Functions and Patterns by Lauren McCluskey

Functions and Functions and PatternsPatterns by Lauren McCluskeyby Lauren McCluskey

Exploring the connection between input / Exploring the connection between input / output tables, patterns, and functions…output tables, patterns, and functions…

CreditsCredits

Function RulesFunction Rules by Christine Berg by Christine Berg Algebra IAlgebra I from Prentice Hall, Pearson from Prentice Hall, Pearson

Education Education The Coordinate PlaneThe Coordinate Plane by Christine Berg by Christine Berg

RelationRelation

According to Prentice Hall: “A According to Prentice Hall: “A relationrelation is a set of ordered pairs.” is a set of ordered pairs.”

OrOr

A A relationrelation is a set of input (x) and is a set of input (x) and output (y) numbers. output (y) numbers.

inin outout

11 44

22 88

FunctionFunctionAccording to Prentice Hall: According to Prentice Hall:

““A A functionfunction is a relation that is a relation that assigns exactly one value in assigns exactly one value in the range (y) to each value in the range (y) to each value in

the domain (x).”the domain (x).”

FunctionsFunctions

What does this mean? What does this mean?

It means that for every input value It means that for every input value there is only there is only oneone output value. output value.

More on that later, but first let’s More on that later, but first let’s review coordinate planes…review coordinate planes…

The Coordinate PlaneThe Coordinate Plane““You can use a graph to show the You can use a graph to show the

relationship between two variables…. relationship between two variables…. When one variable depends on When one variable depends on another, show the dependent quantity another, show the dependent quantity on the vertical axis (y).” Prentice Hallon the vertical axis (y).” Prentice Hall

Always show time on the horizontal Always show time on the horizontal axis (x), because it is an independent axis (x), because it is an independent variable. variable.

Remember:Remember:• The x-axis is a horizontal number The x-axis is a horizontal number

line.line.

• It is positive to the right and It is positive to the right and negative to the left.negative to the left.

The Coordinate PlaneThe Coordinate Plane by Christine Berg by Christine Berg

+-

Y-axisY-axis• The y-axis is a vertical number The y-axis is a vertical number

line.line.

• It is positive upward and negative It is positive upward and negative downward.downward.


+

-

OriginOrigin• The origin is where the x and y The origin is where the x and y

axes intersect. This is (0, 0).axes intersect. This is (0, 0).

(0, 0)(0, 0)The Coordinate PlaneThe Coordinate Plane by Christine Berg by Christine Berg

QuadrantsQuadrantsThe x and y axes divide the The x and y axes divide the coordinate plane into 4 parts coordinate plane into 4 parts

called called quadrants.quadrants.III

III IV


Ordered PairOrdered PairA pair of numbers (x , y) assigned A pair of numbers (x , y) assigned

to a point on the coordinate to a point on the coordinate plane. plane.


Tests for Functions: Tests for Functions: ““One way you can tell whether a One way you can tell whether a

relation is a function is to analyze the relation is a function is to analyze the graph of the relation using the graph of the relation using the vertical-line test. If any vertical line vertical-line test. If any vertical line passes through more than one point passes through more than one point of the graph, the relation is not a of the graph, the relation is not a function.” Prentice Hallfunction.” Prentice Hall

Vertical-Line TestVertical-Line Test

This is a function because a vertical line hits it only once.

Function Tests:Function Tests:““Another way you can tell whether a Another way you can tell whether a

relation is a function is by making a relation is a function is by making a mapping diagram. List the domain mapping diagram. List the domain values and the range values in order. values and the range values in order. Draw arrows from the domain values Draw arrows from the domain values to their range values.” Prentice Hall to their range values.” Prentice Hall

Mapping DiagramMapping Diagram

(0, -6), (4, 0), (2, -3), (6, 3) are all points on (0, -6), (4, 0), (2, -3), (6, 3) are all points on the previous graph. List all of the domain to the previous graph. List all of the domain to the left; all of the range to the right (in order):the left; all of the range to the right (in order):

DomainDomain: : Range: Range:

0 -60 -6

2 -32 -3

4 04 0

6 3 6 3

Mapping DiagramMapping DiagramThen draw lines between the coordinates.Then draw lines between the coordinates.DomainDomain: : RangeRange:: 0 -60 -6 2 -32 -3 4 04 0 6 3 6 3

If there are no values in the domain that have If there are no values in the domain that have more than one arrow linking them to values in the more than one arrow linking them to values in the range, then it is a function. range, then it is a function.

So this So this isis a function. a function.

