Understanding Functions - rcsdk12.org

Functions

Essential Understandings and How These

Important Ideas Relate to One Another

Opening Exercises

1. The function 𝑓 has a domain of {9, 11, 13, 15} and a range of {8, 10, 12}.

Could 𝑓 be represented by {(9, 8), (11, 10), (13, 12)}? Justify your reasoning.

2. The graph below shows the circulation of newspapers in a town for integer years. According to the

graph, is the circulation of newspapers a function of the years shown? Justify your reasoning.

Self-Assessment

5 – Exemplary (I have exceeded that target.)

4 – Accomplished (I have met the target.)

3 – Emerging (I am more than half-way there.)

2 – Beginning (I have started making progress.)

1 – No Understanding (I have not yet started making progress.)

Pre-assessment Learning Target Post-Assessment

1. I can articulate essential understandings about functions and how important mathematical ideas about functions relate to one another.

2. I can identify problems that align to the standards within the Functions conceptual category.

Norms of Collaboration

(Garmston & Wellman, 1999)

The following norms support a community of adult learning: 1. Promoting a Spirit of Inquiry 2. Pausing 3. Paraphrasing 4. Probing 5. Putting ideas on the table 6. Paying attention to self and others 7. Presuming positive intentions

Part 1

The Concept of Function

The concept of function is intentionally broad and flexible, allowing it to apply to a wide range of

situations. The notion of concept encompasses many types of mathematical entities in addition to

“classical” functions that describe quantities that vary continuously. For example, matrices and

arithmetic and geometric sequences can be viewed as functions.

~Developing Essential Understanding of Functions, NCTM, 2010

Consider all possible circular animal pens enclosed by a length of fencing. For each length of fencing,

there is a corresponding area that the fencing will enclose when it is formed into a circle. Describe

the relationship between the area of a circular pen and the length of fencing that it takes to enclose

the pen.

The following are quotes taken from different sources. Select one quote that strikes you as the

most interesting or surprising. Share your thoughts with a partner.

1. A function from one set (called the domain) to another set (called the range) assigns to each

element of the domain exactly one element of the range. If f is a function and x is an element of its

domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of

the equation y = f(x). ~Standard F-IF.1

2. A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched

to one and only one element of 𝑌. The set 𝑋 is called the domain of the function.

~Algebra 1 Module 3 Lesson 9

3. By definition, functions are “single-valued”. In other words, for each element of the domain, there

is exactly one element of the range of the function. ~Developing Essential Understanding of Functions, NCTM, 2010

4. Functions apply to a wide range of situations. They do not have to be described by any specific

expressions or follow a regular pattern. They apply to cases other than those of “continuous

variation.” For example, sequences are functions. ~Developing Essential Understanding of Functions, NCTM, 2010

5. The domain and range of functions do not have to be numbers. ~Developing Essential Understanding of Functions, NCTM, 2010

6. Functions describe situations where one quantity determines another. Because we continually

make theories about dependencies between quantities in nature and society, functions are important

tools in the construction of mathematical models. ~ New York State P-12 Common Core Learning Standards for Mathematics

7. These ideas become semi-formal in Grade 8 with the introduction of the concept of function: a

rule that assigns to each input exactly one output. ~Standard 8.F.1

8. Traditional pattern activities, where students are asked to continue a pattern through observation,

are not a mathematical topic, and do not appear in the Standards in their own right. ~ Progressions for the Common Core State Standards, 2013

9. Time normally spent on exercises involving the vertical line test, or searching lists of ordered pairs

to find two with the same x-coordinate and different y-coordinate, can be reallocated elsewhere.

Indeed, the vertical line test is problematic. ~ Progressions for the Common Core State Standards, 2013

Exercises

1. The table below shows the cost of parking in a 24-hour garage for a given number of hours

0 < 𝑡 < 24. Does this correspondence represent a function? Justify your reasoning.

2. Is the relation 𝑦 = 𝑔(𝑥) = √2 − 𝑥 a function from the Real numbers to the Real numbers?

Justify your reasoning.

3. Is the relation below a function from the Real numbers to the Real numbers?

Justify your reasoning.

𝑦 = 𝑝(𝑥) = {𝑥 + 2 𝑥 ≤ 2

𝑥 𝑥 ≥ 0

4. Is 𝑦 a function of 𝑥? Justify your reasoning.

𝑓(𝑥) = {−𝑥 𝑥 < 0𝑥 𝑥 ≥ 0

5. Does each element of the domain correspond to exactly one element in the range in the following

sequence? Justify your reasoning.

𝑓(1) = 2

𝑓(𝑛) = 𝑓(𝑛 − 1) + 2𝑛 for all natural numbers 𝑛 > 1

6. If 𝑓(𝑥) =1

3𝑥 + 9, which statement is always true?

(1) 𝑓(𝑥) < 0 (3) If 𝑥 < 0, then 𝑓(𝑥) < 0.

(2) 𝑓(𝑥) > 0 (4) If 𝑥 > 0, then 𝑓(𝑥) > 0.

FUNCTION

Definition

Nonexamples Examples

Facts/Characteristics

Part 2

Interpreting Functions

Interpreting Functions F-IF

Understand the concept of a function and use function notation.

1. Understand that a function from one set (called the domain) to another set (called the range)

assigns to each element of the domain exactly one element of the range. If f is a function and x is an

element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is

the graph of the equation y = f(x).

The purpose of this task is to help students learn to read information about a function from its

graph, by asking them to show the part of the graph that exhibits a certain property of the

function.

Use the graph (for example, by marking specific points) to illustrate the statements in (a)–(d). If

possible, label the coordinates of any points you draw.

a. 𝑓(0) = 2

b. 𝑓(−3) = 𝑓(3) = 𝑓(9) = 0

c. 𝑓(2) = 𝑔(2)

d. 𝑔(𝑥) > 𝑓(𝑥) for 𝑥 > 2

https://www.illustrativemathematics.org/illustrations/636




2. Use function notation, evaluate functions for inputs in their domains, and interpret statements

that use function notation in terms of a context.

This task assesses whether students can interpret function notation. The four parts of the

task provide a logical progression of exercises for advancing understanding of function notation

and how to interpret it in terms of a given context.

Let 𝑓(𝑡) be the number of people, in millions, who own cell phones years after 1990. Explain the

meaning of the following statements.

a. 𝑓(10) = 100.3

b. 𝑓(𝑎) = 20

c. 𝑓(20) = 𝑏

d. 𝑛 = 𝑓(𝑡)





3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset

of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1,

f(n+1) = f(n) + f(n-1) for n ≥ 1.

A sequence is simply a function whose domain is restricted. For example, arithmetic sequences are

restrictions of linear functions to the positive integers, and geometric sequences are restrictions of

exponential functions to the positive integers.

These are two examples of sequences expressed using function notation.

August 2014 Algebra 1 Regents Exam #24

If 𝑓(1) = 3 and 𝑓(𝑛) = −2𝑓(𝑛 − 1) + 1, then 𝑓(5) =

(1) –5 (3) 21

(2) 11 (4) 43

June 2014 Algebra 1 Regents Exam #21

A sunflower is 3 inches tall at week 0 and grows 2 inches each week. Which function(s) shown below

can be used to determine the height, 𝑓(𝑛), of the sunflower in 𝑛 weeks?

I. 𝑓(𝑛) = 2𝑛 + 3

II. 𝑓(𝑛) = 2𝑛 + 3(𝑛 − 1)

III. 𝑓(𝑛) = 𝑓(𝑛 − 1) + 2 where 𝑓(0) = 3

(1) I and II (3) III, only

(2) II, only (4) I and III


Interpret functions that arise in applications in terms of the context.

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.★

This task helps reinforce the idea that when a variable represents time 𝑡 = 0, is chosen as an

arbitrary point in time and positive times are interpreted as times that happen after that.

The figure shows the graph of 𝑇, the temperature (in degrees Fahrenheit) over one particular

20-hour period in Santa Elena as a function of time 𝑡.

a. Estimate (14) .

b. If 𝑡 = 0 corresponds to midnight, interpret what we mean by 𝑇(14) in words.

c. Estimate the highest temperature during this period from the graph.

d. When was the temperature decreasing?

e. If Anya wants to go for a two-hour hike and return before the temperature gets over 80 degrees,

when should she leave?





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.★


Officials in a town use a function, 𝐶, to analyze traffic patterns. 𝐶(𝑛) represents the rate of traffic

through an intersection where 𝑛 is the number of observed vehicles in a specified time interval.

What would be the most appropriate domain for the function?

(1) {… − 2, −1, 0, 1, 2, 3, … } (3) {0,1

2, 1, 1

1

2, 2, 2

1

2}

(2) {−2, −1, 0, 1, 2, 3} (4) {0, 1, 2, 3, …}


The function ℎ(𝑡) = −16𝑡2 + 144 represents the height, ℎ(𝑡), in feet, of an object from the ground

at 𝑡 seconds after it is dropped. A realistic domain for this function is

(1) −3 ≤ 𝑡 ≤ 3 (3) 0 ≤ ℎ(𝑡) ≤ 144

(2) 0 ≤ 𝑡 ≤ 3 (4) all real numbers

Algebra 1 Module 3 End of Module Assessment #3a

A boy bought 6 guppies at the beginning of the month. One month later the number of guppies in his tank had doubled. His guppy population continued to grow in this same manner. His sister bought some tetras at the same time. The table below shows the number of tetras, t, after n months have passed since they bought the fish.

Create a function g to model the growth of the boy’s guppy population, where g(n) is the number of guppies at the beginning of each month and n is the number of months that have passed since he bought the 6 guppies. What is a reasonable domain for g in this situation?

Functions provide a means to describe how related quantities vary together. We can classify, predict,

and characterize various kinds of relationships by attending to the rate at which one quantity varies

with respect to the other.


Consider the table of the function 𝑓 below. What do you notice? What stands out?

𝑥 𝑓(𝑥)

–1 1

0 1

1 3

2 7

3 13

4 21



6. Calculate and interpret the average rate of change of a function (presented symbolically or as a

table) over a specified interval. Estimate the rate of change from a graph.★

Algebra 1 Module 4 Lesson 10 Example 2e-f

The table below represents the value of Andrew’s stock portfolio, with 𝑉 representing the value of

the portfolio, in hundreds of dollars, and 𝑡 is the time, in months, since he started investing. How

fast is Andrew’s stock value decreasing between [10, 12]? Find another two-month interval where

the average rate of change is faster than [10, 12] and explain why. Are there other two-month

intervals where the rate of change is same as [10, 12]? Explain your answer.

𝑡 (months) 𝑉(𝑡) (hundreds of dollars)

2 325

4 385

6 405

8 385

10 325

12 225

14 85

16 −95

18 −315

Algebra 1 Module 4 End of Module Assessment #3d

An arrow is shot into the air. A function representing the relationship between the number of

seconds it is in the air, 𝑡, and the height of the arrow in meters, ℎ, is given by:

What is the average rate of change for the interval from 𝑡 = 1 to 𝑡 = 2 seconds? Compare your answer

to the average rate of change for the interval from 𝑡 = 2 to 𝑡 = 3 seconds and explain the difference in

the context of the problem.

Functions can be represented in multiple ways, including algebraic (symbolic), graphical, verbal, and

tabular representations. Links among these different representations are important to studying

relationships and change.


Consider the relationship represented in four different ways below. When might each representation

be more useful than the others?

a. A movie theater has operating costs of $1025 per day. Tickets cost $7.50 each. The movie

theater’s profit each day depends on the number of tickets sold.

b. 𝑃 = 7.5𝑇 − 1025

c.

d.

𝑇 𝑃 0 −$1025

50 −$650

100 −$275

150 $100

200 $475

250 $850

300 $1225


Analyze functions using different representations.

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.


The value of the 𝑥-intercept for the graph of 4𝑥 − 5𝑦 = 40 is

(1) 10 (3) −4

5

(2) 4

5 (4) −8


Draw the graph of 𝑦 = √𝑥 − 1 on the set of axes below.



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.

Suppose ℎ(𝑡) = −5𝑡2 + 10𝑡 + 3 is an expression giving the height of a diver above the water (in

meters), 𝑡 seconds after the diver leaves the springboard.

A. How high above the water is the springboard? Explain how you know.

B. When does the diver hit the water?

C. At what time on the diver's descent toward the water is the diver again at the same height as the

springboard?

D. When does the diver reach the peak of the dive?





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.

Algebra 1 Module 4 Lesson 22 Exit Ticket

Two people, in two different apartment buildings, have buzzers that don’t work. They both must

throw the keys to their apartments out of the window to their guests, who will then use the keys to

enter. Tenant one throws the keys such that the relationship between the height of the keys (in feet),

and the time that has passed (in seconds) can be modeled by ℎ(𝑡) = −16𝑡2 + 18𝑡 + 9.

Tenant two throws the keys such that the height/time relationship can be modeled by the graph

below.

Whose window is higher? Explain how you know.

Part 3

Building Functions

Building Functions F-BF

Build a function that models a relationship between two quantities.

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.

On June 1, a fast growing species of algae is accidentally introduced into a lake in a city park. It starts

to grow and cover the surface of the lake in such a way that the area covered by the algae doubles

every day. If it continues to grow unabated, the lake will be totally covered and the fish in the lake

will suffocate. At the rate it is growing, this will happen on June 30.

(a) When will the lake be covered half-way?

(b) On June 26, a pedestrian who walks by the lake every day warns that the lake will be completely

covered soon. Her friend just laughs. Why might her friend be skeptical of the warning?

(c) On June 29, a clean-up crew arrives at the lake and removes almost all of the algae. When they are

done, only 1% of the surface is covered with algae. How well does this solve the problem of the algae

in the lake?



Functions can be combined by adding, subtracting, multiplying, dividing, and composing them.

Functions sometimes have inverses. Functions can often be analyzed by viewing them as made from

other functions.


Consider the following question:

What is the effect (on the graph of a function) of adding a constant function?

Building Functions F-BF

Build new functions from existing functions.

3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific

values of k (both positive and negative); find the value of k given the graphs. Experiment with cases

and 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.

Consider the functions listed below. What does 𝑥 need to be in order for the argument to be 0?

𝑓(𝑥) = |𝑥| = 0 𝑥 =

𝑓(𝑥 + 1) = |𝑥 + 1| = 0 𝑥 =

𝑓(𝑥 + 2) = |𝑥 + 2| = 0 𝑥 =

𝑓(𝑥 + 3) = |𝑥 + 3| = 0 𝑥 =

𝑓(𝑥 − 1) = |𝑥 − 1| = 0 𝑥 =

𝑓(𝑥 − 2) = |𝑥 − 2| = 0 𝑥 =

𝑓(𝑥 − 3) = |𝑥 − 3| = 0 𝑥 =

The graph of 𝑓(𝑥 + 4) is a _____________shift. It translates the graph of 𝑓(𝑥) _____ units _______.

The graph of 𝑓(𝑥 − 5) is a _____________shift. It translates the graph of 𝑓(𝑥) _____ units _______.

In general, 𝑓(𝑥 + 𝑎) translates 𝑓(𝑥) _______________. If a > 0, the graph slides ______ and if a < 0,

the graph slides ______.

Can you justify why this is true?

Part 4

Linear and Exponential Models

Functions can be classified into different families of functions, each with its own unique

characteristics. Different families can be used to model different real-world phenomena.


