Secondary II
Functions
Teacher Edition
Unit 7
Northern Utah Curriculum Consortium
Project Leader
Sheri Heiter
Weber School District
Project Contributors
Ashley Martin
Bonita Richins
Craig Ashton
Davis School District
Cache School District
Cache School District
Gerald Jackman
Jeff Rawlins
Jeremy Young
Box Elder School District
Box Elder School District
Box Elder School District
Kip Motta
Marie Fitzgerald
Mike Hansen
Rich School District
Cache School District
Cache School District
Robert Hoggan
Sheena Knight
Teresa Billings
Cache School District
Weber School District
Weber School District
Wendy Barney
Helen Heiner
Susan Summerkorn
Weber School District
Davis School District
Davis School District
Lead Editor
Allen Jacobson
Davis School District
Technical Writer/Editor
Dianne Cummins
Davis School District
NUCC| Secondary II Math i
Table of Contents
7.1 EXPLORING FUNCTIONS WITH SARAH ................................................................................................4
Teacher Notes ..................................................................................................................................................4
Mathematics Content .......................................................................................................................................7
Exploring Functions with Sarah A Solidify Understanding Task 1..................................................................8
Ready, Set, Go! ............................................................................................................................................. 12
7.2 FALLING OBJECTS.................................................................................................................................. 15
Teacher Notes ............................................................................................................................................... 15
Mathematics Content .................................................................................................................................... 18
Falling Objects A Practice Understanding Task 2a ...................................................................................... 19
Fences and Function A Practice Understanding Task 2b ............................................................................. 21
Ready, Set, Go! ............................................................................................................................................. 24
7.3 BASIC FUNCTION SHAPES .................................................................................................................... 26
Teacher Notes ............................................................................................................................................... 26
Mathematics Content .................................................................................................................................... 29
Basic Function Shapes and Graphs Student Notes ........................................................................................ 30
Basic Function Shapes A Develop Understanding Task 3 ............................................................................ 32
Ready, Set, Go! ............................................................................................................................................. 33
7.4 DOMAIN AND RANGE ............................................................................................................................ 35
Teacher Notes ............................................................................................................................................... 35
Mathematics Content .................................................................................................................................... 40
Domain and Range Functions Student Notes ................................................................................................ 41
Domain and Range Functions A Develop Understanding Task 4 ................................................................. 44
Ready, Set, Go! ............................................................................................................................................. 45
7.5 PIECEWISE FUNCTIONS ........................................................................................................................ 50
Teacher Notes ............................................................................................................................................... 50
Mathematics Content .................................................................................................................................... 56
Getting Ready for a Pool Party A Develop Understanding Task 5a ............................................................. 57
Piecewise Functions A Develop Understanding Task 5b .............................................................................. 59
A Taxing Situation A Solidify Understanding Task 5c ................................................................................ 62
Ready, Set, Go! ............................................................................................................................................. 64
Functions Quiz .............................................................................................................................................. 69
NUCC| Secondary II Math ii
7.6 REVIEWING TRANSFORMATIONS ...................................................................................................... 72
Teacher Notes ............................................................................................................................................... 72
Mathematics Content .................................................................................................................................... 75
Mathematics Content .................................................................................................................................... 76
Absolute Value Graphs & Transformations A Develop Understanding Task 6............................................ 77
Ready, Set, Go! ............................................................................................................................................. 82
7.7 PARKING DECK PANDEMONIUM........................................................................................................ 86
Teacher Notes ............................................................................................................................................... 86
Mathematics Content .................................................................................................................................... 89
Mathematics Content .................................................................................................................................... 90
Parking Deck Pandemonium A Develop Understanding Task 7................................................................... 91
Ready, Set, Go! ............................................................................................................................................. 96
7.8 APPLESAUCE STOCK ............................................................................................................................. 99
Teacher Notes ............................................................................................................................................... 99
Mathematics Content .................................................................................................................................. 101
Applesauce Stock A Solidify Understanding Task 8 ................................................................................... 102
Ready, Set, Go! ........................................................................................................................................... 104
NUCC| Secondary II Math iii
Unit 7.1
7.1 EXPLORING FUNCTIONS WITH SARAH
Teacher Notes
Time Frame:
Materials Needed:
Purpose: This task’s purpose is to review concepts learned in Secondary Math I so that students
can use that background knowledge to build on.
Concepts that will be reviewed:
 Drawing graphs from a table of data
 Continuous graphs vs. discrete graphs
 Function notation
 Extrapolating data
 Evaluating functions
 Increasing and Decreasing functions
Core Standards Focus Missing: Please find the standard you used to base this lesson.
Launch (Whole Class): As a class, go through the directions in Part 1 of the task. I would also
go over 1a and decide which is the ‘independent variable’ and which is the ‘dependent variable.’
You could even get students started on the graph and labeling the different axes.
NUCC| Secondary II Math 4
Unit 7.1
Explore (Individual, small group or pairs): Have students work through the rest of Part 1 and
Part 2 of the task.
Discuss (Whole Class or Group): Discuss and review function notation and how to evaluate
functions.
NUCC| Secondary II Math 5
Unit 7.1
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 6
Unit 7.1
Mathematics Content
Cluster Title: Construct and compare linear, quadratic, and exponential models and solve
problems. (Need the standard that belong with this lesson)
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


Use a table to observe that exponential functions grow more quickly than quadratic
functions.
Use a graph to observe that exponential functions grow more quickly than quadratic
functions.
Critical Background Knowledge

Graph quadratic and exponential functions.
Academic Vocabulary
exponential, quadratic, rate of change
Suggested Instructional Strategies


Examine contexts of quadratic vs. exponential functions, comparing values at specified
points.
Use technology to explore, predict, and model. Emphasize appropriate viewing windows.
Skills Based Task:
Problem Task:

Find a quadratic and exponential function that:
 Do not intersect.
 Intersect once.
 Intersect twice.
 Intersect more than twice.
Graph the functions y  x 2 and y  2 x
on the same coordinate axes. Compare
the values of the functions over various
intervals.
Some Useful Websites:
Resource: NCTM Illuminations: The Devil and Daniel Webster
NUCC| Secondary II Math 7
Unit 7.1
Exploring Functions with Sarah
A Solidify Understanding Task 1
Name_____________________________________
Hour___________
Part 1
1.
While visiting her grandmother, Sarah found markings on the inside of a closet door
showing the heights of her mother, Tammy, and her mother’s brothers and sisters on their
birthdays growing up. From the markings in the closet, Sarah wrote down her mother’s
height each year from ages 2 to 16. Her grandmother found the measurements at birth and
one year by looking in her mother’s baby book. The data is provided in the table below, with
heights rounded to the nearest inch.
Age (yrs.)
X 0
1
2
3
4
5
6
7
8 9 10 11 12 13 14 15 16
Height (in.)
Y 21 30 35 39 43 46 48 51 53 55 59 62 64 65 65 66 66
a. Which variable is the independent variable, and which is the dependent variable?
Explain your choice.
b. Make a graph of the data.
c. Should you connect the dots on your graph? Explain.
d. Describe how Tammy’s height changed as she grew up.
NUCC| Secondary II Math 8
Unit 7.1
e. How tall was Tammy on her 11th birthday?
f. What do you think happened to Tammy’s height after age 16? Explain. How could
you show this on the graph?
NUCC| Secondary II Math 9
Unit 7.1
Part 2
2.
Function notation gives us another way to write about ideas that you began learning
in middle school, as shown in the table below. In the case of the table above, h(2)
means the y-value when x is 2, which is Tammy’s height (in inches) at age 2, or 35.
Thus, h(2) = 35.
Statement
Type
At age 2, Tammy was 35 inches tall.
Natural language
When x is 2, y is 35.
Statement about variables
When the input is 2, the output is 35.
Input-output statement
h(2) = 35.
Function notation
a. What is h(11)? What does this mean?
b. When x is 3, what is y? Express this fact using function notation.
c. Find an x so that h(x) = 53. Explain your method. What does your answer mean?
d. From your graph or your table, estimate h(6.5). Explain your method. What does your
answer mean?
e. Find an x so that h(x) = 60. Explain your method. What does your answer mean?
f. Describe what happens to h(x) as x increases from 0 to 16.
NUCC| Secondary II Math 10
Unit 7.1
g. What can you say about h(x) for x greater than 16?
h. Describe the similarities and differences you see between these questions and the
questions in #1.
NUCC| Secondary II Math 11
Unit 7.1
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
1. What does it mean to evaluate a function?
2. Complete the following table using the equation: 𝑦 = 𝑥 − 2
x
y
Point
(x, y)
-1
0
1
2
3. Evaluate the function 𝑓(𝑥) = −𝑥 2 + 1 at 𝑥 = −1, 𝑥 =
1, 𝑥 = 2.
4. What “points” on the curve/graph of 𝑓(𝑥) = −𝑥 2 + 1 did you find?
5.
f ( x)  x  2 , find f  0 
7.
g  x   4 g  5 , find g  5
6.
f  x    x  2 , find f  1
NUCC| Secondary II Math 12
Unit 7.1
Set
8. Evaluate the following function at the following values:
a)
b)
c)
d)
e)
f)
g)
f (1) 
f ( 5) 
f ( 4) 
f (6) 
f (0) 
f ( 2) 
f ( 7) 
9. Evaluate the following function at:
a)
b)
c)
d)
e)
f)
f (0) 
f (2) 
f ( 2) 
f (3) 
f (6) 
f (4.5) 
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
y
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
-1
-2
-3
-4
-5
NUCC| Secondary II Math 13
Unit 7.1
Go!
10.
Evaluate:
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
a)
b)
c)
f ( 5) 
f ( 3) 
f ( 1) 
3
y
2
1
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
-7
-8
What points on this graph did you find?
NUCC| Secondary II Math 14
Unit 7.2
7.2 FALLING OBJECTS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: The purpose of this task is to practice the ideas of functions that have been developed;
in this case we will be discussing a quadratic function.
Core Standards Focus Missing: Please find the standard you used to base this lesson.
Launch (Whole Class): As a class, go through the directions in Falling Objects Task 2a.
As a class, go through the directions in Falling Objects Task. 2b.
NUCC| Secondary II Math 15
Unit 7.2
Explore (Individual, small group or pairs): Allow time for the students to finish on their own.
Nothing placed here by author, so I put what might work… needs help.
NUCC| Secondary II Math 16
Unit 7.2
Discuss (Whole Class or Group): Bring class together and discuss their findings from the task.
Clarify any misunderstandings. Nothing placed here by author, so I put what might work…
needs help.
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 17
Unit 7.2
Mathematics Content
Cluster Title: Construct and compare linear, quadratic, and exponential models and solve
problems. (Need the standard that belongs with this lesson.)
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


Use a table to observe that exponential functions grow more quickly than quadratic
functions.
Use a graph to observe that exponential functions grow more quickly than quadratic
functions.
Critical Background Knowledge

Graph quadratic and exponential functions.
Academic Vocabulary
exponential, quadratic, rate of change
Suggested Instructional Strategies


Examine contexts of quadratic vs. exponential functions, comparing values at specified
points.
Use technology to explore, predict, and model. Emphasize appropriate viewing windows.
Skills Based Task:
Problem Task:

Find a quadratic and exponential function that:
 Do not intersect.
 Intersect once.
 Intersect twice.
 Intersect more than twice.
Graph the functions y  x 2 and y  2 x
on the same coordinate axes. Compare
the values of the functions over various
intervals.
Some Useful Websites:
Resource: NCTM Illuminations: The Devil and Daniel Webster
NUCC| Secondary II Math 18
Unit 7.2
Falling Objects
A Practice Understanding Task 2a
Name_____________________________________
Hour___________
Sarah is taking physics. Her sister, Chelsea, is taking physical science. Sarah decided to use
functions to help Chelsea understand one basic idea related to gravity and falling objects. Sarah
explained that, if a ball is dropped from a high place, such as the Tower of Pisa in Italy, then
there is a formula for calculating the distance the ball has fallen. If y, measured in meters, is the
distance the ball has fallen and t, measured in seconds, is the time since the ball dropped, then y
is a function of t, and the relationship can be approximated by the formula y = d(t) = 4.9t2. Here
we name the function d because the outputs are distances.
t(in seconds)
0
1
2
3
t2
0
1
4
9
2
Y = d(t) = 4.9t
0
5
20
a. Fill in the missing values in the table above.
4
5
6
…
…
…
b. Suppose the ball is dropped from a building at least 100 meters high. Measuring from the
top of the building, draw a picture indicating the position of the ball at the times indicated
in your table values.
c. Draw a graph of t versus y for this situation. Should you connect the dots? Explain.
NUCC| Secondary II Math 19
Unit 7.2
d. What is the relationship between the picture (part b) and the graph (part c)?
e. Explain what happens to the speed of the ball as the ball falls. Use your table and your
picture to help you justify your reasoning
f.
What is f(4)? What does this mean?
g. Estimate t such that f(t) = 50. Explain your method. What does it mean?
h.
In this context, y is proportional to t2. Explain what that means. How can you see this in
the table?
NUCC| Secondary II Math 20
Unit 7.2
Fences and Function
A Practice Understanding Task 2b
Name_____________________________________
Hour___________
1. Claire wanted to plant a rectangular garden in her back yard using 30 pieces of fencing that were
given to her by a friend. Each piece of fencing was a vinyl panel 1 yard wide and 6 feet high. Claire
needed to determine the possible dimensions of her garden, assuming that she used all of the fencing
and did not cut any of the panels. She began by placing ten panels (10 yards) parallel to the back side
of her house and then calculated that the other dimension of her garden would be 5 yards, as shown in
the diagram below.
Claire looked at the 10 fencing panels lying on the ground and decided that she wanted to
consider other possibilities for the dimensions of the garden. In order to organize her
thoughts, she let x be the garden dimension parallel to the back of her house measured in
yards, and let y be the other dimension perpendicular to the back of the house measured in
yards. She recorded the first possibility for the dimensions of the garden as follows:
When x = 10, y = 5.
10 yds
Garden
5 yds
House
a. Explain why y must be 5 when x is 10.
b. Make a table showing the possibilities for x and y.
c. Find the perimeter and area of each of the possible gardens you listed in part b. What do you
notice? Explain why this happens.
NUCC| Secondary II Math 21
Unit 7.2
d. Did you consider x = 15 in part b? If x = 15, what must y be? What would the garden look like
if Claire chose x = 15?
e. Can x be 16? What is the maximum possible value for x? Explain.
f.
Write a formula relating the y-dimension of the garden to the x-dimension.
g. Make a graph of the possible dimensions of Claire’s garden.
NUCC| Secondary II Math 22
Unit 7.2
h. What would it mean to connect the dots on your graph? Does connecting the dots make sense for
this context? Explain.
i.
As the x-dimension of the garden increases by 1 yard, what happens to the y-dimension? Does it
matter what x-value you start with? How do you see this in the graph? In the table? In your
formula? What makes the dimensions change together in this way?
Make a graph of the x-dimension compared with the area.
NUCC| Secondary II Math 23
Unit 7.2
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
Sarah is paid $7 per hour in her part-time job at the local Dairy Stop. Let t be the amount time
that she works, in hours, during the week, and let P(t) be her gross pay (before taxes), in dollars,
for the week.
a. Make a table showing how her gross pay depends upon the amount of time she works during the
week.
b. Make a graph illustrating how her gross pay depends upon the amount of time that she works.
Should you connect the dots? Explain.
c. Write a formula showing how her gross pay depends upon the amount of time that she works.
NUCC| Secondary II Math 24
Unit 7.2
d. What is P(9)? What does it mean? Explain how you can use the graph, the table, and the formula
to compute P(9).
e. If Sarah works 11 hours and 15 minutes, what will her gross pay be? Show how you know.
Express the result using function notation.
f.
If Towanda works 4 hours and 50 minutes, what will her gross pay be? Show how you know.
Express the result using function notation.
g. One week Sarah’s gross pay was $42. How many hours did she work? Show how you know.
Another week Sarah’s gross pay was $57.19. How many hours did she work? Show how you know.
NUCC| Secondary II Math 25
Unit 7.3
7.3 BASIC FUNCTION SHAPES
Teacher Notes
Time Frame:
Materials Needed:
Purpose: The purpose of today’s lesson is to introduce and review basic function graph shapes.
Students have seen the following
functions previously:
 Linear 𝑓(𝑥) = 𝑥
 Exponential 𝑓(𝑥) = 2𝑥
 Quadratic 𝑓(𝑥) = 𝑥 2
In Secondary Math II we will be focusing on:






Square root 𝑓(𝑥) = √𝑥
3
Cube Root 𝑓(𝑥) = √𝑥
Piecewise
Step Functions 𝑓(𝑥) = ⟦𝑥⟧
Absolute Value 𝑓(𝑥) = |𝑥|
Quadratics (introduced in Unit 5) 𝑓(𝑥) = 𝑥 2
You can review/introduce the idea of CORE graphs and PARENT functions and how
transformations can be used, if functions happen to be in a certain form. Although this lesson is
more of an umbrella activity, it is used to introduce the different graphs. Students will be able to
identify the “basic shape” of a graph based on the equation.
For example: A square root graph draws a certain shape. (arc, shooting star, eyebrow, etc.). You
know that an equation like 𝑦 = 2√𝑥 + 5 − 1 is going to have that basic shape because of the
square root.
Standards: F.IF.4 Interpret key features of graphs and equations in terms of quantities.
Compare properties of two functions each represented in a different way.
Launch (Whole Class): As a whole class, fill in the notes for the following basic function
shapes and graphs. Get as detailed as you want to during this introduction.
NUCC| Secondary II Math 26
Unit 7.3
Explore (Individual, small group or pairs): Have students complete the Basic Function Shapes
Task 3 to practice the concepts covered in class. Perhaps check student answers or have students
display answers on the board.
NUCC| Secondary II Math 27
Unit 7.3
Discuss (Whole Class or Group): Review the basic function shapes and discuss other possible
function graphs.
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 28
Unit 7.3
Mathematics 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



Distinguish linear, quadratic, and exponential relationships based on equations, tables, and
verbal descriptions.
Given a function in a table or in algebraic or graphical form, identify key features such as xand y-intercepts; intervals where the function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; and end behavior.
Use key features of an algebraic function to graph the function.
Critical Background Knowledge

Graph linear and exponential functions from a table or equation.
Academic Vocabulary
increasing, decreasing, interval, intercept, maximum, minimum, symmetry, end behavior,
quadratic, vertex
Suggested Instructional Strategies




Given key features of a quadratic function, sketch the function by hand.
Use graphing technology to explore and identify key features of a quadratic function.
Compare key features of linear, exponential, and quadratic functions.
Use interval notation or symbols of inequality to communicate key features of graphs.
Skills Based Task:
Find the maximum height of the path of an
arrow modeled by the function
h(t )  162  96t . During what interval is the
arrow going up? Going down? When does it
hit the ground?
Problem Task:
f (t )
time
0
300
5
777.5
10
1010
15
997.5
20
740
25
237.5
Create a situation that could
have produced the given data.
Use appropriate vocabulary
and key features to tell the
story.
Some Useful Websites:
NUCC| Secondary II Math 29
Unit 7.3
Basic Function Shapes and Graphs
Student Notes
Name_____________________________________
Hour___________
Linear
Greatest Integer/Step Function
Quadratic
Exponential
NUCC| Secondary II Math 30
Unit 7.3
Absolute Value
Square Root
Piecewise-Functions
Cube Root Functions
NUCC| Secondary II Math 31
Unit 7.3
Basic Function Shapes
A Develop Understanding Task 3
Name_____________________________________
Hour___________
Based on what you learned in class, sketch the following graphs. They do not have to be
perfect but they need to agree with the equation. Create tables, use calculators, and work
as a group to be as accurate as possible.
1.
f ( x )  3x 1
4. y  x  3
2.
f ( x)  3 x  1
3.
5.
y   x  4
6. y  3x
2
f ( x)  3 x
NUCC| Secondary II Math 32
Unit 7.3
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
Draw a sketch of the following graph descriptions.
1. y  x Linear
2. f ( x )  x Greatest Integer/Step
4.
f  x   x Root
5.
3. y  x AbsoluteValue
f  x   x2 Quadratic/Parabola
6. 𝑦 = 𝑎 𝑥 Exponential
Set
State which basic function shape corresponds to the following equation.
3
7. 𝑦 = 3 √𝑥 − 3
8. 𝑓(𝑥) = −5 + 𝑥
9. 𝑦 = −√𝑥 + 4
10. 𝑓(𝑥) = 7𝑥 + 5
11. 𝑦 = −5|𝑥 + 1| − 4
12. 𝑔(𝑥) = 4𝑥−3
NUCC| Secondary II Math 33
Unit 7.3
Go
Match the name with the equation and the graph by connecting them with lines.
13.
Name
Quadratic
Equation
14.
Linear
y  x2  3
15.
Square Root
f ( x)  2 x
16.
Absolute Value
f ( x)  x  1  2
Graph
y  x 4
NUCC| Secondary II Math 34
Unit 7.4
7.4 DOMAIN AND RANGE
Teacher Notes
Time Frame:
Materials Needed:
Purpose: To review and develop understanding on functions, domain, and range. Students have
been introduced to the idea of domain and range in Math 1. The idea was developed mostly with
tables and finite sets. We will be extending this idea to graphs of all kinds – both graphs of
functions and non-functions. This activity should also introduce the vertical line test for
functions and how to tell if a relation is a function in multiple forms.
Students learned about domain and range in Secondary Math 1. They focus on tables and discrete
tables on an interval. This lesson is meant to build on that knowledge and take it to the level of
graphs.
Vocabulary: At the beginning of the notes, there are “boxes” to review and talk about the
following terms.