Function NotationFunction Notation

f(x) = 3x + 5f(x) = 3x + 5

Output InputFunction RulesFunction Rules by Christine Berg by Christine Berg

FunctionFunctionFunction Notation:Function Notation:

f(x) = 3x + 5f(x) = 3x + 5

Rule for FunctionFunction RulesFunction Rules by Christine Berg by Christine Berg

FunctionFunctionSet up a table using the rule: Set up a table using the rule:

f(x)= 3x+5 f(x)= 3x+5

xx

(Input)(Input)

11 22 33 44 55

yy

(Output)(Output)

88

Function RulesFunction Rules by Christine Berg by Christine Berg

FunctionFunctionEvaluate this rule for these x Evaluate this rule for these x

values: f(x)= 3x+5 values: f(x)= 3x+5

So 3(2) + 5 = 11…So 3(2) + 5 = 11…

xx

(Input)(Input)

11 22 33 44 55

yy

(Output)(Output)

88 1111 1414 1717 2020

Function RulesFunction Rules by Christine Berg by Christine Berg

FunctionsFunctions““You can model functions using rules, You can model functions using rules,

tables, and graphs.” Prentice Halltables, and graphs.” Prentice Hall

Each one shows the relationship from Each one shows the relationship from a different perspective. A table shows a different perspective. A table shows the input / output numbers, a graph is the input / output numbers, a graph is a visual representation, a function a visual representation, a function rule is concise and easy to use. rule is concise and easy to use.

PatternsPatternsPatterns are functions. Patterns are functions.

They’re predictable. They’re predictable.

Patterns may be seen in:Patterns may be seen in:• Geometric FiguresGeometric Figures• Numbers in TablesNumbers in Tables• Numbers in Real-life SituationsNumbers in Real-life Situations• Linear GraphsLinear Graphs• Sequences of NumbersSequences of Numbers

Patterns with TrianglesPatterns with TrianglesJian made some designs using Jian made some designs using

equilateral triangles, as shown equilateral triangles, as shown below. He noticed that as he added below. He noticed that as he added new triangles, there was a new triangles, there was a relationship between relationship between nn, the number , the number of triangles, and of triangles, and pp, the outer , the outer perimeter of the design.perimeter of the design.

from the MCAS

P=3

P= 4

P=5

P=6

Number ofNumber of TrianglesTriangles Outer PerimeterOuter Perimeter

(in units)(in units) 1 31 3 2 4 2 4 3 5 3 5 4 6 4 6 ... … ... … N pN p

from the MCAS

P = 3

P = 4

P = 5

P = 6

TrianglesTriangles

* Write a rule for finding * Write a rule for finding pp, the outer , the outer perimeter for a design that uses perimeter for a design that uses nn triangles.triangles.

from the MCAS

P= 3

P= 4

P= 5

P= 6

P = 3 P = 5

How to Write a Rule:How to Write a Rule:

1)1) Make a table.Make a table.2)2) Find the constant difference.Find the constant difference.3)3) Multiply the constant difference Multiply the constant difference

by the term number (x).by the term number (x).4)4) Add or subtract some number in Add or subtract some number in

order to get y. order to get y. 5)5) Check your rule for at least 3 Check your rule for at least 3

values of x.values of x.

*Does it work? *Does it work?

# of# of TrianglesTriangles Outer PerimeterOuter Perimeter (in units)(in units)

1 3 (+1) 1 3 (+1) 2 4 (+1+1) 2 4 (+1+1) 3 5 (+1+1+1)3 5 (+1+1+1)

****The constant difference is +1. The constant difference is +1. So multiply x by 1 So multiply x by 1 then add 2 then add 2 to get the output number.to get the output number.

from the MCAS

P=3

P=4

P=5

P= 6

f(x)= X + 2f(x)= X + 2

So evaluate and you get:So evaluate and you get:2+1= 3; 2+1= 3; 2+2 = 4; 2+2 = 4; and 3+2 = 5. and 3+2 = 5.

It works!It works!

P = 3

P= 4

P = 5

P = 6

Brick WallsBrick Walls

What’s my rule?

from the MCAS

Now you try one:


1)1) Make a table. Make a table. 2)2) Find the constant difference.Find the constant difference.3)3) Multiply the constant difference Multiply the constant difference





StepsStepsxx f(x) or yf(x) or y

11 77

22 1313

33 1919

The constant difference is +6, so the rule is 6x + 1.

StepsStepsYou can see the You can see the

constant difference.constant difference.

You’re adding 6 blocks each time.

6 blocks

6 blocks

6 blocks

6 blocks

6 blocks

6 blocks

Square TilesSquare Tiles The first four figures in a pattern are The first four figures in a pattern are

shown below.shown below.

