Content Strand: Interpreting Functions

Grade 9 Algebra 1, Q2

Content Strand: Interpreting Functions

Standard: A1.FIF.4 Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.)

Related Standards:

SCCCR A1.AREI.10

SCCCR A1.NQ.2

SCCCR A1.FIF.2

SCCCR A1.FIF.5

SCCCR A1.FIF.7

SCCCR A1.ACE.2

CCSSM F.IF.4

Vocabulary: function, quantities, graphical, tabular, coordinate plane, ordered pairs, intercepts, intervals, increasing, decreasing, constant, positive, negative, relative, maximums, minimums, symmetries, end behavior, periodicity, vertex, linear, quadratic, exponential, continuous, discrete, horizontal, vertical

Example:

A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet.

What is a reasonable domain restriction for t in this context?
Determine the height of the rocket two seconds after it was launched.
Determine the maximum height obtained by the rocket.
Determine the time when the rocket is 100 feet above the ground.
Determine the time at which the rocket hits the ground.
How would you refine your answer to the first question based on your response to the second and fifth questions?

Strategies/Activities:

SREB Unit 4 Linear Functions / Process Standards
1. Make sense of problems and persevere in solving them.
2. Reason both contextually and abstractly.
3. Use critical thinking skills to justify mathematical reasoning and critique the reasoning of others.
4. Connect mathematical ideas and real-world situations through modeling.
6. Communicate mathematically and approach mathematical situations with precision.
7. Identify and utilize structure and patterns.

Resources:

Text:

Holt McDougal Algebra I Common Core Edition, Sections 3-1, 4-2

Exemplar Lessons:

Web Sites:

Videos:

Web Code: ate-0775, Chapter 1-4 Introduction to Functions

Chapter 5-1 Relating Graphs to Events

USATESTPrep: Linear Equations: Finding Intercepts

Sample Assessment-like Questions:

Assessments: Textbook assignments, Worksheet assignments, Quizzes, Tests, Oral responses, Observations

Holt Algebra 1 Test Prep: 3-1 p. 168-169; 4-2 p. 242

Multiple Choice:

Open Ended: Timmy goes to the fair with $40. Each ride costs $2. Using $40 as the y-intercept and -$2 as the slope, sketch a graph to show much money Timmy will have left after riding n rides?

Content Strand: Interpreting Functions

Standard: A1.FIF.1 Extend previous knowledge of a function to apply to general behavior and features of a function.

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.

Related Standards:

SCCCR A1.FIF.1 B and C

SCCCR A1.FIF.2

SCCCR A1.FIF.4

SCCCR A1.FIF.5

SCCCR A1.FIF.7

SCCCR A1.FLEQ.2

CCSSM F.IF.1

Vocabulary: function, input, output, domain, range, ordered pair, mapping, graphing, vertical line test

Example:

For numbers 1a – 1d, determine whether each relation is a function. Explain your answer.

Solutions:

1a. Yes – All x-coordinates are unique, so it meets the definition of a function.

1b. No – An input of x = 1 has two corresponding outputs, y = 3 and y = 3, so it fails to meet the definition of a function.

1c. Yes – This is a function since for each value chosen along the x-axis, there is exactly one y-value on the graph that corresponds to it.

1d. No – This is not a function since the input of 5 has two corresponding output values, 3 and 2.

Resources:

Text:

Holt McDougal Algebra I Common Core Edition, Section 3-2, 3-2 Algebra Lab: The Vertical Line Test, 3-3, 3-4, 3-4 Technology Lab

Exemplar Lessons:

Web Sites:

Relationships in 2 Variable

Videos:

USATestPrep: Functions and Non-Functions: Graphically(01:47)

Functions and Non-Functions:Algebraically (02:08), Patterns: Functions or Relations (01:34)

Functions: Definitions and Examples (01:23), Relations: Definitions and Examples (02:14)

Domain and Range of Functions, Graphing y=x (01:28), Graphing y=x2 (01:26), Graphing y=x3(01:23)

Web Code: ate-0775, Chapter 5-2 Relations and Functions

Sample Assessment-like Questions:

Assessments: Textbook assignments, Worksheet assignments, Quizzes, Tests, Oral responses, Observations

Holt Algebra 1 Test Prep: p. 175, 184-185, 191-192, 194

Multiple Choice:

If f(x) = 4x - 2 and D = {-1, 0, 1}, what is the range?

{-6, 2}
{2}
{-6,2}
{-6, -2}

Open Ended:

Express the relation as a table, a graph, and a mapping. Then determine the domain and range.

