations. At this level, limit to vertical and horizontal
translations of linear and exponential functions. Even and odd functions
are not addressed. Relate the vertical translation of a linear function to
its y-intercept.
Identify the effect transformations have on functions in multiple
modalities or ways. Student should be fluent with representations of
functions as equations, tables, graphs, and descriptions. They should
also understand how each representation of a function is related to the
others. For example, the equation of the line is related to it’s graph,
table, and when in context, the problem being solved.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Sample Assessment Tasks
Skill-based task
Graph the following on a single set of axes:
f(x) = x2
f(x) = x2 + 2
f(x) = (x+2)2
f(x) = 2x2
Unit # 7
Unit Title: Functions and Their Graphs Part II
Problem Task
1. Compare and contrast the graph of any function f(x) with
the graph of f(x) + k, f(x +k) and kf(x).
2. Given g(x) = x2 describe the changes in the graph of g(x),
that occurred to create f(x) = 2(x – 5)2 + 7.
3. If f(x) represents a diver’s position from the edge of a pool
as he dives from a 5ft. long board 25ft. above the water. If
his second dive was from a 10ft. long board that is 10ft
above the water, what happens to my equation of f(x) to
model the second dive?
4. Fill in all missing components to the below table.
Description
of Change
Original
Function
Output
x
-3
-2
-1
0
1
2
3
f(x)
9
4
1
0
1
4
9
Multiply the
original
function
output by 5
f(x)+5
5
9
a. Graph and label each of the function outputs with the
corresponding x-values on the same set of axis in three
different colors.
b. Explain the relationship that exists between the original
function and the transformed functions.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Interpret functions that arise in applications in terms of a context.
Standard: F.IF.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.
Concepts and Skills to Master:




Given a graph, identify key features such as x-and y-intercepts; intervals where the function is increasing, decreasing,
positive or negative; maxima or minima; symmetry and end-behavior
Given a table of values, identify key features such as x- and y-intercepts; intervals where the function is increasing,
decreasing, positive or negative; maxima or minima; symmetry and end-behavior
Find key features of a function and use them to graph the function
Use interval notation and symbols of inequality to communicate key features of graphs
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Ability to graph a quadratic, absolute value or piecewise-defined function from a table or equation
Academic Vocabulary
Increasing, decreasing, positive, negative, intervals, intercepts, interval notation, maximum, minimum
Suggested Instructional Strategies:
Resources:
 Use graphing technology to explore and identify key features
 Textbook Correlation: 4-2, 4-3, 5-3, 5-4, 5-5, 7-6,
of a function.
7-7, 9-1, 9-2, 9-7, 11-7

Use key features of a function to graph functions by hand
NCDPI Unpacking
F.IF.4: When given a table or graph of a function that models a reallife situation, explain the meaning of the characteristics of the graph in
the context of the problem. The characteristics described should
include rate of change, intercepts, maximums/minimums, symmetries,
and intervals of increase and/or decrease. At this level, focus on linear,
exponential, and quadratic functions; no end behavior or periodicity.
When given a verbal description of the relationship between two
quantities, sketch a graph of the relationship, showing key features.
Note – This standard should be seen as related to F-IF.7
with the key difference being students can interpret
from a graph or sketch graphs from a verbal description
of key features.
This standard should be revisited with every function your class is
studying. Students should be able to move fluidly between graphs,
tables, words, and symbols and understand the connections between
the different representations. For example, when given a table and
graph of a function that models a real-life situation, explain how the
table relates to the graph and vice versa. Also explain the meaning of
the characteristics of the graph and table in the context of the problem
as follows:
At the course one level, the focus is on linear, exponentials, and
quadratics
• Linear – x/y-intercepts and slope as increasing or decreasing at a
constant rate.
• Exponential- y-intercept and increasing at an increasing rate or
decreasing at a decreasing rate.
• Quadratics – x-intercepts/zeroes, y-intercepts, vertex, intervals of
increase/decrease, the effects of the coefficient of x2 on the concavity
of the graph, symmetry of a parabola.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
Sample Assessment Tasks
Skill-based task
Identify the intervals where the function is increasing and decreasing.
Problem Task
1. Create a story that would generate a quadratic, absolute value
or piecewise-defined function and describe the meaning of key
features of the graph as they relate to the story.