Domain
Range
Relation
Function
They should have learned a little bit about these terms in Secondary Math 1.
Core Standards Focus Missing: Please find the standard you used to base this lesson.
Launch (Whole Class): As a whole class, fill in the notes for the following domain and range
functions. Get as detailed as you want to during this introduction.
NUCC| Secondary II Math 35
Unit 7.4
NUCC| Secondary II Math 36
Unit 7.4
Explore (Individual, small group or pairs): Have students complete the Domain and Range
Function Task 4 to practice the concepts covered in class. Perhaps check student answers or have
students display answers on the board. Nothing place here by author, so I put what might work…
needs help.
Discuss (Whole Class or Group): Review the basic function shapes and discuss other possible
function graphs. Nothing place here by author, so I put what might work… needs help.
NUCC| Secondary II Math 37
Unit 7.4
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 38
Unit 7.4
NUCC| Secondary II Math 39
Unit 7.4
Mathematics Content
Cluster Title: Construct and compare linear, quadratic, and exponential models and solve
problems. (Need the standard that generated this lesson.)
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


Use a table to observe that exponential functions grow more quickly than quadratic
functions.
Use a graph to observe that exponential functions grow more quickly than quadratic
functions.
Critical Background Knowledge

Graph quadratic and exponential functions.
Academic Vocabulary
exponential, quadratic, rate of change
Suggested Instructional Strategies


Examine contexts of quadratic vs. exponential functions, comparing values at specified
points.
Use technology to explore, predict, and model. Emphasize appropriate viewing windows.
Skills Based Task:
Problem Task:

Find a quadratic and exponential function that:
 Do not intersect.
 Intersect once.
 Intersect twice.
 Intersect more than twice.
Graph the functions y  x 2 and y  2 x
on the same coordinate axes. Compare
the values of the functions over various
intervals.
Some Useful Websites:
Resource: NCTM Illuminations: The Devil and Daniel Webster
NUCC| Secondary II Math 40
Unit 7.4
Domain and Range Functions
Student Notes
Name_____________________________________
Domain –
Range –
Relation –
Function –
Hour___________
NUCC| Secondary II Math 41
Unit 7.4
Examples 1-6: Find the domain and range of the following relations and determine if they are
functions.
1.
(1,3),(2,5),(1,0),(3, 4)
2.
Domain:
Domain:
Range:
Range:
Function?
Function?
3.
4.
Domain:
Domain:
Range:
Range:
Function?
Function?
NUCC| Secondary II Math 42
Unit 7.4
5.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
6.
y
-4
-3
-2
-1
y
5
5
4
4
3
3
2
2
1
1
-5
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
-1
x
x
1
2
3
4
5
-5
-4
-3
-2
-1
-1
1
2
3
4
5
-2
-2
-3
-3
Domain:
Domain:
Range:
Range:
Function?
Function?
NUCC| Secondary II Math 43
Unit 7.4
Domain and Range Functions
A Develop Understanding Task 4
Name_____________________________________
Hour___________
With a partner – or in a group, decide if the following graphs are functions. Then,
determine: a) domain, b) range, and c) Function?
1.
2.
3.
a)
a)
a)
b)
b)
b)
c)
c)
c)
4.
5.
6.
a)
a)
a)
b)
b)
b)
c)
c)
c)
NUCC| Secondary II Math 44
Unit 7.4
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
State the domain and range:
1. {(0, 3), (2, 4), (6, −2), (2, −3)}
Domain:
2. {(−1, 1), (−2, 2), (−3, 3), (1, −1)}
Domain:
Range:
Range:
State the domain and range, then determine if the following relations are functions.
3.
4.
x
y
0
-5
2
-4
2
0
5.
-8 -1
State the domain and range:
6.
7.
NUCC| Secondary II Math 45
Unit 7.4
Set
8.
9.
Domain:
Domain:
Range:
Range:
Function?
Function?
10.
11.
Domain:
Domain:
Range:
Range:
Function?
Function?
NUCC| Secondary II Math 46
Unit 7.4
12.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
13.
y
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
5
3
4
2
3
1
2
x
1
x
-3
-2
-1
1
2
-3
-2
-1
1
2
3
-1
3
-1
-2
-2
-3
Domain:
Domain:
Range:
Range:
Function?
Function?
14.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
15.
y
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
3
3
2
2
1
1
x
x
-3
-3
-2
-1
1
2
3
-2
-1
1
2
3
-1
-1
-2
-2
-3
-3
Domain:
Domain:
Range:
Range:
Function?
Function?
NUCC| Secondary II Math 47
Unit 7.4
Go
16.
Tara’s car travels about 25 miles on one gallon of gas. She has between 10 and 12 gallons
of gas in the tank. Find the reasonable domain and range values.
17.
Sal and three friends plan to bowl one or two games each. Each game cost $2.50. Find the
reasonable domain and range values.
18.
Which of the following points, when deleted from the coordinate grid, will result in a
relationship that represents a function?
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
A. Point A
B. Point B
C. Point D
D. Point F
5
F
4
3
A
B
2
1
-5 -4 -3 -2 -1
-1
x
1
E
2
3
4
5
-2
C
-3
D
-4
-5
NUCC| Secondary II Math 48
Unit 7.4
19-22. Draw two functions and two non-functions on the graphs below. Determine their
domains and ranges.
19.
20.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
7
7
6
6
5
5
4
4
3
3
2
2
1
-7
-6
-5
-4
-3
-2
1
x
-1
1
2
3
4
5
6
-7
7
-2
1
-4
-4
-5
-5
-6
-6
-7
-7
y
7
6
6
5
5
4
4
3
3
2
2
-1
2
3
4
5
6
7
2
3
4
5
6
7
y
1
x
1
x
-1
-3
1
-3
-2
-3
22.
-4
-3
-2
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
-5
-4
-2
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
-6
-5
-1
21.
-7
-6
-1
7
y
-7
-6
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
2
3
4
5
6
7
NUCC| Secondary II Math 49
Unit 7.5
7.5 PIECEWISE FUNCTIONS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: This task will develop the idea of a piecewise function. The task is best done in groups
of 2-4 students. It is designed to carry students through the thinking of piecewise functions. At
the end of the task, there is an example (#5) might be best done as a class. The next example (#6)
can be done as a class or in pairs/groups/individually. There are 3 Tasks and nothing in any of
the tasks have a problem #6,… needs help.
Standards: F.IF.4 . . . 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; maximums and minimums, symmetries, end behavior and
periodicity.)
Launch (Groups): Split students into groups. Have them work through the first page of the task
and answer the questions. It may be helpful to then review what students discovered before they
move on. Look for words like: linear, quadratic, function, etc. If some groups get far ahead of
others you can ask review questions like: What is the domain? What is the range? When is the
function increasing? When is it decreasing? etc.
Task 5a
NUCC| Secondary II Math 50
Unit 7.5
Task 5b
NUCC| Secondary II Math 51
Unit 7.5
Task 5c
Explore (Individual, small group or pairs): Complete example 5 as a class. Only in task 5b.
Discuss (Whole Class or Group): Discuss example 6. I don’t see example 6 in any task.
Assignment: Ready, Set, Go! Assign after Task 5b. Nothing place here by author, so I put what
might work… needs help.
NUCC| Secondary II Math 52
Unit 7.5
NUCC| Secondary II Math 53
Unit 7.5
Functions Quiz
NUCC| Secondary II Math 54
Unit 7.5
NUCC| Secondary II Math 55
Unit 7.5
Mathematics 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



Distinguish linear, quadratic, and exponential relationships based on equations, tables, and
verbal descriptions.
Given a function in a table or in algebraic or graphical form, identify key features such as xand y-intercepts; intervals where the function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; and end behavior.
Use key features of an algebraic function to graph the function.
Critical Background Knowledge

Graph linear and exponential functions from a table or equation.
Academic Vocabulary
increasing, decreasing, interval, intercept, maximum, minimum, symmetry, end behavior,
quadratic, vertex
Suggested Instructional Strategies




