Functions and Functions and Patterns Patterns by Lauren McCluskey by Lauren McCluskey Exploring the connection between Exploring the connection between input / output tables, patterns, input / output tables, patterns, and functions… and functions…
Mar 31, 2015
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.
The Coordinate PlaneThe Coordinate Plane by Christine Berg by Christine Berg
+
-
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
The Coordinate PlaneThe Coordinate Plane by Christine Berg by Christine Berg
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.
The Coordinate PlaneThe Coordinate Plane by Christine Berg by Christine Berg
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:
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?
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
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?
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.)
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?
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
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?
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
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?
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
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?
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
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?
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?
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?
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?
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 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
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?
Make a Table of the CoordinatesMake a Table of the Coordinates
(x)(x) (y)(y) -2-2
-1-1
00
11
22from the MCAS
Patterns in Graphs #1Patterns in Graphs #1
f(x) = x - 4f(x) = x - 4
Patterns in Graphs #2Patterns in Graphs #2
from the MCAS
What’s my rule?
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?
Make a Table of the Make a Table of the Coordinates:Coordinates:
(x) (x) (y)(y) -1-1 00 11 22
from the MCAS
Patterns in Graphs #2Patterns in Graphs #2
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?
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?
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:
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.
*Then ask: Does it work? *Then ask: Does it work?