2. Below is a table that represents the relationship between daily
profit, P for an amusement park and the number of paying visitors
in thousands, n.
a. What are the x-intercepts and y-intercepts and explain them in
the context of the problem.
b. Identify any maximums or minimums and explain their meaning
in the context of the problem.
c. Determine if the graph is symmetrical and identify which shape
this pattern of change develops.
d. Describe the intervals of increase and decrease and explain
them in the context of the problem.
3. A rocket is launched from 180 feet above the ground at time t =
0. The function that models this situation is given by h(t) = – 16t2
+ 96t + 180, where t is measured in seconds and h is height
above the ground measured in feet.
a. What is the practical domain for t in this context? Why?
b. What is the height of the rocket two seconds after it was
launched?
c. What is the maximum value of the function and what does it
mean in context?
d. When is the rocket 100 feet above the ground?
e. When is the rocket 250 feet above the ground?
f. Why are there two answers to part e but only one practical
answer for part d?
g. What are the intercepts of this function? What do they mean in
the context of this problem?
h. What are the intervals of increase and decrease on the
practical domain? What do they mean in the context of the
problem?
4. Elizabeth and Joshua tried to get a monthly allowance from
their mother. If their mother initially paid them a penny and 2
pennies for the first day of the month, 4 pennies for the second
day, and so on. How much would their mother have to pay on the
10th, 20th, and 30th day of the month? Sketch the graph of the
relationship between the two quantities and explain what the point
(0,1) represents.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Interpret functions that arise in applications in terms of a context.
Standard F.IF.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.
Concepts and Skills to Master:
 Identify domains of functions given a graph
 Graph a function given a restricted domain
 Identify reasonability of a domain in a particular context
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Familiarity with function notation and domain
 Knowledge of independent and dependent variables
Academic Vocabulary
Domain, function, integers, independent variables, dependent variable, restricted domain
Suggested Instructional Strategies:
 Discuss contexts where the domain of a function
should be limited to a subset of integers, positive
or negative values, or some other restriction to the
real numbers
 Find examples of functions with limited domains
from other curricular areas (science, physical
education, social studies, consumer science…)
 Focus on quadratic functions and compare with
linear and exponential functions studied earlier in
the course.
Resources:
Textbook Correlation: 4-4, 7-6, 9-1, 11-6
NCDPI Unpacking
F.IF.5: From a graph students will identify the domain. In
context, students will identify the domain, stating any
restrictions and why they are restrictions. At this level,
focus on linear and exponential functions.
Given a function, determine its domain. Describe the
connections between the domain and the graph of the
function. Know that the domain taken out of context is a
theoretical domain and that the practical domain of a
function is found based on a contextual situation given, and
is the input values that make sense to the constraints of the
problem context.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Sample Assessment Tasks
Skill-based task
You are hoping to make a profit on the school play and
have determined the function describing the profit to be
f(t) = 8t – 2654 where t is the number of tickets sold. What
is a reasonable domain for this function? Explain.
Unit Title: Functions and Their Graphs Part II
Problem Task
1. Create functions in context where the domain would be:
a. All real numbers
b. Integers
c. Negative integers
d. Rational numbers
e. (10, 40)
2. If Jennifer buys a cell phone and the plan she decided
upon charged her $50 for the phone and $0.10 for each
minute she is on the phone. What would be the appropriate
domain that describes this relationship? Describe what
is meant by the point (10, 51).
3. Graph the function f(x) = 4x + 7 and determine the domain
and range, identifying any restrictions on that exist.
4. A rocket is launched from 180 feet above the ground at
time t = 0. The function that models this situation is given by
h(t) = – 16t2 + 96t + 180, where t is measured in seconds
and h is height above the ground measured in feet.
a. What is the theoretical domain for the function? How do
you know this?
b. What is the practical domain for t in this context? Explain.
c. What is the height of the rocket two seconds after it was
launched?
d. What is the maximum value of the function and what does
it mean in context?
e. When is the rocket 100 feet above the ground?
f. When is the rocket 250 feet above the ground?
g. Why are there two answers to part e but only one
practical answer for part d?
h. What are the intercepts of this function? What do they
mean in the context of this problem?
i. What are the intervals of increase and decrease on the
practical domain? What do they mean in the context of the
problem?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: : Analyze functions using different representations
Standard F.IF.7: 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 show intercepts, maxima, and minima
Concepts and Skills to Master:
 Graph parabolas expressed in vertex form or factored form by hand.
 Identify intercepts, maxima, minima, end behavior, increasing/decreasing intervals, and axis of symmetry in
graphs of quadratic functions
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Graph points on the coordinate plane
Academic Vocabulary
Quadratic, increasing, decreasing, maximum, minimum, axis of symmetry, y-intercept, x-intercept, vertex
Suggested Instructional Strategies:
 Allow students to develop graphs from tables and
use those graphs to generalize graphing
strategies.
 Graph equations generated from real-life contexts
Resources:
 Textbook Correlation: 9-1, 9-2, CC-6, CC-16,
CB5-8, 10-5
 Geogebra (free online) problem task
 Flying T-shirt Task
NCDPI Unpacking
F.IF.7a: Students should graph functions given by an
equation and show characteristics such as but not limited
to intercepts, maximums, minimums, and intervals of
increase or decrease. Students may use calculators or a
CAS for more difficult cases.
This standard should be seen as related to F-IF.4 with the
key difference being students can create graphs, by hand
and using technology, from the symbolic function in this
standard.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Sample Assessment Tasks
Skill-based task
1. Graph the function y = (x – 3)(x + 2), identify the axis of
symmetry and the vertex.
Unit Title: Functions and Their Graphs Part II
Problem Task
1. The all-star kicker kicks a field goal for the team and the
path of the ball is modeled by f (x) = −4.9t2 + 20t.
Find the realistic maximum and minimum values for the path
of the ball and describe what each means in the context of
this problem.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Analyze functions using different representations
Standard F.IF.8: 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 = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as
representing exponential growth or decay.
Concepts and Skills to Master:
 Factor quadratic equations and identify zeros
 Complete the square to show extreme values and symmetry of the graph
 Use properties of exponents to interpret expressions for exponential functions
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Multiplying binomials, perfect square trinomials
Academic Vocabulary
Factor, completing the square, quadratic, zero, extreme value (minimum, maximum), symmetry, properties of exponents,
growth, decay
Suggested Instructional Strategies:
Resources:
 Have students “discover” the process of
 Textbook Correlation: 7-7, CC-15
completing the square by using algebra tiles (or
 A visual representation of completing the square
blocks)
Proof without words: completing the square
NCDPI Unpacking
F.IF.8a: Students should take a function and manipulate it
in a different form so that they can show and explain
special properties of the function such as; zeros, extreme
values, and symmetries. Students should factor and
complete the square to find special properties and interpret
them in the context of the problem. Keep in mind when
completing the square, the coefficient on the x2 variable
must always be one and what you add in to the problem,
you must also subtract from the problem. In other words,
we are adding zero to the problem in order to manipulate it
and get it in the form we want.
Students should manipulate a quadratic function to identify
its different forms (standard, factored, and vertex) so that
they can show and explain special properties of the
function such as; zeros, extreme values, and symmetry.
Students should be able to distinguish when a particular
form is more revealing of special properties given the
context of the situation.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Sample Assessment Tasks
Skill-based task
Unit # 7
Unit Title: Functions and Their Graphs Part II
Problem Task
1. The Falling Freely Skydiving Company charges a basic price of
$150 per person for each jump. However, business is slow and to
attract more clients, the company reduces the price of each jump
by $5 for each person in the group. The larger the group, the less
each person pays.
a. Define variables and write an equation for the price of a single
jump.
b. If you and a group of your friends decided to go skydiving, what
would the equation be for the total price the company charges?
c. What is the total price of a jump for a group of 6 people?
d. The company reports that the cost of the skydiving trip was
$1000.00, How many people were on the trip?
e. What limitations on group size should the skydiving company
use in order to make a profit?
2. Suppose you have a rectangular flower bed whose area is
24ft2. The shortest side is (x-4)ft and the longest side is (2x)ft.
Find the length of the shortest side.
3. In the cartoon, Coyote was chasing the Road Runner, seeing
no easy escape, Road Runner jumped off a cliff towering above
the roaring river below. Molly mathematician was observing the
chase and obtained a digital picture of this fall. Using her
mathematical knowledge, Molly modeled the Road Runner’s fall
using several quadratic functions:
h(t) = -16t2 + 32t + 48
h(t) = -16(t+ 1)(t – 3)
h(t) = -16(t – 1)2 + 64
a. Does Molly have three unique equations that model the same
situation? Explain.
b. Explain what it means for expressions or equations to be
mathematically equivalent?
c. Which of the equivalent equations would be most helpful in
answering each of these questions? Explain.
i. What is the maximum height the Road Runner reaches
and when will it occur?
ii. When would the Road Runner splash into the river?
iii. At what height was the Road Runner when he jumped
off the cliff?