* What’s my rule?* What’s my rule?

from the MCAS







Square TilesSquare Tilesxx f(x) or yf(x) or y

11 88

22 1212

33 1616

The constant difference is +4 so the rule is

4x + 4.

+4 blue +4 red +4 green+4 corners

You can see this: You can see this:

Square TilesSquare Tiles

+4 blue +4 red +4 green

+ 4 blue + 4 red + 4 green etc…

+ 4 corners

Extending Patterns in TablesExtending Patterns in TablesBased on the pattern in the input-output table Based on the pattern in the input-output table

below, what is the value of below, what is the value of yy when when xx = 4? = 4?

Input (Input (x)x) OutputOutput(y)(y)

11 77

22 1414

33 2121

44 ??

from the MCAS

Hint: (Write a rule then evaluate.)Hint: (Write a rule then evaluate.)







Extending Patterns in TablesExtending Patterns in TablesBased on the pattern in the input-output table Based on the pattern in the input-output table

below, what is the value of below, what is the value of yy when when xx = 4? = 4?

Input (Input (x)x) OutputOutput(y)(y)

11 77

22 1414

33 2121

44 2828from the MCAS

Patterns in TablesPatterns in TablesA city planner created a table to show the A city planner created a table to show the

total number of seats for different numbers total number of seats for different numbers of subway cars. Copy the table.of subway cars. Copy the table.

What is the rule?What is the rule?

from the MCAS







Subway CarsSubway CarsNumber of Subway Number of Subway CarsCars

Total Number of SeatsTotal Number of Seats

66 180180

88 240240

1010 300300

… … ……

nn ssfrom the MCAS

First, make a table…

Subway CarsSubway Cars

f(x) = 30xf(x) = 30x

Try it!Try it!Write a rule that describes the Write a rule that describes the

relationship between the input (relationship between the input (xx) and ) and the output (the output (yy) in the table below.) in the table below.

Input (Input (xx)) 22 55 1010 1111

Output (Output (yy)) 55 1111 2121 2323

from the MCAS







Input / Output TableInput / Output Table

f(x)=2x + 1f(x)=2x + 1

Patterns in Real-life SituationsPatterns in Real-life Situations

Lucinda earns $20 each week. She Lucinda earns $20 each week. She spends $5 each week and saves the rest. spends $5 each week and saves the rest. The table below shows the total amount The table below shows the total amount that she saved at the end of each week for that she saved at the end of each week for 4 weeks.4 weeks.

What’s the rule?What’s the rule?

from the MCAS







Lucinda’s SavingsLucinda’s Savings

f(x) = $15xf(x) = $15x

from the MCAS

Write a rule Write a rule for the cost of for the cost of nn rides: rides:

from the MCAS







Fall CarnivalFall Carnival

f(x) = $10 + $2xf(x) = $10 + $2x

Patterns in Real-Life Patterns in Real-Life Situations:Situations:

The local library charges the same fine per The local library charges the same fine per day for each day a library book is day for each day a library book is overdue. The table below shows the overdue. The table below shows the amount of the fine for a book that is amount of the fine for a book that is overdue for different numbers of days.overdue for different numbers of days.

Fines for Overdue Fines for Overdue Library BooksLibrary Books

22 44 66 ……

Amount of FineAmount of Fine $0.30$0.30 $0.60$0.60 $0.90$0.90 ……

from the MCASWhat’s the rule? What do they charge for 1 day?







Library FinesLibrary Fines

f(x) = $0.15xf(x) = $0.15x

from the MCAS

Patterns in Graphs #1Patterns in Graphs #1

from the MCAS

What’s the What’s the rule?rule?


1)1) Make a table.Make a table.2)2) Find the constant difference.Find the constant difference.3)3) Multiply the constant difference by Multiply the constant difference by

the term number (x).the term number (x).4)4) Add or subtract some number in Add or subtract some number in




Make a Table of the CoordinatesMake a Table of the Coordinates

(x)(x) (y)(y) -2-2

-1-1

00

11

22from the MCAS


f(x) = x - 4f(x) = x - 4


from the MCAS

What’s my rule?







Make a Table of the Make a Table of the Coordinates:Coordinates:

(x) (x) (y)(y) -1-1 00 11 22

from the MCAS


f(x) = 2x -1f(x) = 2x -1

Patterns in Sequences of Patterns in Sequences of Numbers: Numbers:

12, 16, 20, 24…12, 16, 20, 24…

What’s my rule? What’s my rule?







Patterns in Sequences of NumbersPatterns in Sequences of Numbers

f(x) = 4x + 8f(x) = 4x + 8

Remember: to Write a Rule:Remember: to Write a Rule:





*Then ask: Does it work? *Then ask: Does it work?

Documents

Functions and Patterns by Lauren McCluskey