Math-3

Lesson 1-1

Relations and Functions

Vocabulary

Relation: A “mapping” or pairing of input values to output values.

Function: A relation where each input has exactly one output.

Your turn:

Describe how a relation is

1) Similar to

1) Different from

a function?

Both have inputs matched to outputs.

One input to a relation can be matched with

two or more outputs.

Is it a function or a relation?

input output

2

3

-4

-5

2

3

4

5

input output

2

3

-4

-5

2

3

4

5

Something to think about:

in math OR doesn’t mean

one or the other. It means

one or the other or both!!

Both

Both

Is it a function or a relation?

input output

2

3

-4

-5

2

3

4

5

input output

2

3

-4

-5

2

3

4

5

Something to think about:

in math OR doesn’t mean

one or the other. It means

one or the other or both!!

Both

Relation only

The input value “2” has

two outputs (“2” and “5”)

Is it a function or a relation?

input output

2

3

-4

-5

2

3

4

5

input output

2

3

-4

-5

2

3

4

5

(There aren’t any pairings.)

Neither

Both

Each input has exactly one

output (even though it’s

the same output for each).

Which is a “mapping”?

http://www.flickr.com/photos/sanchome/525890022/

Name 6 ways to show a relation between inputand output values.

Ordered Pairs: (2, 4), (3, 2), (-4, 3), etc.

Data table: x 2 3 -4y 4 2 3

Graph:

Equation: y = 2x + 1

Mappinginput output

2

3

-4

2

3

4

Function notation: f(2) = 4

Note: not all of the relations above are the same.

Your turn:What are 6 ways you can show a relation

between input and output ?

There are actually more ways to show a relation. Next time we’ll talk about sequences.

Does anyone remember the two different ways sequences can be defined?

Domain: the set made up of the input values.

Range: the set made up of the output values.

This is too simplistic!!!

Domain: the set made up of the input values for

which there is a corresponding output value.

Identify the Domain

1. (2, 4), (3, 5), (-4, 2)

2. x 6 9 -2y 4 7 3

3.4.

input output

2

3

-4

-5

2

3

4

-3 1 3

2

3

-4

-5

Determining if the relation is linear.

What are 6 ways you can show a relationbetween input and output ?

Graph:

The slope needs to be constant.

Data table: Is the data linear?

x f(x)-2 -70 -52 -34 -16 18 3

10 512 715 9

1st “difference”

2 x

in change

2 x

In engineering we often

refer to the change (or

difference) as the “delta”,

(using the Greek letter)

2 x

2 x

2 x

2y

2y

2y

2y

2y

If the 1st difference for

both input and output

(“x” and “y”) is always

the same then the

relation is linear.

Why is that?3 x

Your turn: Which data set is linear?

x f(x)-4 -7-3 -5-2 -3-1 -10 11 32 53 74 9

x g(x)-4 32-3 18-2 8-1 20 01 22 83 184 32

x f(x)0 01 12 1.43 1.74 2.05 2.26 2.47 2.68 2.89 3

A B C

Your turn: Is the data linear? If so, what is the equation that “fits” the data?

x f(x)-4 -7-3 -5-2 -3-1 -10 11 32 53 74 9

bmxy

What is this number?

The value of ‘y’ when x = 0.

bmy )0(

by

),0( b

1 mxy

Your turn: What is the equation that fits the linear data?

x f(x)-4 -7-3 -5-2 -3-1 -10 11 32 53 74 9

1 mxy

What is this number?

Slope:

12 xy

2 y

in change

x''in change

y''in changem

1 x

in change

21

2m

Vocabulary:

Parent function: The most basic function in

a family of functions.

For lines: y = x is the “parent function”

Graph the simplest line of all.

y = x

1

1

Your turn:

Your turn: What is the equation that fits the data?

x f(x)

-4 -9

-2 -6

0 -3

2 0

4 3

6 6

8 9

10 12

12 15

Your turn: What is the difference between the two representations?

x f(x)

-4 -7

-3 -5

-2 -3

-1 -1

0 1

1 3

2 5

3 7

4 9

1

1

3

What is the difference between the three representations?

x f(x)

-2 -3

-1 -1

0 1

1 3

1 1

3

1

3

1

Discrete Discrete Continuous

What is the domain of each?

x f(x)

-2 -3

-1 -1

0 1

1 3

1 1

3

1

3

1

D = {x = -2, -1, 0, 1}Discrete Continuous

D = {-2 ≤ x ≤ 1}

Linear Relationships

Does the grade a person earns vary linearly with the number of hours he/she studies?

Linear Relationships

Does the amount of natural gas used by a family vary linearly with the outside temperature?

Does the height of person relate linearly to his/her weight?

Using this data, how could we predict the weight of a person who is 7 feet tall?

bmxy

)75 ,35(),( yx

2535-60 x

55

75130

y

Slope: 2.225

55

x

y

m

bxy 2.2

b )35(2.275

2)"84(2.2 y

)"84'7(

b )35(2.2752b

lb. 189y

Linear RelationshipsIs height of a falling object linear with time?

Linear “Correlation” between the quantities being measured

“Positive” refers to what in the upper left graph?

“Negative” refers to what in the lower left graph?

Pure Math: Equations, graphs, tables of numbers, ordered pairs, mappings, and proofs that are just math and are not being used to relate to the physical world around us

Applied Math: The use of equations, graphs, tables of numbers, ordered pairs, and mappings that are used to model relationships between quantities in the real world.

Engineering: The use of applied math and science to design machines and tools for use in the real world.

Graphing Points on your calculator

Push the “stat” button

There are 3 “pull

down” menu’s.

You are in the

“edit” menu.

Push “enter”

You will enter the

x-values into “L1”

(list 1) and y-values

into “L2”.

You have to

clear L1

Move cursor until L1

heading is highlighted.

Push “clear”

Notice that only the

bottom box was cleared.

Push “enter”

Your are now ready to enter values into the two lists.

With the cursor

in L1, type “-4”

Move the cursor to L2

x f(x)

-4 -7

-3 -5

-2 -3

-1 -1

0 1

1 3

2 5

3 7

4 9

Push “enter”

Enter the rest of the

x-values into L1.

Enter the y-value into L2

Turn on the ability to graph points

Push “y = “Move the cursor to

“Plot1” then hit

enter.

Clear the equation.

The minimum x-value that

will be graphed is -2.5.Check to see if the window

is “big” enough.

x f(x)

-4 -7

-3 -5

-2 -3

-1 -1

0 1

1 3

2 5

3 7

4 9

Push the “zoom” button

Push “6”

Will all the

points show in

the window?

Push “graph”

x f(x)

-4 -7

-3 -5

-2 -3

-1 -1

0 1

1 3

2 5

3 7

4 9

22xy

12 xy

xy

Which equation is it?

Your turn: Enter the data into L1 and L2

x f(x)

-4 -9

-2 -6

0 -3

2 0

4 3

6 6

8 9

10 12

12 15

Your turn: Plot the data on your calculator.

Your turn: Is the data linear?