4. Stephanie is considering moving to Carmel or Lunsford.
Carmel’s population curve in thousands of people in year x can
be shown by the equation y = 200(1.015)x. Lunsford’s population
curve in thousands of people in year x can be shown by the
equation y = 60(0.998)x.
a. What is the rate of change in Carmel? And what is the rate of
change in Lunsford?
b. Which population model represents decay and which one
represents growth. Justify your answer.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Analyze functions using different representations
Standard F.IF.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.
Concepts and Skills to Master:
Compare extrema and intercepts of two quadratic functions where one is represented algebraically, graphically,
numerically, in tables, or in a description and the other is modeled using a different form of representation.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Factor quadratic expressions
Academic Vocabulary
Quadratic function, maximum, minimum, intercepts, vertex, axis of symmetry
Suggested Instructional Strategies:
Resources:
 Compare two functions expressed in different
 Textbook Correlation: CC-4, CC-11, CC-13
representations. Ask: How can you determine
 Geogebra (free online)
whether the graph of the function will have a
 Graphing calculators
maximum or minimum? How can you find the
MARS Task: Sorting Functions
vertex of a quadratic from its graph, table,
algebraically?
 Match functions expressed using different
representations that have the same
properties.
NCDPI Unpacking
F.IF.9: Students should compare the properties of two
functions represented by verbal descriptions, tables,
graphs, and equations. For example, compare the
growth of two linear functions, two exponential
functions, or one of each. At this level, limit to linear,
exponential, and quadratic functions.
This standard compares different functions when they
have different representations. For example, one may
be represented using a graph and the other an
equation.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Sample Assessment Tasks
Skill-based task
1. Compare the functions represented below. Which
has the lowest minimum?
a. f(x) = 3x2 +13x +4
b.
Unit Title: Functions and Their Graphs Part II
Problem Task
1. A herd of horses at Corolla Beach was first counted at 100
heads. Repopulation efforts have yielded a net growth of 16%
yearly of the existing horse population. Simultaneously, biologists
have recorded the sea turtle population growth in the following
table: Which population is growing at a faster rate? Explain your
reasoning.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Construct and compare linear, quadratic, and exponential models and solve problems.
Standard: F.LE.3: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Concepts and Skills to Master:
 Compare linear and exponential growth to quadratic growth.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Identify linear, quadratic and exponential functions
 Graph linear, quadratic and exponential functions
 Rate of change
Academic Vocabulary
Linear, exponential, quadratic, factor, difference
Suggested Instructional Strategies:
 This standards should be taught in conjunction
with others in this cluster
Resources:
 Textbook Correlation: 9-7, CC-14
NCDPI Unpacking
F.LE.3: When students compare graphs of various
functions, such as linear, exponential, quadratic, and
polynomial they should see that any values that
increase exponentially eventually increases or grows at
a faster rate than values that increase linearly,
quadratically, or any polynomial function. At this level,
limit to linear, exponential, and quadratic functions;
general polynomial functions are not addressed.
When comparing values of various functions that
increase as the x-value gets larger, the values taken on
by the exponential function will eventually be greater
than the values of a linear, quadratic, or polynomial
function.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Sample Assessment Tasks
Skill-based task
Let P = (0, 1) and Q = (1, 2) in the (x, y) plane.
a. Show that there is a unique linear function described
by the equation y = mx + b whose graph contains P
and Q: find m and b.
b. Show that there is a unique exponential function
described by the equation y = abx whose graph
contains P and Q: find a and b.
c. Show that there is more than one quadratic function
described by the equation y = ax2 + bx + c whose graph
contains P and Q .
answers:
1. y = x +1
2. y = 2x
3. answers will vary. Ex. y = x2 + 1, y = -x2 + 2x + 1
Unit Title: Functions and Their Graphs Part II
Problem Task
1. Carrie and Elizabeth applied for a job at the local seafood market.
Carrie asked for $2 and hour, but the boss proposed giving them $.010
for the first hour, $.020 for the second hour, $.040 for the third hour,
$.080, and so on. Below is a table of the hours worked per week and the
pay for each hour for each of the two pay plans. When would you want to
use Carrie’s plan and when would you use the Boss’s plan? Why?
2. Freddy and Fannie Frugal has $10 to invest and is considering three
different investment plans.
Plan 1: Guarantees that the Frugals will earn $10.00 in interest every
year.
Plan 2: Guarantees that the Frugals will earn 10% interest compounded
annually on the account.
Plan 3: This plan has a lump sum payout at the day of withdraw that
depends on the total time the investment is in the bank. The Frugals will
receive the original $10 investment plus an interest payment equal to the
number of years invested squared in.
I. Utilizing technology create sketch of all three plans on the same set of
axis. Indicate key intersection point(s) and the meaning of each.
II .Making sense of the quantities and relationships that occur in the other
three quadrants, answer the following problem situations.
a. If the Frugals know that their anticipated twins (Mae and Mac) are
going to arrive in the next 10 years, which plan would make the most
money for the Frugals to renovate the nursery? What are the key
characteristics of this plan?
b. Rather than using the investment for nursery renovations, they elect to
save for the twin’s college education. Which would be the best plan?
Why?
c. The Frugals then consider providing a retirement account for the twins.
Which plan would be the most beneficial if the retirement age is 65? How
do you know this?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Unit Title: Functions and Their Graphs Part II
CORE CONTENT
Cluster Title: Write expressions in equivalent forms to solve problems
Standard A.SSE.3a – b: Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines
b. Complete the square in a quadratic expression to reveal the maximum or minimum values of the function it
defines.
Concepts and Skills to Master:
 Given a quadratic function explain the meaning of the zeros of the function. e.g. if f(x) = (x – c)(x – a), then f(a) = 0
and f(c) = 0
 Given a quadratic expression, explain the meaning of the zeros graphically. e.g. for an expression (x – c)(x – a), a
and c correspond to the x-intercepts (if a and c are real numbers).
 Write the vertex form of a quadratic expression by completing the square.
 Use the vertex form to find the maximum or minimum of a quadratic function and explain the meaning of the
vertex
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Identifying and factoring perfect square trinomials
Academic Vocabulary
Completing the square, quadratic function, perfect square trinomials
Suggested Instructional Strategies:
Resources:
Use algeblocks or algebratiles to explore the meaning of
 Textbook Correlation: 7-7, CC-15
“completing the square” and discover the algorithm
NCDPI Unpacking
A.SSE.3a – b: Students factor quadratic expressions and
find the zeros of the quadratic function they represent.
Zeroes are the x-values that yield a y-value of 0. Students
should also explain the meaning of the zeros as they relate
to the problem. For example, if the expression x2 – 4x + 3
represents the path of a ball that is thrown from one person
to another, then the expression (x – 1)(x – 3) represents its
equivalent factored form. The zeros of the function,
(x –1)(x – 3) = y would be x = 1 and x = 3, because an
x-value of 1 or 3 would cause the value of the function to
equal 0. This also indicates the ball was thrown after 1
second of holding the ball, and caught by the other person
2 seconds later. At this level, limit to quadratic expressions
of the form ax2 + bx + c.
Students rewrite a quadratic expression in the form
a = 1 (x − h)2 + k, with a = 1, to identify the vertex of the
parabola (h, k), and explain its meaning in context.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Middle School Math I
Unit # 7
Sample Assessment Tasks
Skill-based task
1. How are the following functions related? State the form
of each equation.
y1= (x– 3)(x + 1)
y2= x2– 2x– 3
y3= (x– 1)2– 4
Determine the vertex, y-intercept, and x-intercept(s) for
each function. Describe which form is each point most
easily determined.
Unit Title: Functions and Their Graphs Part II
Problem Task
1. The expression 3m2 – 15m is the income gathered by
promoters of a rock concert based on the ticket price, m.
For what value(s) of m would the promoters break even?
2. The profit that a company makes selling an item (in
thousands of dollars) depends on the price of the item (in
dollars). If p is the price of the item, then three equivalent
forms for the profit are:
Standard form: −2p2 + 24p − 54
Factored form: −2(p − 3)(p − 9)
Vertex form: −2(p − 6)2 + 18.
Which form is most useful for finding
a. The prices that give a profit of zero dollars?
b. The profit when the price is zero?
c. The price that gives the maximum profit?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.