Given key features of a quadratic function, sketch the function by hand.
Use graphing technology to explore and identify key features of a quadratic function.
Compare key features of linear, exponential, and quadratic functions.
Use interval notation or symbols of inequality to communicate key features of graphs.
Skills Based Task:
Find the maximum height of the path of an
arrow modeled by the function
h(t )  162  96t . During what interval is the
arrow going up? Going down? When does it
hit the ground?
Problem Task:
f (t )
time
0
300
5
777.5
10
1010
15
997.5
20
740
25
237.5
Create a situation that could
have produced the given data.
Use appropriate vocabulary
and key features to tell the
story.
Some Useful Websites:
NUCC| Secondary II Math 56
Unit 7.5
Getting Ready for a Pool Party
A Develop Understanding Task 5a
Name_____________________________________
Hour___________
Marie has a small pool full of water that needs to be emptied and cleaned, then refilled for a pool
party. During the process of getting the pool ready, Marie did all of the following activities, each
during a different time interval.
Removed water with a single bucket
Filled the pool with a hose (same rate as
emptying the pool)
Cleaned the empty pool
Drained water with a hose (same rate as
filling the pool)
Marie and her two friends removed the water Took a break
with three buckets
1. Sketch a possible graph showing the height of the water level in the pool over time. Be sure
to include all of the activities Marie did to prepare the pool for the party. Remember that only
one activity happened at a time. Think carefully about how each section of your graph will
look, labeling where each activity occurs.
NUCC| Secondary II Math 57
Unit 7.5
2. Create a story connecting Marie’s process for emptying, cleaning, and then filling the pool to
the graph you have created. Do your best to use appropriate math vocabulary.
3. Does your graph represent a function Why or why not? Would all graphs created for this
reason represent a function?
NUCC| Secondary II Math 58
Unit 7.5
Piecewise Functions
A Develop Understanding Task 5b
Name_____________________________________
Hour___________
1. Look for patterns in the function below. Would it be possible to mistakenly judge the nature
of this function if you based your analysis on just one part? __________________________
Explain.
2.
a) Let’s assume that someone described the function above as linear. Explain how is their
statement is partially correct.
b) Explain how that same statement is also false.
c) Explain how you could accurately identify the parts of the function that are linear.
NUCC| Secondary II Math 59
Unit 7.5
3. Let’s look at the individual pieces of the previous piecewise function. Identify each type
of function and provide its equation.
a.
10
8
function type
6
4
f(x) =
2
2
b.
4
6
8
10
12
14
16
18
equation
20
10
8
function type
6
4
f(x) =
2
2
c.
4
6
8
10
12
14
16
18
equation
20
10
8
function type
6
4
f(x) =
2
2
4
6
8
10
12
14
16
18
20
equation
NUCC| Secondary II Math 60
Unit 7.5
4. Only a piece of each of the functions above was used to form the piecewise function in
question 1. Using numbers and symbols, identify the piece of each function that is used.
a. ____________________
b. ____________________
c. ____________________
Piecewise functions are functions that cannot be represented by just one equation. Each equation
corresponds to a different part of the domain.
Let’s practice with more piecewise functions.
5. (Complete as a class)
Write the piecewise function for the figure at right. You should identify the domain that
corresponds with each equation.
f(x) =
NUCC| Secondary II Math 61
Unit 7.5
A Taxing Situation
A Solidify Understanding Task 5c
Name_____________________________________
Hour___________
1. Piecewise functions are used to describe a wide variety of data sets. One good example of a
piecewise function is income tax. The 2007 Federal Tax Rate Schedule for a single person
filing taxes is as follows.
Taxable Income
Tax
$0 - $7,825
10%
$7,825 - $31,850
782.50 plus 15% of amount over $7,825
$31,850 - $77,100
$4,386.25 plus 25% of amount over $31,850
$77,100 - $160,850
$15,698.75 plus 28% of the amount over $77,100
$160,850 - $349,700
$39,148.75 plus 33% of the amount over $160,850
$349,700 +
$101,469.25 plus 35% of the amount over $349,700
a. Write the equation for a piecewise function, c, that would accurately represent the income
tax for a single person in the United States according to this current tax plan.
b. Compare the salaries and the taxes owed by each of these single US taxpayers in 2007.
Include in your discussion the percent of their income they retain after taxes.
1. A teacher who made $36,000
2. An attorney who made $80,000
3. A dental assistant who made $28,000
4. A radiologist who made $200,000
5. A professional athlete who made $1.5 million
NUCC| Secondary II Math 62
Unit 7.5
c. Graph the function. Be sure to label your axis. Is this a continuous or discontinues
function? Explain how you know.
2. Jacob Jones has made a proposal for a flat tax for US taxpayers. He has proposed that every
taxpayer should pay 17% of their taxable income in taxes.
a. Write an equation for the function, f, to represent Mr. Jones’s proposal.
b. Graph this equation on the same coordinate plane as #1c.
c. At what income level would a flat tax be the same as our current tax rate? Explain.
NUCC| Secondary II Math 63
Unit 7.5
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
Part I. Find the domain and range for each piecewise function. Then, evaluate the graph at the
specified domain value.
1.

2x  1
f x    2

x  3
x 1
x 1
Domain:________________
Range:_________________
f 2  
f 6  
f 1  
2.
2x  1
f x   
5x  4
x 2
x 2
Domain:________________
Range:_________________
f 4  
f 8  
f 2  
NUCC| Secondary II Math 64
Unit 7.5
3.
x 2  1

f x   2x  1
3

x 0
0x 5
x 5
Domain:________________
Range:_________________
f 2  
f 0  
f 5  
4.
2

x
f x    2

x  4
x 0
x 0
Domain:________________
Range:_________________
f 4  
f 0  
f 3 
NUCC| Secondary II Math 65
Unit 7.5
5.
5
f x   
2x  3
x  3
x  3
Domain:________________
Range:_________________
f 4  
f 0  
f 3 
Set
Write a piecewise function for each graph. Also, give the domain and range.
6.
NUCC| Secondary II Math 66
Unit 7.5
7.
Go
Answer the following questions for the given graph:
8.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
Relative Maxima/Minima:
y
5
4
Intervals Increasing:
3
2
Intervals Decreasing:
1
x
-3
-2
-1
1
-1
2
3
x-intercepts:
-2
y-intercepts:
NUCC| Secondary II Math 67
Unit 7.5
9.
Relative Maxima/Minima:
Intervals Increasing:
Intervals Decreasing:
x-intercepts:
y-intercepts:
10.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
y
Relative Maxima/Minima:
3
2
Intervals Increasing:
1
x
Intervals Decreasing:
-3
-2
-1
1
2
3
-1
-2
x-intercepts:
-3
y-intercepts:
NUCC| Secondary II Math 68
Unit 7.5
Name ________________________________________________
Period _________
Date _____________________
Functions Quiz
1. Explain the difference between a relation and a function.
2. Explain difference between domain and range.
For numbers 3-6, determine whether each relation is a function. Write yes or no.
D
3.
R
2
3.________________
21
25
8
30
4.
D
D
4.________________
5
105
10
110
15
5.
x
-3
-1
0
2
3
y
0
-1
0
-2
4
5. _______________
NUCC| Secondary II Math 69
Unit 7.5
6.
6. _______________
y
x
Graph the relation and answer questions 7-9.
𝐿 = {(−4, −1)(4,0)(0,3)(2,0)}
7. Domain:
8. Range:
9. Function?
Write the equation, draw the graph, and then answer questions.
Name: Quadratic/Parabola
10.
Write your own Equation:
y
11. Domain _____________________________
12. Range ______________________________
NUCC| Secondary II Math 70
Unit 7.5
Given the following graph and equation, find:
13. f (5) 
14. f ( 2) 
15. f (0) 
16. f ( 4) 
y  x2  8
NUCC| Secondary II Math 71
Unit 7.6
7.6 REVIEWING TRANSFORMATIONS
Teacher Notes
Time Frame:
Materials Needed: Task Worksheet and pencils
Background Knowledge Needed: Evaluating basic absolute value expressions; plotting points
on a coordinate plane
Purpose: This task is going to review transformations (learned in the quadratics unit) and apply
the concepts to absolute value graphs.
Standards: F.IF.4 . . . sketch graphs showing key features given a verbal description of the
relationship. . .
F.IF.7 – …Graph absolute value functions.
Launch (Whole Class): Review Part A on the task worksheet with the students.
Explore (Individual, small group or pairs): Have students work through part B on the task
worksheet. They will be discovering absolute value transformations and building on the
background knowledge they used for quadratics. Pay particular attention to:

𝑦 = −|𝑥| 𝑎𝑛𝑑 𝑦 = |−𝑥|.
Be sure to remind students to make tables as the directions ask.
NUCC| Secondary II Math 72
Unit 7.6
Discuss (Whole Class or Group): Go through part C as a whole class. Here we will tie together
piecewise (learned previously in this unit) and absolute value graphs.
NUCC| Secondary II Math 73
Unit 7.6
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 74
Unit 7.6
Mathematics 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



Distinguish linear, quadratic, and exponential relationships based on equations, tables, and
verbal descriptions.
Given a function in a table or in algebraic or graphical form, identify key features such as xand y-intercepts; intervals where the function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; and end behavior.
Use key features of an algebraic function to graph the function.
Critical Background Knowledge

Graph linear and exponential functions from a table or equation.
Academic Vocabulary
increasing, decreasing, interval, intercept, maximum, minimum, symmetry, end behavior,
quadratic, vertex
Suggested Instructional Strategies




Given key features of a quadratic function, sketch the function by hand.
Use graphing technology to explore and identify key features of a quadratic function.
Compare key features of linear, exponential, and quadratic functions.
Use interval notation or symbols of inequality to communicate key features of graphs.
Skills Based Task:
Find the maximum height of the path of an
arrow modeled by the function
h(t )  162  96t . During what interval is the
arrow going up? Going down? When does it
hit the ground?
Problem Task:
f (t )
time
0
300
5
777.5
10
1010
15
997.5
20
740
25
237.5
Create a situation that could
have produced the given data.
Use appropriate vocabulary
and key features to tell the
story.
Some Useful Websites:
NUCC| Secondary II Math 75
Unit 7.6
Mathematics 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.
b. Graph square root, cube root, and piecewise-defined functions, including step functions
and absolute value functions.
Concepts and Skills to Master



Graph quadratic functions expressed in various forms by hand.
Use technology to model quadratic functions, when appropriate.
Graph and find key features of square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions.
Critical Background Knowledge



Graph linear and exponential functions showing key features (I.2.F.IF.7).
Interpret key features of a graph (II.2.F.IF.4).
Identify and use transformation of functions (II.2.F.BF.3).
Academic Vocabulary
piecewise, step function, axis of symmetry, absolute value, x
Suggested Instructional Strategies


Find real-world contexts that motivate the use of step functions.
Compare the absolute value function to its piecewise definition.
Skills Based Task:
Graph the function and identify the key
features.
 x  2 x 1
f ( x)   2
x 3 x 1
Problem Task:
Write and graph three different functions whose
minimum is (-1,5).
Some Useful Websites:
NUCC| Secondary II Math 76
Unit 7.6
Absolute Value Graphs & Transformations
A Develop Understanding Task 6
Name_____________________________________
Hour___________
Part A
Graph and label x and y intercepts (using ordered pairs) on the graph.
1. 𝑦 = 𝑥
2. 𝑓(𝑥) = 𝑥 − 3
3. 𝑔(𝑥) = 𝑥 + 3
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
y
5
4
4
3
3
3
2
1
-4
-3
-2
5
4
2
-5
y
x
-1
1
2
3
4
2
1
5
-5
-1
-2
-4
-3
-2
-1
y
1
x
1
2
3
4
5
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
2
3
4
5
-3
-4
-5
4. Explain your knowledge about absolute value using words.
5. Using past knowledge to create new knowledge, try graphing the following function:
𝑦 = |𝑥|
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
y
5a. Explain your reasoning for the graph you
created.
4
3
2
1
-5
-4
-3
-2
-1
x
1
-1
-2
-3
2
3
4
5
5b. Please justify this method (using another
method).
-4
-5
NUCC| Secondary II Math 77
Unit 7.6
Part B
Now try graphing the following absolute value equations. Create your own table to justify
values.
6. 𝑓(𝑥) = |𝑥 − 3|
𝒙
𝒇(𝒙)
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𝒙
𝒇(𝒙)
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y
6
6
5
5
4
4
3
3
2
2
1
-6
𝑔(𝑥) = |𝑥 + 3|
7.
-5
-4
-3
-2
1
x
-1
1
2
3
4
5
6
y
-6
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
8. Compare the graphs from problems 5, 6, and 7. Make a conjecture about functions that come
in the form 𝑦 = |𝑥 − ℎ|.
Use a table to create the following graph.
9. 𝑦 = |𝑥| + 3
𝒙
6
𝒇(𝒙)
9a. Explain the difference between this
graph and the graph of
𝑦 = |𝑥|.
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y
5
4
3
2
1
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
NUCC| Secondary II Math 78
Unit 7.6
Now try graphing the following absolute value equations. Create a table to justify values.
10. 𝑦 = |𝑥| + 1
𝒙
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𝒇(𝒙)
11. 𝑓(𝑥) = |𝑥| − 2
6
6
5
5
4
4
3
3
2
-5
-4
-3
-2
𝒇(𝒙)
y
2
1
-6
𝒙
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y
1
x
-1
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
12. Make a conjecture about functions that come in the form: 𝑦 = |𝑥| + 𝑘.
Use a table to create the following graph:
13. 𝑦 = −|𝑥|
𝒙
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6
𝒇(𝒙)
13b. Explain the difference
between this graph and the
graph of 𝑦 = |𝑥|.
y
5
4
3
2
1
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
NUCC| Secondary II Math 79
Unit 7.6
14. 𝑦 = −|𝑥| + 1
15. 𝑓(𝑥) = −|𝑥| − 2
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6
y
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
y
6
1
x
-1
1
2
3
4
5
6
-6
-5
-4
-3
-2
x
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
16. Explain what happens to the graph if the absolute value is multiplied by a negative.
Graph the following:
3
17. 𝑦 = 2|𝑥|
𝒙
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6
y
𝒇(𝒙)
𝒙
18. 𝑓(𝑥) = 2 |𝑥|
𝒇(𝒙)
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6
y
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
x
-1
1
2
3
4
5
1
x
6
-1
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
1
𝒙
19. 𝑦 = 2 |𝑥|
𝒇(𝒙)
M at h Com poser 1. 1. 5
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6
y
5
4
3
2
1
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
NUCC| Secondary II Math 80
Unit 7.6
20. What is the effect on the graph of multiplying the absolute value by a number? Does it matter
what the number is?
Part C
Absolute value functions can be written without absolute value bars if they are separated into
two equal parts.
Example:
𝑦 = |𝑥| can be written as two different linear functions.
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6
y
5
4
3
2
1
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
21. Sketch the graph.
22. Where in the domain do you think the graph will change from one function to the next?
21. If you answered 𝑥 = 0 for number 22, thank you. Now, let’s break it apart and write the
equation:
𝑓(𝑥) =
NUCC| Secondary II Math 81
Unit 7.6
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
1.
What will be the equation of the resulting graph if the graph of y = ∣x∣ is shifted 4 units
down?
A) y=∣x+4∣
C) y=∣x∣+4
B) y=∣x - 4∣
D) y=∣x∣ - 4
2.
What will be the equation of the resulting graph if the graph of y=∣x∣ is shifted 3 units
up?
A) y= ∣x - 3∣
C) y=∣x∣+3
B) y=∣x+3∣
D) y=∣x∣ - 3
3.
What will be the equation of the resulting graph if the graph of y=∣x∣ is shifted 3 units to
the left?
A) y=∣x∣-3
C) y=∣x+3∣
B) y=∣x∣+3
D) y=∣x-3∣
4.
What will be the equation of the resulting graph if the graph of y=∣x∣ is shifted 4 units to
the right?
A) y=∣x∣+4
C) y=∣x+4∣
B) y=∣x∣-4
D) y=∣x-4∣
5.
What will be the equation of the resulting graph if the graph of y=∣x∣ is reflected in the xaxis?
A) y=∣x∣-1
C) y=-∣x∣
∣𝑥∣
B) y= 2
D) y=2∣x∣
6.
When compared to the graph of y=∣x∣, the graph of y=∣x∣ - 8 is
A) Shifted to the left 8 units
C) shifted down 8 units
B) Shifted up 8 units
D) shifted to the right 8 units
7.
When compared to the graph of y=∣x∣, the graph of y=∣x+5∣ is
A) Shifted up 5 units
C) shifted down 5 units
B) Shifted to the right 5 units
D) shifted to the left 5 units
NUCC| Secondary II Math 82
Unit 7.6
Set
Graph each absolute value equation. State the domain (D) and range (R) of each:
8.
𝑦 = |−𝑥 − 3|
f  x  x  3
9.
D:
D:
R:
R:
10.
𝑓(𝑥) = −|𝑥 + 3|
11.
D:
1
2
𝑦 = |𝑥 − 4|
D:
R:
R:
12.
D:
f  x  1 x
f  x  4  x
13.
D:
R:
R:
NUCC| Secondary II Math 83
Unit 7.6
Graph each of the following absolute value equations on the graph provided.
14.
𝑦 = 4|2 − 𝑥|
15.
y  x2 4
16.
y  4 x3
17.
f  x  3  5  x
18.
y  2 x
19.
f  x  2  x
NUCC| Secondary II Math 84
Unit 7.6
Go!
Write your own 5 absolute value equations and graph them. YOUR EQUATIONS MUST
CONTAIN AT LEAST 3 TRANSFORMATIONS OF THE PARENT GRAPH. (i.e.,
Left/right, up/down, shrink/stretch, reflections)
20.
21.
22.
23.
24.
NUCC| Secondary II Math 85
Unit 7.7
7.7 PARKING DECK PANDEMONIUM
Teacher Notes
Time Frame:
Materials Needed:
Purpose: The purpose of this task is to introduce the idea of a step function. (greatest integer,
ceiling function, floor function, etc.) Students will work through a four part task about parking
garage rates and make connections through tables and graphs.
 Be sure to introduce step function notation. 𝒇(𝒙) = ‖𝒙‖ 𝒐𝒓 𝒇(𝒙) = ⟦𝒙⟧
Standards: F.IF.7 – …graph functions including step functions. Analyze key features of the
graph by hand in simple cases and using technology for more complicated cases.
F.IF.9 - ….compare properties of two functions each represented a different way.
Launch (Pairs/Groups): Have students work through Part A of the task. After most students are
finished go through and review what was discovered. Students may have trouble graphing at first
because it is very different from anything else that they have graphed up to this point.
Explore (Individual, small group or pairs): Have students work through the rest of the task.
They should be making tables and drawing graphs. They will use this information to compare
quantities and the differences between Part A, B and C’s problems.
NUCC| Secondary II Math 86
Unit 7.7
Discuss (Pairs/Groups): Review with students the differences between the step functions and
others.
NUCC| Secondary II Math 87
Unit 7.7
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 88
Unit 7.7
Mathematics 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.
c. Graph linear and quadratic functions and show intercepts, maxima, and minima.
d. Graph square root, cube root, and piecewise-defined functions, including step functions
and absolute value functions.
Concepts and Skills to Master



