Functional RelationshipsKaryn Baize and Rachael Roeckel

Trexler MS / Allen HS

Functions As Mappings

Objective:SWBAT develop a

rule/definition of a function from examples and non-examples; identify domain and range of a function.

Activity:There are multiple definitions of functions. In this investigation we will look at one of those definitions. One definition of a relation is a mapping from the elements one set to the elements of a second set. As an example, the Caesar cipher, which Julius Caesar was said to use to encrypt messages, maps each letter of the alphabet to the next letter with the last letter being mapped to the first letter. The following mapping shows part of this relation for the capital letters in the English alphabet.

A function is a relation with a special property. Your mission is to determine that property from the examples and non-examples of functions below. Functions

Non-FunctionsEx 1

Ex 2

Look for patterns in the mappings above. Create a definition for a relation to be a function.

In order for a relation to be a function, for every

value on the left (input), it maps to ONLY one value on the right (output).

Domain and Range Notice that with this definition a function, there are two sets: the first set or set of inputs and the second set or set of outputs. The domain of a function will be the set of all inputs for the function. The range of a function will be the set of outputs for the function. Find the domain and range for the function in example 1. Domain : { -2, -1, 0, 1}

Range: {1}

Find the domain and range for the function in example 2.

Domain : { -2, -1, 0, 1}

Range: {0, 1, 4}

Animal PensObjective:

SWBAT find and recognize patterns; create a table of values and scatterplot that models a real-world situation; analyze the representations and create an explicit rule.

Materials:Graph paper, Ruler, Tiles (optional)

Activity:Farmer Jim uses fence panels of the same

length to create pens for his animals. He decides to arrange the pens in a single row with all the pens being connected as shown.

Find a rule that tells you how many panels are needed to build a certain number of pens, if you know how many panels are needed to build one less pen. Why does your rule work?

If you know how many panels are needed to build one less pen than you are asked to

build, then all you have to do is add 3 more panels. Pens share one panel so you only need 3 to make

another one. Create a scatterplot of the ordered pairs in the table. Draw a line through these points. This line is a graphical representation of the model. What is the y-intercept of this line? Use the line to find the number of panels after 9 iterations.

Find an explicit rule for finding the number of panels needed to build n pens.

Journal Reflection: Why does your rule work? How did you use the graph of the line to find

how many panels were needed for 9 iterations? How are your two rules related to each other

and to the graph you created?

Verbal Representations of Functions

Objective:SWBAT analyze

verbal representations of real-world situations to

determine: if the situation represents a function (Linear function, Exponential function, etc.)

Independent/dependent variables Rates of change/Common ratio/constant

percent change

Activity 3: In each case determine if the words represent a function. Explain why each description is or is not a function. The input is the person’s first name and the output is the number of pets they have.

This is not a function. You can have two people

with the same name with a different number of

pets. Here one input goes to two outputs.

The input is the individual’s social security number and the output is the person’s name.

This is a function. Each Social Security number is individualized and doesn’t duplicate. Each input only goes to one output.

The input is the height of a person and the output is the shoe size of the person.

This is not a function. Two people may have the same height with different shoe sizes. Here one input goes to two outputs.

The input is an individual’s height in inches and the output is the individual’s height in centimeters.

This is a function. When you convert from one unit to another, there is only one conversion ratio. Each input only goes to one output. Recognizing Linear Functions Represented Using Words Which of the following verbal descriptions represent a linear function? Identify the independent and dependent variables and the rate of change for the linear functions.

Initially the perimeter of the triangle train was 3 inches. The perimeter increases by 1 inch per triangle added.

This is a linear function. The constant rate of change is 1 inch per triangle. The independent variable is number of triangles and the dependent variable is the perimeter.

The frog started at the 10 inch mark. Its average rate was 20 inches per jump for the first 10 jumps.

This is not a linear function. There is no constant rate of change instead there is an average rate of change. While there is an average rate of change, we don’t know the rate of change for each jump. 1

With no weight attached the length of the spring was 52 inches. Each time a 1 gram weight was attached, the spring stretched another 5 inches.

This is a linear function. The rate of change is 5

inches for every gram attached. The independent variable is the weight and the dependent value is the spring length.

The plumber charged $120 for a 1 hour job. Her hourly rate is $80 per hour.

This is a linear function. The starting point is at

$120 instead of 0, and the rate of change is $80. The independent variable is hours and the dependent variable is money charged. 2

Number of pens

Number of panels

1 4

2 7

3 10

4 13

5 16

6 19

7 22

A

B

C

Z

B

C

D

A

-2

-1

01

1 1

13

3

-9

4

5

6

78

-2-101

4

0

1

1

2

3

4

45678

Connections, Extensions, and

Notes1Extension: Students must create a table with an average rate of change of 20 inches per jump and then create a scatter plot using the data. When done the class would discuss how a linear function can be created to approximate the solution.

2Note: Students may struggle to see the pattern. Encourage them to create a table to see the pattern.

3Recognizing Exponential Functions Represented Using Words Exponential functions are multiplicative, meaning that they have a common ratio. For example, if the number of bacteria doubled every hour, the common ratio is 2 and we have an exponential function. A second way of looking at exponential functions is to say that the percent change per unit change in the input values is constant. For example, in recent years the population of the United States has increased at a rate of 1% per year. Since the percent change is a constant 1%, we have an exponential model. Which of the following verbal descriptions represent an exponential function? Identify the independent and dependent variables and either the common ratio or the constant percent change for the exponential functions.

The area of the original square is 64 square centimeters. At each step a new square is obtained with an area that is half of the area of the previous square.This is an exponential function. The independent variable is the number of steps/cuts to the square. The dependent variable is the area of the square. The common ratio is ½. The number of tiles needed to surround the pool was 10 for the smallest pool and increased by 6 with each 1 yard increase in the width of the pool.This is not an exponential function. The rate of change is constant (6 tiles for each 1 yard) making it a linear function. Julie invested $2,000 in a certificate of deposit that pays 3% interest per year. Julie reinvested all of the interest earned.The key word in this problem is reinvested. This makes it an exponential function with two independent variables (years invested and amount invested) and a dependent variable of interest earned. The constant percent change is 3%. If the problem didn’t include the word reinvested the function would be linear. The rate of change would be 3/100.