Consider which function would best model each of the following:

a. Gregory plans to purchase a video game player. He has $500 in his savings account, and plans to

save $20 per week from his allowance until he has enough money to buy the player. He needs to

figure out how long it will take.

b. c.

d. Margie got $1000 from her grandmother to start her college fund. She is opening a new savings

account and finds out that her bank offers a 2% annual interest rate, compounded monthly. She

wants to calculate the amount of money in the bank compounded monthly.

e. The function that associates to the input 𝑥 the output 𝑦 given by 𝑦 = 5(𝑥 + 5)(𝑥 − 6).

Linear, Quadratic, & Exponential Models F-LE

Construct and compare linear, quadratic, and exponential models and solve problems.

1. Distinguish between situations that can be modeled with linear functions and with exponential

functions.

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential

functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative

to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit

interval relative to another.


A population that initially has 20 birds approximately doubles every 10 years. Which graph represents this population growth?


Which situation could be modeled by using a linear function? (1) a bank account balance that grows at a rate of 5% per year, compounded annually (2) a population of bacteria that doubles every 4.5 hours (3) the cost of cell phone service that charges a base amount plus20 cents per minute (4) the concentration of medicine in a person’s body that decays by a factor of one-third every hour


2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a

graph, a description of a relationship, or two input-output pairs (include reading these from a table).


The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is 𝑎1, which is

an equation for the nth term of this sequence?

(1) 𝑎𝑛 = 8𝑛 + 10 (3) 𝑎𝑛 = 16𝑛 + 10

(2) 𝑎𝑛 = 8𝑛 − 14 (4) 𝑎𝑛 = 16𝑛 − 38


The table below represents the function F.

The equation that represents this function is

(1) 𝐹(𝑥) = 3𝑥 (3) 𝐹(𝑥) = 2𝑥 + 1

(2) 𝐹(𝑥) = 3𝑥 (4) 𝐹(𝑥) = 2𝑥 + 3


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.

Algebra 1 Module 3 Lesson 5 Opening Exercise

Two equipment rental companies have different penalty policies for returning a piece of equipment

late:

Company 1: On day 1, the penalty is $5. On day 2, the penalty is $10. On day 3, the penalty is $15.

On day 4, the penalty is $20 and so on, increasing by $5 each day the equipment is late.

Company 2: On day 1, the penalty is $0.01. On day 2, the penalty is $0.02. On day 3, the penalty is

$0.04. On day 4, the penalty is $0.08 and so on, doubling in amount each additional day late.

Jim rented a digger from Company 2 because he thought it had the better late return policy. The job

he was doing with the digger took longer than he expected, but it did not concern him because the

late penalty seemed so reasonable. When he returned the digger 15 days late, he was shocked by

the penalty fee. What did he pay, and what would he have paid if he had used Company 1 instead?

Company 1 Company 2 Day Penalty Day Penalty

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 10

11 11

12 12

13 13

14 14

15 15

1. Which company has a greater 15 day late charge?

2. Describe how the amount of the late charge changes from any given day to the next successive

day in both companies 1 and 2.

3. How much would the late charge have been after 20 days under Company 2?


Interpret expressions for functions in terms of the situation they model.

5. Interpret the parameters in a linear or exponential function in terms of a context.


A company that manufactures radios first pays a start-up cost, and then spends a certain amount of

money to manufacture each radio. If the cost of manufacturing r radios is given by the function

𝑐(𝑟) = 5.25𝑟 + 125, then the value 5.25 best represents

(1) the start-up cost

(2) the profit earned from the sale of one radio

(3) the amount spent to manufacture each radio

(4) the average number of radios manufactured


The breakdown of a sample of a chemical compound is represented by the function 𝑝(𝑡) = 300(0.5)𝑡, where p(t) represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent.

Exit Ticket: How have these sessions helped to deepen your understanding of functions?

We appreciate your feedback, as it will help us reflect on our practice.

What worked well? What needs improvement?

What are you walking away with? What more do you need?

Understanding Functions - rcsdk12.org

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