{(4, 3), (-1,4), (3, -2), (2, 3), (-2,1)}

Content Strand: Interpreting Functions

Standard: A1.FIF.1 Extend previous knowledge of a function to apply to general behavior and features of a function.

B. Represent a function using function notation and explain that f(x) denotes the output of function that corresponds to the input x.

Related Standards:

SCCCR A1.FIF.2

SCCCR A1.FIF.1 A and C

SCCCR A1.FIF.4

SCCCR A1.FIF.5

SCCCR A1.FIF.7

SCCCR A1.FLEQ.2

CCSSM F.IF.1

Vocabulary: function, function rule, function notation, output, input, domain, range, independent variable, dependent variable

Example:

Strategies/Activities:

SREB Unit 4 Linear Functions / Process Standards
1. Make sense of problems and persevere in solving them.
2. Reason both contextually and abstractly.
3. Use critical thinking skills to justify mathematical reasoning and critique the reasoning of others.
4. Connect mathematical ideas and real-world situations through modeling.
5. Use a variety of mathematical tools effectively and strategically.
6. Communicate mathematically and approach mathematical situations with precision.
7. Identify and utilize structure and patterns.

Resources:

Text:

Holt McDougal Algebra I Common Core Edition, Section 3-3, 3-4, 3-4 Technology Lab

Exemplar Lessons:

Web Sites:

Videos:

USATestPrep: Functions and Non-Functions: Graphically(01:47)

Functions and Non-Functions:Algebraically (02:08), Patterns: Functions or Relations (01:34) Functions: Definitions and Examples (01:23), Relations: Definitions and Examples (02:14)

Domain and Range of Functions, Graphing y=x (01:28), Graphing y=x2 (01:26), Graphing y=x3(01:23)

Web Code: ate-0775, Chapter 5-2 Relations and Functions

Sample Assessment-like Questions:

Assessments: Textbook assignments, Worksheet assignments, Quizzes, Tests, Oral responses, Observations

Holt Algebra 1 Test Prep: p. 184-185, 191-192, 194

Multiple Choice:

A florist delivers flowers to anywhere in town. d is the distance from the delivery address to the florist shop in miles. The cost to deliver flowers, based on the distance d, is given by Evaluate C(d) for d=6 and d=11, and describe what the values of the function represent.

represents the cost, $15.24, of delivering flowers to a destination that is 6 miles from the shop.

represents the cost, $22.09, of delivering flowers to a destination that is 11 miles from the shop.

represents the cost, $62.04, of delivering flowers to a destination that is 6 miles from the shop.

represents the cost, $179.39, of delivering flowers to a destination that is 11 miles from the shop.

represents the cost, $23.43, of delivering flowers to a destination that is 6 miles from the shop.

represents the cost, $49.62, of delivering flowers to a destination that is 11 miles from the shop.

represents the cost, $22.09, of delivering flowers to a destination that is 6 miles from the shop.

represents the cost, $15.24, of delivering flowers to a destination that is 11 miles from the shop.

Open Ended:

The base of a cardboard box is a square with side length c centimeters. The volume of the box in cubic centimeters is given by the function .
Hint: The formula for the volume of a rectangular prism is base * height
Part A: Show how to find all of the box’s dimension given that c = 4.
Part B: Evaluate V(1), V(2), and V(3) for this function. Show your work. What do these values mean in the context of this problem?
Part C: Should the domain for this function be restricted? If not, explain why not. If so, explain why and give the restricted domain.

Content Strand: Interpreting Functions

Standard: A1.FIF.1 Extend previous knowledge of a function to apply to general behavior and features of a function.

C. Understand that the graph of a function labeled as f is the set of all ordered pairs (x,y) that satisfy the equation y = f(x).

Related Standards:

SCCCR A1.AREI.10

SCCCR A1.FIF.1 A and B

SCCCR A1.FIF.2

SCCCR A1.FIF.4

SCCCR A1.FIF.5

SCCCR A1.FIF.7

SCCCR A1.FLQE.2

CCSSM F.IF.1

Vocabulary: function, coordinate plate, ordered pair, vertical line test, function notation, function rule

Example:

Strategies/Activities:

Scroll down to The Vertical Line Test

SREB Unit 4 Linear Functions / Process Standards
1. Make sense of problems and persevere in solving them.
2. Reason both contextually and abstractly.
3. Use critical thinking skills to justify mathematical reasoning and critique the reasoning of others.
4. Connect mathematical ideas and real-world situations through modeling.
5. Use a variety of mathematical tools effectively and strategically.
6. Communicate mathematically and approach mathematical situations with precision.
7. Identify and utilize structure and patterns.