Graph quadratic functions expressed in various forms by hand.
Use technology to model quadratic functions, when appropriate.
Graph and find key features of square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions.
Critical Background Knowledge



Graph linear and exponential functions showing key features (I.2.F.IF.7).
Interpret key features of a graph (II.2.F.IF.4).
Identify and use transformation of functions (II.2.F.BF.3).
Academic Vocabulary
piecewise, step function, axis of symmetry, absolute value, x
Suggested Instructional Strategies


Find real-world contexts that motivate the use of step functions.
Compare the absolute value function to its piecewise definition.
Skills Based Task:
Graph the function and identify the key
features.
 x  2 x 1
f ( x)   2
x 3 x 1
Problem Task:
Write and graph three different functions whose
minimum is (-1,5).
Some Useful Websites:
NUCC| Secondary II Math 89
Unit 7.7
Mathematics 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 intercepts maxima and minima, rates of change, and end behavior of two quadratic
functions, where one is represented algebraically, graphically, numerically in tables, or by
verbal descriptions, and the other is modeled using a different representation.
Critical Background Knowledge
 Find intercepts, rates of change, end behavior, extreme values, and symmetry of quadratic
functions (II.2.F.IF.4).
Academic Vocabulary
intercepts, rates of change, end behavior, extreme values, symmetry
Suggested Instructional Strategies



Use technology to transition between forms of a function.
Match functions expressed using different representations that have the same properties.
Compare two functions expressed in different representations. Ask: Which is growing at a
faster rate? Which one has a higher initial value? Why does it increase faster than the other?
How do you know?
Skills Based Task:
Which has a greater average rate of change
over the interval [5,10]?
f ( x )  x 2  4 or
Time f(t)
0
300
5
777.5
10
1010
15 997.5
20
740
25 237.5
Some Useful Websites:
Problem Task:
Represent two quadratic functions with a
minimum of (0,2), one expressed in function
notation and the other in a table.
NUCC| Secondary II Math 90
Unit 7.7
Parking Deck Pandemonium
A Develop Understanding Task 7
Name_____________________________________
Hour___________
Part A
Directions: The fee schedule at parking decks is often modeling using a step function. Let’s look
at a few different parking deck rates to see the step functions in action. (Most parking decks have
a maximum daily fee. However, for our exploration, we will assume that this maximum does not
exist.)
1. As you drive through town, Pete’s Parking Deck advertises free parking up to the first hour
(i.e., the first 59 minutes). Then, the cost is $1 each additional hour or part of an hour. (If you
park for 1 ½ hours, you owe $1; 2 hours costs $2.)
a. Make a table listing the fees for parking at Pete’s for up to 5 hours. Be sure to include
some non-integer values. Then draw the graph that illustrates the fee schedule at Pete’s.
b. Use your graph to determine the fee, if you park for 3 ½ hours. What is the fee if you
park 3 hours, 55 minutes? What is the fee for 4 hours, 5 minutes?
c. What are the x- and y-intercepts of this graph? What do they represent?
NUCC| Secondary II Math 91
Unit 7.7
d. What do you notice about the time and the corresponding fees? Make a conjecture about
the fee, if you were to park at Pete’s Parking Deck for 10 ½ hours (assuming no
maximum fee). Similar to the other situations explored in this unit, this graph could be
represented by a piecewise function.
e. Write a piecewise function to model the fee schedule at Pete’s Parking Deck.
Part B
2. Paula’s Parking Deck is down the street from Pete’s. Paula recently renovated her deck to
make the parking spaces larger, so she charges more per hour than Pete. Paula’s Parking
Deck offers free parking up to the first hour (i.e., the first 59 minutes). Then, the cost is $2
each additional hour or part of an hour. (If you park for 1 ½ hours, you owe $2.)
a. Graph the fee schedule for Paula’s Parking Deck for up to the first 5 hours.
b. How does this graph compare with the graph of Pete’s Parking Deck? To what graphical
transformation does this change correspond?
NUCC| Secondary II Math 92
Unit 7.7
c. If you were to connect the left endpoints of the steps in 1a, what would be the equation of
the resulting function? If you connected the left endpoints of the steps in 2a, what would
be the equation of the resulting function? How do these answers relate to your answer in
2b?
Part C
3. Pablo’s Parking Deck is across the street from Paula’s deck. Pablo decided to make his fee
schedule even more straightforward than Pete’s and Paula’s. Rather than provide any free
parking, Pablo charges $1 for each 0 – 59 minutes. (If you park for 59 minutes, you owe $1;
if you park for 1 hour, you owe $2; etc.)
a. Graph the fee schedule for Pablo’s Parking Deck for up to the first 5 hours.
NUCC| Secondary II Math 93
Unit 7.7
b. How does this graph compare with the graph of Pete’s Parking Deck? To what two
different graphical transformations does this change correspond?
c. Write the function, h, in terms of the greatest integer function, that represents this graph.
(What are the two different forms that this function could take?)
Part D
4. Padma’s Parking Deck is the last deck on the street. To be a bit more competitive, Padma
decided to offer parking for each full hour at $1/hour. (If you park for 59 minutes or exactly
one hour, you owe $1; if you park for up to and including 2 hours, you owe $2.)
a. Graph the fee schedule for Padma’s Parking Deck for up to the first 5 hours.
NUCC| Secondary II Math 94
Unit 7.7
b. To which of the graphs of the other parking deck rates is this graph most similar? How
are the graphs similar? How are they different?
NUCC| Secondary II Math 95
Unit 7.7
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
Evaluate the following greatest integer expressions.
1. ⟦7.1⟧
2. ⟦1.8⟧
⟦−6.8⟧
4.
5. ⟦−2.1⟧
3.
⟦𝜋⟧
6.
⟦0⟧
Solve the following equations for x and write the answers in interval notation.
2𝑥
7. ⟦ 7 ⟧ = 1
8. ⟦3𝑥⟧ = 12
Set
Using what you learned about translations/transformations of the form: y  a x  h  k and
y  a( x  h )2  k , graph the following by hand and check your answer on your calculator.
𝑓(𝑥) = ⟦𝑥⟧ + 2
9.
10. 𝑔(𝑥) = ⟦𝑥 + 2⟧
M at h Com poser 1. 1. 5
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6
y
6
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
-1
y
1
x
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
Explain the shift in each graph and how they differ.
NUCC| Secondary II Math 96
Unit 7.7
12. 𝑔(𝑥) = ⟦2𝑥⟧
11. 𝑓(𝑥) = 2⟦𝑥⟧
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6
y
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
1
x
-1
y
6
1
2
3
4
5
6
-6
-5
-4
-3
-2
x
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
Explain the dilation in each graph and how they differ.
14. 𝑔(𝑥) = ⟦−𝑥⟧
13 𝑓(𝑥) = −⟦𝑥⟧
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6
y
6
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
-1
1
x
1
2
3
4
5
6
y
-6
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
Explain the relection in these graphs and how they differ.
NUCC| Secondary II Math 97
Unit 7.7
Go!
Prior to September 2000, taxi fares from Washington DC to Maryland were described as follows:
$2.00 up to and including ½ mile, $0.70 for each additional ½ mile incremement.
15. Desribe the independent and dependent variables and explain your choices.
16. Graph the fares for the first 2 miles. Make sure to label the axes.
17. Write the piecewise function for 0 to 2 miles.
18. Discuss why this step function is different from the greatest integer parent function
𝑓(𝑥) = ⟦𝑥⟧
NUCC| Secondary II Math 98
Unit 7.8
7.8 APPLESAUCE STOCK
Teacher Notes
Time Frame:
Materials Needed:
Purpose: The purpose is to help students remember how to analyze and graph linear, exponential and
quadratic functions, then analyze how they are related and specifically, how the compare when
increasing.
Standards: 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.
Launch (Whole Class): Give the students the task; make sure they understand that they are
looking at each of the options, comparing them to see which one is better.
Explore (Pairs/Groups): Let the students read through the problem and think about the options
for a couple of minutes, then let them work in partners or groups to discuss the options and how
they will determine which one is better. Students should be able to complete the chart.
NUCC| Secondary II Math 99
Unit 7.8
Discuss (Whole Class): The students should be able to identify that they are all increasing at
different rates, and then should use the info in the table to produce equations and graphs.
Option 1
0
100 +100
200 +100
300 +100
400 +100
500 +100
600 +100
700 +100
800 +100
900 +100
Since 1st
differences are
constant, it is a
linear equation:
y = 100x
0
1
2
3
4
5
6
7
8
9
Option 2
0
0
1 25
+25
2 100
+75
+50
3 225
+125 +50
4 400
+175 +50
5 625
+225 +50
6 900
+275 +50
7 1225 +325 +50
8 1600 +375 +50
9 2025 +425 +50
Since 1st differences are
not constant, but 2nd
differences are, it is a
quadratic equation
y = 25x2
Option 3
5
10
+5
20
+10
*2
40
+20
*2
80
+40
*2
160
+80
*2
320
+160
*2
640
+320
*2
1280 +640
*2
2560 +1280
*2
st
Since 1 differences are
similar to original values,
the multiplier is 2, and it is
an exponential equation
y = 5(2)x
0
1
2
3
4
5
6
7
8
9
From the values and the graph, students should be able to see that Option 1 is the greatest until
x=4, then Option 2 has a greater value until x >8, then Option 3 is greater, and continues to grow
rapidly.
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 100
Unit 7.8
Mathematics 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


Use a table to observe that exponential functions grow more quickly than quadratic
functions.
Use a graph to observe that exponential functions grow more quickly than quadratic
functions.
Critical Background Knowledge

Graph quadratic and exponential functions.
Academic Vocabulary
exponential, quadratic, rate of change
Suggested Instructional Strategies


Examine contexts of quadratic vs. exponential functions, comparing values at specified
points.
Use technology to explore, predict, and model. Emphasize appropriate viewing windows.
Skills Based Task:
Problem Task:

Find a quadratic and exponential function that:
 Do not intersect.
 Intersect once.
 Intersect twice.
 Intersect more than twice.
Graph the functions y  x 2 and y  2 x
on the same coordinate axes. Compare
the values of the functions over various
intervals.
Some Useful Websites:
Resource: NCTM Illuminations: The Devil and Daniel Webster
NUCC| Secondary II Math 101
Unit 7.8
Applesauce Stock
A Solidify Understanding Task 8
Name_____________________________________
Hour___________
What stock option should he choose?
Tommy has the rare opportunity to purchase some stock of a very promising technology
company, Applesauce. Applesauce has the legal rights to take all developed technology and
“mash” them together. He has $100 to invest and then has the choice of three options. The
options’ returns are shown in the table below.
Option 1
1 year
2
3
4
5
6
Option 2
100
200
300
400
1 year
2
3
4
5
6
Option 3
25
100
225
400
1 year
2
3
4
5
6
10
20
40
80
Work Space:
NUCC| Secondary II Math 102
Unit 7.8
1. What questions would you need to ask Tommy before you can give him his desired advice?
2. Based on the information in the table, what option would you choose and why?
3. What is your choice dependent on?
4. Explain the terms for which each option is the better choice.
NUCC| Secondary II Math 103
Unit 7.8
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
Graph the given equations:
1. y = 2x – 4
2.
y = -3x + 2
3. 3x – 4y = 12
4. y = -x2
5.
y = x2 + 2x + 5
6
7. y = 3x
8.
y = 2 (2)x
9. y = 2x+2
y = 2(x – 4)2 + 1
NUCC| Secondary II Math 104
Unit 7.8
Set
Graph the system of equations:
10. y = -2x + 5
11. y = 1/5x + 2
y = 2x + 1
y = x2
Go!
Graph the system of equations:
12. y = -3/2x + 1
y = 2x – 3
y = -x2
13. Which equation eventually will have the highest value?
NUCC| Secondary II Math 105