Resources:

Text:

Holt McDougal Algebra I Common Core Edition, Section 3-2, 3-3, 3-4, 3-4 Technology Lab

Exemplar Lessons:

Web Sites:

Videos:

USATestPrep: Functions and Non-Functions: Graphically(01:47)

Functions and Non-Functions:Algebraically (02:08), Patterns: Functions or Relations (01:34)

Functions: Definitions and Examples (01:23), Relations: Definitions and Examples (02:14), Function Notation(02:22), Evaluate Linear Functions (01:14)

Web Code: ate-0775, Chapter 5-4 Writing a Function Rule

Sample Assessment-like Questions:

Assessments: Textbook assignments, Worksheet assignments, Quizzes, Tests, Oral responses, Observations

Holt Algebra 1 Test Prep: p. 175, 184-185, 191-192, 194

Multiple Choice:

The graph of function f is shown below:

Which of the following equations represents the function?

y = 2x + 3
y = 2x + 2
y = 3x + 3
y = 3x + 5

Open Ended:

The directions on a turkey tell you to cook the turkey 20 minutes per pound.

a. Write a function in function notation for this situation where x is the number of

pounds.

b. Describe the domain of this function. Assume turkeys can weigh up to 30 pounds.

c. Graph this function and state the range.

d. Use this graph to determine the cooking time for an 18 pound turkey.

Content Strand: Interpreting Functions

Standard: A1.FIF.2* Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.

Related Standards:

SCCCR A1.FIF.1

SCCCR A1.FIF.1 B and C

SCCCR A1.FIF.4

SCCCR A1.FIF.5

SCCCR A1.FIF.7

SCCCR A1.FLQE.2

CCSSM F.IF.2

Vocabulary: function, function notation, evaluate, interpret, expression

Example:

Let . Find , , , and

You put a yam in the oven. After 45 minutes, you take it out. Let f be the function that assigns to each minute after you placed the yam in the oven, its temperature in degrees Fahrenheit.

A. Write a sentence explaining what f(0)=65 means in everyday language.

B. Write a sentence explaining what f(5)<f(10) means in everyday language.

C. Write a sentence explaining what f(40)=f(45) means in everyday language

D. Write a sentence explaining what f(45)>f(60) means in everyday language.

Sample Response:

f(0)=65 means that when you placed the yam in the oven, its temperature was 65 degrees Fahrenheit.
f(5)<f(10) means that the temperature of the yam 5 minutes after you placed it in the oven was less than its temperature 10 minutes after you placed it in the oven. This would be because the yam's temperature will increase from 65 degrees Fahrenheit during the first few minutes its in the oven.
f(40)=f(45) means that the temperature of the yam 40 minutes after you placed it in the oven was the same as its temperature 45 minutes after you placed it in the oven. This would be because the temperature of the yam eventually plateaus.
f(45)>f(60) means that the temperature of the yam 45 minutes after you placed it in the oven was greater than its temperature 60 minutes after you placed it in the oven. This would be because the yam began to cool down after you removed it from the oven.

Strategies/Activities:

SREB Unit 4 Linear Functions / Process Standards
1. Make sense of problems and persevere in solving them.
2. Reason both contextually and abstractly.
3. Use critical thinking skills to justify mathematical reasoning and critique the reasoning of others.
4. Connect mathematical ideas and real-world situations through modeling.
5. Use a variety of mathematical tools effectively and strategically.
6. Communicate mathematically and approach mathematical situations with precision.
7. Identify and utilize structure and patterns.

Resources:

Text:

Holt McDougal Algebra I Common Core Edition, Sections 3-4, 4-2

Exemplar Lessons:

Web Sites:

Videos:

USATestPrep:USATestPrep: Functions and Non-Functions: Graphically(01:47)

Functions and Non-Functions:Algebraically (02:08), Patterns: Functions or Relations (01:34)

Functions: Definitions and Examples (01:23), Relations: Definitions and Examples (02:14) Function Notation(02:22), Linear Equations: Finding Intercepts, Graphing y=x,

Graphing Linear Functions

Web Code: ate-0775, Chapter 1-4 Introduction to Functions

Chapter 5-4 Writing a Function Rule

Sample Assessment-like Questions:

Assessments: Textbook assignments, Worksheet assignments, Quizzes, Tests, Oral responses, Observations

Holt Algebra 1 Test Prep: p. 191-192, 194, 242

Multiple Choice:

The following table represents a linear function f(x).

x / f(x)
-5 / -30
-2 / -12
1 / 6
9 / ?

What is f(9)?

Open Ended:

The distance traveled (in meters) by the Oregon slug can be modeled by the function f(t) = 0.9t, where t is the time in minutes. Find the distance traveled in 27.5 minutes.

Content Strand: Interpreting Functions

Standard: A1.FIF.5* Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)

Related Standards:

SCCCR A1.ACE.2

SCCCR A1.RE1.10

SCCCR A1.FIF.1 B and C

SCCCR A1.FIF.2

SCCCR A1.FIF.4

SCCCR A1.FIF.7

SCCCR A1.FLQE.2

CCSSM F.IF.1

CCSSM F.IF.2

CCSSM F.IF.5

Vocabulary: function, domain, range, input, output, ordered pair, linear

Example:

Example 1: Oakland Coliseum, home of the Oakland Raiders, is capable of seating 63,026 fans. For each game, the amount of money that the Raiders' organization brings in as revenue is a function of the number of people,n, in attendance. If each ticket costs $30.00, find the domain and range of this function.

Solution: Let r represent the revenue that the Raider's organization makes, so that r=f(n). Since n represents a number of people, it must be a nonnegative whole number. Therefore, since 63,026 is the maximum number of people who can attend a game, we can describe the domain of f as follows:

Domain ={n:0≤n≤63,026 and n is an integer}

The range of the function consists of all possible amounts of revenue that could be earned. To explore this question, note that r=0 if nobody comes to the game, r=30 if one person comes to the game, r=60 if two people come to the game, etc. Therefore, r must be a multiple of 30 and cannot exceed 30⋅63,026=1,890,780, so we see that

Range={r:0≤r≤1,890,780 and r is an integer multiple of 30}.

Note that the representations used above are just sample ways of writing down the domain and range, using set-builder notation. Other options for writing down descriptions of the same sets abound.

Example 2:

Solution:

Resources:

Text:

Holt McDougal Algebra I Common Core Edition, Sections 3-2, 3-3, 3-4, 4-1, 4-1 Career Application, 4-2, 4-5

Exemplar Lessons:

Web Sites:

Videos:

USATestPrep: Function Notation (02:22), Evaluate Linear Functions (01:14), Functions and Non-Functions: Numerically (02:08),Functions and Non-Functions: Graphically (01:47), Functions: Definitions and Examples, (01:25), Recognizing Functions(03:34), Relations:Definitions and Examples (02:14)

Web Code: ate-0775, Chapter 1-4 Introducing Functions,

Chapter 5-2, Relations and Functions

Chapter 5-3 Writing a Function Rule

Sample Assessment-like Questions:

Assessments: Textbook assignments, Worksheet assignments, Quizzes, Tests, Oral responses, Observations

Holt Algebra 1 Test Prep: p. 175, 184-185, 191-192, 194, 235-236, 242, 264-266

Multiple Choice:

What is the range for the function, f (x), shown here?

all real numbers
{(0, 2), (1, 2), (–1, 2), (2, 2), (–2, 2)}
{2}
{(2, 2)}

Open Ended:

When Aidan had his picture taken, the photographer charged a $10 sitting fee and $6 for each sheet of pictures purchased.

a. Write the function for the situation, where x is the number of sheets purchased.

b. Graph this function.

c. Explain how the graph of the function can be used to find the cost of pictures if

Aidan bought 10 sheets of pictures. How could you use the function to check this

answer?

Content Strand: Interpreting Functions

Standard: A1.FIF.6* Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.)

Related Standards:

SCCCR A1.ACE.2

SCCCR A1.ASE.1

SCCCR A1.FIF.7

SCCCR A1.FLQE.2

SCCCR A1.FLQE.5

CCSSM F.IF.6

Vocabulary: function, graphical, symbolic, tabular, average rate of change, slope, table of values, interval, linear, slope formula, slope-intercept form

Example:

Strategies/Activities:

SREB Unit 4 Linear Functions Lesson 2 and 3 / Process Standards
1. Make sense of problems and persevere in solving them.
2. Reason both contextually and abstractly.
3. Use critical thinking skills to justify mathematical reasoning and critique the reasoning of others.
4. Connect mathematical ideas and real-world situations through modeling.
5. Use a variety of mathematical tools effectively and strategically.
6. Communicate mathematically and approach mathematical situations with precision.
7. Identify and utilize structure and patterns.

Resources: