Three-Column Assessment
Solve
8𝑥 − 5 4𝑥 + 1 = −1 + 2 −7𝑥 − 3
Agenda
Details
Test
Books
INB
Website Accounts
INB
Front of 1st page – Express yourself
Back of 1st page – Table of Contents (I) or (i)
Front of 2nd page – Table of Contents (II) or (ii)
Back of 2nd page – Table of Contents (III) of (iii)
Front of 3rd page – Table of Contents (IV) or (iv)
Back of 3rd page – Table of Contents (V) or (v)
Front of 4th page – Algebra 2 Formula Sheet (VI) or (vi)
Back of 4th page – Geometry Reference Page (VII) or (vii)
Front of 5th page – Geometry Reference Page (VIII) or (viii)
Back of 5th page – Frayer Model (page 1) or (pg. 1) or (1)
Model
Front of 6th page – Power Cornell Notes (page 2) or (pg. 2) or (2)
Website accounts
www.mangahigh.com
first_last – password - initials
www.quizlet.com
www.khanacademy.com
www.blendspace.com
www.brainscape.com
Algebra 2
1.1
Topic: Input/Output
Tables
Objectives: SWBAT
1) Identify and describe specific patterns in input-output
tables.
2) Determine whether a linear function matches a table.
Tool Kit
Relations
Input
Output
Function
Domain
Range
Activate Prior Knowledge
What is a relation?
What is a function?
What is an input of a function?
What is an output of a function?
Are all relations functions?
Are all functions relations?
How can you tell if a relation is a function or not?
Terminology
Relation:
-is just a relationship between sets of numbers
Example:
-People (x) in this class and their heights (y)
You name off someone and I can give you a height.
Or you give me a height and I can give you the name of
everyone that is that tall. (Hint: there can be more than one
person with the same height)
The set of all the inputs (x-values) is called the _________.
Domain
The set of all the outputs (y-values) is called the _________.
Range
More Terminology
Function:
-is a relation where each element in the domain is
paired up with exactly one element in the range
Yes or No?
Just Remember the 3 D’s:
Don’t
Double
the Domain
Things you have done in the
past.
Find the value of, y = 5𝑥 − 9 , when 𝑥 = 3, 7, −5, 0.
Input
Process
Output
3
7
-5
0
How does the value you got “out” RELATE to the value
you put “in”?
Function Table Game
Rules:
Draw all tables
Find a simple rule that agrees with each table
Ways to describe (or relate) each:
Input to Output
Output to output
Examples:
In Words – take away 3 from each output to get the next
output
In Words – the output is half of the input
Algebraically – y = 3x + 1
Combination – output = input x 5
Me
Table A
Input, n
Output, A(n)
0
0
1
2
2
4
3
6
4
8
Possible Answers:
 Each output is 2 more than the previous
output.
 𝑂𝑢𝑡𝑝𝑢𝑡 = 𝑖𝑛𝑝𝑢𝑡 × 2
 𝑦 = 2𝑥
 𝐴 𝑛 = 2𝑛
function notation
We
Table B
Input, n
Output, B(n)
0
0
1
2
2
6
3
12
4
20
Two
Table C
Input, n
Output, C(n)
0
2
1
1
2
0
3
-1
4
-2
You
Table D
Input, n
Output, D(n)
0
0
1
3
2
8
3
15
4
24
Upper Level
Table E
Input
Output
0
6
1
12
2
20
3
30
4
42
What are all the possible
operations that can be done to
the input?
Stand Up, Board Up, Pair
Up
Input
Output
0
0
1
4
2
16
3
36
4
64
Stand Up, Board Up, Pair
Up
Input
Output
0
2
1
3
2
6
3
11
4
18
Stand Up, Board Up, Pair
Up
Input
0
1
2
3
4
Output
-3
2
7
12
17
Stand Up, Board Up, Pair
Up
Input
0
1
2
3
4
Output
-16
-15
-12
-7
0
Question
Why are tables important?
Exit Slip
Verbal: You have a job washing
dishes. You are paid $7.50 an hour.
Graph:
Algebraic:
𝑝 = 7.5ℎ
Where p is your pay and h
is the number of hours
you work.
Table:
# of hours
2
4
5
8
10
Wages
Exit Slip
Verbal: Your sister’s hair is 6 inches Algebraic:
long and grows an inch each month.
ℎ=𝑚+6
Graph:
Where h is hair length
and m is the number of
months.
Table:
# of
months
0
1
2
3
4
Length of
hair
Exit Slip
Verbal: Two rabbits live in the new
park. The rabbit population doubles
each year.
Algebraic:
𝑟 = 2𝑦 ∙ 2 𝑜𝑟 2𝑦+1
Where r is the rabbit
population and y is the
number of years.
Graph:
Table:
# of
years
# of
rabbits
0
1
2
3
4
Exit Slip
Verbal: You have a $10 iTunes gift
card. Each song now costs $1.50.
Algebraic:
𝑏 = 10 − 1.5𝑠
Where b is the balance on
your gift card and s is the
number of songs you have
purchased.
Graph:
Table:
# of songs
purchased
Balance on
Card
1
2
3
5
7
Assignment
Make sure you copy all tables
Pgs. 6 – 7 (3-13,16, 17-20*)
Algebra 2
1.2
Topic: Domain and
Range
Objectives: SWBAT
1)
Differentiate between a closed-form and a recursive function and write
both definitions of a function.
2)
State the domain and range of each function and describe the
differences between the two types.
Tool Kit
Domain
Range
Function Notation
Closed-form Definition
Recursive Definition
Card Sort
Each group gets a set of cards
Must sort them in groups of 4
Each group must have the corresponding table, graph,
relationship (pattern) and verbal form that represents
them as a whole
When you are done, raise your hand and I will check
to see if you got all of them right
Exploring Tables
Can you describe the pattern (rule) in this table in two
ways?
Input to Output:
Input
Output
0
3
1
8
2
13
3
18
4
23
Output to Output:
Two Different Ways to
Define a Function
Closed Form
is an equation (in function notation) that describes the
pattern (or relationship) from input to output
lets you find any output for any input by direct
calculation
Recursive
is an equation that describes the pattern from output
to output
to find any value using recursive definition you must
have the previous output value to find the next output
value
Thinking of the Real World
Closed Form
Recursive
Exploring Tables
Write the closed-form and the recursive definition of this
table?
Closed Form:
Input
Output
0
3
1
8
2
13
3
18
4
23
𝑓 𝑛 = 5𝑛 + 3
Recursive:
𝑓 𝑛 =
3
𝑖𝑓 𝑛 = 0
𝑓 𝑛 − 1 + 5 𝑖𝑓 𝑛 > 0
Function Notation (ClosedForm)
𝑓 𝑛 = 5𝑛 + 3
Find:
𝑓(0)
𝑓(18)
Recursive Definition
3
𝑖𝑓 𝑛 = 0
𝑓 𝑛 =
𝑓 𝑛 − 1 + 5 𝑖𝑓 𝑛 > 0
Find:
𝑓(0)
𝑓(1)
𝑓(2)
𝑓(18)
We
Write the closed-form and recursive definitions of the table
below.
Closed Form:
Input
Output
0
3
1
5
2
7
3
9
4
11
Recursive:
We
Write the closed-form and recursive definitions of the table
below.
Closed Form:
Input
Output
0
5
1
9
2
13
3
17
4
21
Recursive:
Group
Write the closed-form and recursive definitions of the table
below.
Closed Form:
Input
Output
0
-2
1
-4
2
-6
3
-8
4
-10
Recursive:
All together
Write the closed-form and recursive definitions of the table
below.
Closed Form:
Input
Output
0
0
1
3
2
8
3
15
4
24
5
35
6
48
Recursive:
Terminology
Domain:
-are the values that go “into” a function
-or it is the inputs/x-values
-when they ask for the domain of a function they are
asking for all the possible values you could put into a
function.
Range:
-are the values that come “out” a function
-or it is the output/y-values
-when they ask for the range of a function they are
asking for all the possible values that could come out of
the function.
Closed-Form vs. Recursive
Closed-Form:
Input to Output
Domain: All real #’s
Range: All real #’s (for linear functions)
Recursive:
Output to Output
Domain: All real #’s ≥ 0
Range: Depends on first output value
We
State the domain and range of each function for each
definition.
Closed Form:
Input
Output
0
3
1
5
2
7
3
9
4
11
Recursive:
We
State the domain and range of each function for each
definition.
Closed Form:
Input
Output
0
5
1
9
2
13
3
17
4
21
Recursive:
Group
State the domain and range of each function for each
definition.
Closed Form:
Input
Output
0
-2
1
-4
2
-6
3
-8
4
-10
Recursive:
All together
State the domain and range of each function for each
definition.
Closed Form:
Input
Output
0
0
1
3
2
8
3
15
4
24
5
35
6
48
Recursive:
Fan-and-Pick
Exploring Tables
Can you use each function to find the output when the input is -2?
Closed Form:
Input
Output
0
-1
1
1
2
3
3
5
4
7
Recursive:
Building Tables
Building Tables
Building Tables
Question?
Why should we worry about recursive functions?
Assignment
Writing Functions and Building Tables WS
Bell-Ringer
Given the recursive definition of a function, create an
input/output table for the first 5 values starting with 0.
−7,
𝑓 𝑛 =
𝑓 𝑛 − 1 + 3𝑛,
𝑖𝑓 𝑛 = 0
𝑖𝑓 𝑛 > 0
Algebra 2
1.2 (Part 2)
Objectives: SWBAT
1) Students will use recursive definitions to build tables of
data and integrate that knowledge with spreadsheets.
Closed-Form Vs. Recursive
Without looking think of ways that closed-form and
recursive definition differ:
Direct Calculation vs. Building from previous
Inputs to Outputs vs. Outputs to outputs
2 or more processes vs. 1 process
Domains are different
Ranges are different
What’s the purpose of
recursive functions?
Buying a Car
Buying a Car
Suppose you buy a new car for $15,000. You put no
money down and have no interest. The bank says that
you will need to make a minimum payment of
$300/month.
Table
Function
Months Balance
0
1
2
3
4
Graph
Buying a Car
Suppose you buy a new car for $15,000. You put no
money down and sign up for a 7% interest rate. The
bank says that you will need to make a minimum
payment of $300/month.
Table
Months
0
1
2
3
4
Process
Balance
Months
Process
Balance
0
$15,000
$15, 000
1
$15,000 +
0.07
× 15,000 − 300
12
$14,787.50
2
0.07
$14,787.50 +
× $14,787.50
12
− 300
3
0.07
$14,573.76 +
× $14,573.76
12
− 300
4
$14,573.76
$14,358.77
Programming Functions
Grab a Computer
Assignment
Finish the Programming Assignment
page 12 (1-6, 8, 12-14) – make sure you read about
difference tables before you start on page 12
Fan-and-Pick
-Partner up (Partner A and Partner B)-need boards
-fan out the cards with the tables facing Partner A
-Partner B will pick out a card without looking at it
-Partner A will cover up the bottom (answer) with
another card
-Partner B will write down the closed-form and
recursive definition of the table
-When they get it……put the card off to the side
-Switch roles(you get 6 minutes)
Before Cornell Notes
Leave a page blank for a foldable that we will create
together.
Don’t copy the things in red into notes
Algebra 2
1.3
Topic: Constant
Difference
Objectives: SWBAT
1) Make a connection with the constant difference in
recursive definition and the closed form coefficient.
Tool Kit
Constant Difference
Slope
y-intercept
Closed-Form and Recursive
Connection
Create the recursive definition of the table below.
In (n)
Out, f(n)
0
3
1
8
2
13
3
18
4
23
Recursive:
3,
𝑖𝑓 𝑛 = 0
𝑓 𝑛 =
𝑓 𝑛 − 1 + 5 𝑖𝑓 𝑛 > 0
Today we are going to look at an extra column
of the Input/Output tables……this will help
guide us to a connection between recursive and
closed-form
Input, n
Output, f(n)
∆
0
3
5
1
8
5
2
13
5
3
18
5
4
23
Warning! This only works when the input goes by 1
unit.
This is called
the constant
difference
(or delta)
because the
difference
from one
output to the
next is
constant
Exploring-the table and recursive
definition
Say your looking for 𝑓(4) and all you know is the recursive definition.
𝑓 4 =𝑓 3 +5
But to know 𝑓(3) you need to know 𝑓(2). So, that means you need the
recursive definition again.
𝑓 3 =𝑓 2 +5
Now if I combine those two steps together: 𝑓 4 = 𝑓 2 + 5 + 5. Two steps,
adding two fives. Let’s keep going…..if you want 𝑓(2), you need 𝑓(1).
𝑓 2 =𝑓 1 +5
But to know 𝑓(1) you need to know 𝑓(0). So, that means you need the
recursive definition again.
𝑓 1 =𝑓 0 +5
Now if I combine all those 4 steps: 𝑓 4 = 𝑓 0 + 5 + 5 + 5 + 5.
Four steps of adding 5 to the beginning.
To simplify that instead of adding 5 four times…..do it this way
𝑓 4 =𝑓 0 +4∙5
The definition tells me that 𝑓 0 = 3, so 𝑓(4) is 3 plus 4 times 5.
Does it work?
Making the connection
Using the last exploration
Find 𝑓(17)
That means 3 (𝑓(0)) plus 17 fives or just 3 + 17 ∙ 5.
Now lets make that into a Closed-Form Definition of the
Function
𝑓 𝑛 = 5𝑛 + 3
Does it Work?
Find:
𝑓 15
𝑓 15 = 78
Closed-Form and Recursive
Connection
Create the recursive definition of the table below.
Input, n
Output, f(n)
∆
0
-23
9
1
-14
9
2
-5
9
3
4
9
4
13
9
5
22
9
6
31
Recursive:
−23,
𝑖𝑓 𝑛 = 0
𝑓 𝑛 =
𝑓 𝑛 − 1 + 9,
𝑖𝑓 𝑛 > 0
Now fill out the
difference
column.
Closed-Form-previous table
To get 𝑓(6) I would need to add 9 to 𝑓(0) 6 times….looks like this:
𝑓 6 = −23 + 6 ∙ 9
Because I need to add 6 steps of 9 to the 𝑓(0), the beginning.
Well I don’t like it that way:
𝑓 6 = 6 ∙ 9 − 23
And to find 𝑓
And to find 𝑓
And to find 𝑓
And to find 𝑓
And to find 𝑓
5
4
3
2
1
= 5 ∙ 9 − 23
= 4 ∙ 9 − 23
= 3 ∙ 9 − 23
= 2 ∙ 9 − 23
= 1 ∙ 9 − 23
So, the closed-form would be 𝑓 𝑛 = 9𝑛 − 23
Notice the 9 comes from the constant difference and the -23
comes from the value at 𝑓 0 .
Making the Connection
Input, n
Output, f(n)
∆
0
-23
9
1
-14
9
2
-5
9
3
4
9
4
13
9
5
22
9
6
31
The recursive is adding 9 to
previous output (constant
different.
−23,
𝑖𝑓 𝑛 = 0
𝑓 𝑛 =
𝑓 𝑛 − 1 + 9,
𝑖𝑓 𝑛 > 0
Notice: -23 is the value of the
function (output) when n = 0 (or
x(input), we could say)
The closed-form is 𝑓 𝑛 = 9𝑛 − 23
-the 9 comes from the constant difference
-and the -23 came from when the input is 0
Can you see the connection?????
Something you have seen
before
𝑦 = 𝑚𝑥 + 𝑏
What is this?
Yes, slope-intercept form.
What is the m?
Yes, the slope.
What is the b.
Yes, the y-intercept.
What is the x-coordinate at the y-intercept?
Yes, 0.
Going back to Definition of a
Function
Input, n
Output, f(n)
∆
0
-7
-3
Lets Graph these
points and see
what it creates:
(0, -7)
(1, -10)
1
-10
-3
2
-13
-3
3
-16
-3
4
-19
(2, -13)
(3, -16)
(4, -19)
Graphing
Grab a computer and go to www.desmos.com.
Go to tables of data
Go to add item and put information into table
Fill out table
You may have to zoom out to see what it creates
-Go to next slide after everyone has computer and
information put in
Look at this table in recursive definition:
−7,
𝑖𝑓 𝑛 = 0
𝑓 𝑛 =
𝑓 𝑛 − 1 − 3,
𝑖𝑓 𝑛 > 0
The n’s are our inputs (x-values) so when n (or x) = 0 the output (y-value) is
-7…….so would that be the y-intercept?
What is the slope of your graph?
−3
Every time it runs 1 it falls 3 or 1 𝑜𝑟 − 3. How does the delta column
relate to this?
Does this stay CONSTANT throughout the
whole graph?
Now write the slope-intercept form of this graph:
𝑦 = −3𝑥 − 7
Could this be our closed-form of it?
𝑓 𝑛 = −3𝑛 − 7
Does it work?
When the input is 0, then the output will
be the y-intercept.
Input, n
Output, f(n)
∆
0
-7
-3
1
-10
-3
2
-13
-3
3
-16
-3
4
-19
So the constant
difference is the slope
Fill out the missing pieces
and then graph information
Input
Output
∆
-3
-3
2
-2
-1
2
-1
1
2
0
3
2
1
5
After graphing, answer
these questions with your
group: (don’t need to write
answer just discuss except
for the #4)
1. What does it create?
2. What is the slope?
3. What is the y-intercept?
4. Write the recursive and
closed-form of this
function. Does the
closed-form have any
relationship with slopeintercept form?
Fill out the missing pieces
and then graph information
Input
Output
∆
0
-3
6
1
3
6
2
9
6
3
4
15
21
6
After graphing, answer
these questions with your
group: (don’t need to write
answer just discuss except
for the #4)
1. What does it create?
2. What is the slope?
3. What is the y-intercept?
4. Write the recursive and
closed-form of this
function. Does the
closed-form have any
relationship with slopeintercept form?
Fill out the missing pieces
and then graph information
Input
Output
∆
2
0
-4
3
-4
-4
4
-8
-4
5
-12
-4
6
-16
After graphing, answer
these questions with your
group: (don’t need to write
answer just discuss except
for the #4)
1. What does it create?
2. What is the slope?
3. What is the y-intercept?
4. Write the recursive and
closed-form of this
function. Does the
closed-form have any
relationship with slopeintercept form?
Fill out the missing pieces and then graph informationwarning the inputs don’t go by 1 unit and the table
doesn’t’ start at 0.
Input
Output
∆
-5
8
4
-3
12
4
-1
16
4
1
20
4
3
24
After graphing, answer
these questions with your
group: (don’t need to write
answer just discuss except
for the #4)
1. What does it create?
2. What is the slope?
3. What is the y-intercept?
4. Write the recursive and
closed-form of this
function. Does the
closed-form have any
relationship with slopeintercept form?
Fill out the missing pieces and then graph informationwarning the input doesn’t start at 0.
Input
Output
∆
-1
6.5
1
2
0
7
1
2
1
7.5
1
2
2
8
1
2
3
8.5
After graphing, answer
these questions with your
group: (don’t need to write
answer just discuss except
for the #4)
1. What does it create?
2. What is the slope?
3. What is the y-intercept?
4. Write the recursive and
closed-form of this
function. Does the
closed-form have any
relationship with slopeintercept form?
Assignment
page 19 (1-8, 10-14, 16-17)
Algebra 2
1.4
Topic: Slope
Objectives: SWBAT
1) Link new information of linear functions to
existing knowledge of y = mx + b.
Foldable
Ways to say domain:
Input
x-values
x
n
What goes in
independent variable
x-axis
Ways to say range:
Output
y-values
y
f(n)
What comes out
dependent variable
y-axis
Old Information
Copy the table.
-2
-7
-1
-4
0
-1
1
2
2
5
3
8
Recursive:
Write the slope-intercept form of
this line.
Snap Shot: Graph
−2, −7
−1, −4
0, −1
1,2
Fill out the 2,5
∆ column.
3,8
Closed-form:
What did it create?
What is the slope of this
graph?
Where does it cross the yaxis?
Good Quiz Questions
What type of function is formed when there are constant
differences?
What is the constant difference to the function?
What is f(0)? What is its significance?
What does domain mean?
What does range mean?
What is a closed form function?
What is a recursive function?
What are the 4 differences between closed and recursive
functions?
Exploration of the Constant Difference
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ∆ = 𝑆𝑙𝑜𝑝𝑒 =
Copy the table below.
In
Out
0
1
22
2
3
4
10
∆
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 Δ𝑦
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡𝑠
Δ𝑥
Not much information but
can get plenty from it (ME)
In
Out
-5
9
-2
3
Find the slope or constant
change.
Could you find the closed-form
from this information?
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 Δ𝑦
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ∆ = 𝑆𝑙𝑜𝑝𝑒 =
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡𝑠
Δ𝑥
Not much information but
can get plenty from it (WE)
In
Out
2
14
7
44
Find the slope or constant
change.
Could you find the closed-form
from this information?
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 Δ𝑦
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ∆ = 𝑆𝑙𝑜𝑝𝑒 =
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡𝑠
Δ𝑥
Not much information but
can get plenty from it (TWO)
In
Out
-1
8
5
11
Find the slope or constant
change.
Could you find the closed-form
from this information?
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡𝑠 Δ𝑦
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ∆ = 𝑆𝑙𝑜𝑝𝑒 =
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡𝑠
Δ𝑥
Stand Up, Board Up, Pair
Up
When the input is 4 the output is 21. When the input is
12, the output is 37. What is the slope?
What is the output when the input is 0? What is the
significance of this value?
Can you find the closed-form?
Stand Up, Board Up, Pair
Up
Find slope, Find f(0), Write Closed-form
In
Out
3
6
18
In
Out
-2
-6
8
4
11
Assignment
Pg. 24 (1-15)
QUIZ NEXT CLASS!!!
3-Column Assessment
The table below represents a linear function. Find the
slope ∆ , f(0), and Closed-form of the following table.
In, n
Out, f(n)
14
1
26
7
Algebra 2
1.5(Part 1)
Topic: Second
Difference
Objectives: SWBAT
1) Use the difference column to decide what type of
function can fit a table.
2) Discover the meaning of differences beyond the first ∆.
Use that info to write rules for higher order functions.
Tool Kit
Linear Function
Polynomial
Terms
Coefficient
Leading Coefficient
Quadratic Function
Terminology
Terms – are the parts that make up an expression or
equation (multiplication glues things together)
Coefficient – the number in front of a variable (it is the
numerical factor of a term)
Example:
𝑓 𝑥 = 2𝑥 + 9
Find the ∆, Find f(0), and
Find the Closed-Form
∆:
Input, n
Output,
A(n)
0
0
1
2
2
4
3
6
4
8
𝐴 0 :
𝐶𝑙𝑜𝑠𝑒𝑑 − 𝐹𝑜𝑟𝑚:
How can you tell this is a
linear function?
Find the ∆, Find f(0), and
Find the Closed-Form
Input, n
Output,
C(n)
0
2
1
1
2
0
3
-1
4
-2
∆:
𝐶 0 :
𝐶𝑙𝑜𝑠𝑒𝑑 − 𝐹𝑜𝑟𝑚:
How can you tell this is a
linear function?
Find the ∆, Find f(0), and
Find the Closed-Form
Input
0
1
2
3
4
Output
-3
2
7
12
17
∆:
𝑓 0 :
𝐶𝑙𝑜𝑠𝑒𝑑 − 𝐹𝑜𝑟𝑚:
How can you tell this is a
linear function?
Find the ∆, Find f(0), and
Find the Closed-Form
Input, n
Output,
B(n)
0
0
1
2
2
6
3
12
4
20
∆:
𝐵 0 :
𝐶𝑙𝑜𝑠𝑒𝑑 − 𝐹𝑜𝑟𝑚:
How can you tell this is not a
linear function?
Find the ∆, Find f(0), and
Find the Closed-Form
Input
Output
0
6
1
12
2
20
3
30
4
42
∆:
𝑓 0 :
𝐶𝑙𝑜𝑠𝑒𝑑 − 𝐹𝑜𝑟𝑚:
How can you tell this is not a
linear function?
Problem from 1.2
The table below represents the function 𝑓 𝑥 = 𝑎𝑥 + 𝑏. Fill out the
outputs for the corresponding inputs and then fill out the ∆ column.
Input
(𝒙)
Output
𝒇 𝒙 = 𝒂𝒙 + 𝒃
∆
0
𝑏
𝑎
1
𝑎+𝑏
𝑎
2
2𝑎 + 𝑏
𝑎
3
3𝑎 + 𝑏
𝑎
4
4𝑎 + 𝑏
Ways to tell if it is a linear
function:
The difference from one output to another output is
constant…….meaning it is the same throughout the
whole table (per input)
The highest degree (power) of the variable is 1 in the
polynomial you created from the Closed-Form
Input, n
Output,
C(n)
0
2
1
1
2
0
3
-1
4
-2
∆: -1
𝐶 0 :2
𝐶𝑙𝑜𝑠𝑒𝑑 − 𝐹𝑜𝑟𝑚: 𝑓 𝑛 = −𝑛 + 2
Extending the Differences
In
Out
∆
∆𝟐
0
1
1
2
1
2
3
2
2
5
5
2
3
10
7
4
17
Terminology
Leading Coefficient – is the coefficient of the term with
the highest degree(power)
Quadratic Function - a polynomial with a degree of 2
Example:
𝑓 𝑥 = 4𝑥 2 + 9𝑥 − 15
Second Difference Needed
The table below represents the function 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Fill out
the outputs for the corresponding inputs and then fill out the ∆
column. Does it come out constant?
In
Out
∆
∆𝟐
(𝒙)
𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
0
1
2
3
4
𝑐
𝑎+𝑏
2𝑎
𝑎+𝑏+𝑐
3𝑎 + 𝑏
2𝑎
4𝑎 + 2𝑏 + 𝑐
5𝑎 + 𝑏
2𝑎
9𝑎 + 3𝑏 + 𝑐
7𝑎 + 𝑏
16𝑎 + 4𝑏 + 𝑐
Input
Output
0
6
1
12
2
20
3
30
4
42
Linear or Quadratic??
In
Out
0
8
1
11
2
16
3
23
4
32
∆
∆𝟐
Linear, Quadratic, or
Something else??
In
Out
0
5
1
6
2
13
3
32
4
69
∆
∆𝟐
∆𝟑
Assignment
page 34 (3-5, 7, 13, 17)
#5 - pick an equation where the leading coefficient ≠ 1
and has more than 1 term
Algebra 2
1.5(Part 2)
Topic: Writing
Quadratic Functions
Objectives: SWBAT
1) Discover the meaning of differences beyond the first ∆.
Use that info to write rules for additional functions.
Taking a Look Back
The table below represents the function 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Fill out
the outputs for the corresponding inputs and then fill out the ∆
column. Does it come out constant?
In
Out
∆
∆𝟐
(𝒙)
𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
This will help us
with writing a
Quadratic Function
0
1
2
3
4
𝑐
𝑎+𝑏
2𝑎
𝑎+𝑏+𝑐
3𝑎 + 𝑏
2𝑎
4𝑎 + 2𝑏 + 𝑐
5𝑎 + 𝑏
2𝑎
9𝑎 + 3𝑏 + 𝑐
7𝑎 + 𝑏
16𝑎 + 4𝑏 + 𝑐
What you should have
noticed
The 𝑓(0) is the 𝑐.
The ∆2 is twice as much as the leading coefficient 𝑎 .
The ∆ of 𝑓 1 and 𝑓 0 is equal to 𝑎 + 𝑏.
Taking a Deeper Look
The table below represents the function, 𝑓 𝑥 = 3𝑥 2 + 4𝑥 + 5. Find
the outputs for the corresponding inputs. Find the ∆ and ∆2 . Then
relate the things you find with the 𝑎 − 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡, 𝑏 −
𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡, and 𝑐 − 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
In
(𝒙)
Out
𝒇 𝒙 = 𝟑𝒙𝟐 + 𝟒𝒙 + 𝟓
∆
∆𝟐
5
7
6
1
12
13
6
2
25
19
6
3
44
25
4
69
0
Foldable
Steps
Step #1
-Find the 𝑓 0 value…this is your 𝑐 − 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
Step #2
-Find the ∆ and ∆2 ….the ∆2 is twice your 𝑎 −
𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡….∆2 ÷ by 2
Step #3
-Use your ∆ between 𝑓 1 and 𝑓 0 …this is your 𝑎 + 𝑏,
use this to find your 𝑏 − 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
Step #4
-Write your function in the form, 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐,
with all the missing pieces
Honors
Writing the Recursive Definition of a Quadratic Function
Step #1
-Find your ∆ and ∆2
Step #2
𝑓 𝑛 =
𝑓𝑖𝑟𝑠𝑡 𝑜𝑢𝑡𝑝𝑢𝑡,
𝑖𝑓 𝑛 = 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑖𝑛𝑝𝑢𝑡
𝑓 𝑛 − 1 + ∆2 𝑛 − 1 + ∆(𝑓 1 − 𝑓(0)), 𝑖𝑓 𝑛 > 𝑜𝑟 < 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑖𝑛𝑝𝑢𝑡
Together
Fill out table…..figure out if it is linear or quadratic then write the function that
represents the table
Input
Output
0
-2
1
0
2
16
3
46
4
90
∆
∆𝟐
Your Turn
Fill out table…..figure out if it is linear or quadratic then write the function that
represents the table
Input
Output
0
-6
1
1
2
14
3
33
4
58
∆
∆𝟐
One more
Fill out table…..figure out if it is linear or quadratic then write the function that
represents the table
Input
Output
0
-5
3
-4
6
-3
9
-2
12
-1
∆
∆𝟐
Group Activity
Person #1
Person #2
Person #3
Person #4
In
Out
In
Out
In
Out
In
Out
0
10
0
-8
0
-1
0
11
1
17
1
-13
1
7
1
9
2
32
2
-16
2
21
2
17
3
55
3
-17
3
41
3
35
4
86
4
-16
4
67
4
63
Answers-Closed-Form
#1
𝑓 𝑥 = 4𝑥 2 + 3𝑥 + 10
#2
𝑓 𝑥 = 𝑥 2 − 6𝑥 − 8
#3
𝑓 𝑥 = 3𝑥 2 + 5𝑥 − 1
#4
𝑓 𝑥 = 5𝑥 2 − 7𝑥 + 11
Answers-Recursive-Honors
#1
−10,
𝑓 𝑛 =
𝑓 𝑛 − 1 + 8 𝑛 − 1 + 7,
𝑖𝑓 𝑛 = 0
𝑖𝑓 𝑛 > 0
−8,
𝑓 𝑛 − 1 + 2 𝑛 − 1 − 5,
𝑖𝑓 𝑛 = 0
𝑖𝑓 𝑛 > 0
−1,
𝑓 𝑛 =
𝑓 𝑛 − 1 + 6 𝑛 − 1 + 8,
𝑖𝑓 𝑛 = 0
𝑖𝑓 𝑛 > 0
11,
𝑓 𝑛 =
𝑓 𝑛 − 1 + 10 𝑛 − 1 − 2,
𝑖𝑓 𝑛 = 0
𝑖𝑓 𝑛 > 0
#2
𝑓 𝑛 =
#3
#4
Assignment
page 34 (1-2, 10, 14-15)
Algebra 2
1.6
Topic: Gathering
Data
Objectives: SWBAT
1) Decide whether a linear function reasonably
represents a data set.
Warning!!
You will be having a project due for Unit 1 over
gathering data and making connections with the data
Tool Kit
Correlation
Positive Correlation
Negative Correlation
Line of Best Fit
Balance Point
Linear Regression
Point-Slope Form
Terminology
Correlation
-The word correlation is made of CO-(meaning “together”) and
RELATION. Coming to the conclusion it means finding a relationship
between two or more things…….(in our case inputs-outputs)
Positive Correlation
-a correlation is positive if values increase together
Negative Correlation
-a correlation is negative if one value decreases as the other
increases
Line of Best Fit
-is a line that is drawn through the data on a scatter plot to describe
the trend of the data
Linear Regression
-a statistical procedure in which a straight line is established
through a data set that best represents a relationship between the two sets
of data
Outlier
-a data point that is far away from the rest and looks like it doesn’t
belong in the data
Point-Slope Form
-if given a point ℎ, 𝑘 that is on a line and given the
slope 𝑚 of that line, the following equation can be used
to find the equation of the line:
𝑦 − 𝑘 = 𝑚(𝑥 − ℎ)
Example:
A line passes through the point 5, −3 and has a slope
of -5. Find the equation 𝑦 = 𝑚𝑥 + 𝑏 of the line.
Me, We, Two, You
Given the point 4,3 and the slope 2. Find the equation of the
line that contains this point and has this slope.
Given the point −1,0 and the slope 2. Find the equation of
the line that contains this point and has this slope.
Given the point −12, −5 and the slope 4. Find the equation
of the line that contains this point and has this slope.
1
Given the point 12, −5 and the slope . Find the equation of
2
the line that contains this point and has this slope.
Levels of Correlation
Ice Cream Sales vs
Temperature
Temperature
°C
Ice Cream
Sales
14.2°
$215
16.4°
$325
11.9°
$185
15.2°
$332
18.5°
$406
22.1°
$522
19.4°
$412
25.1°
$614
23.4°
$544
18.1°
$421
22.6°
$445
17.2°
$408
Conclusion: (Making a
connection)…The data above has a high
positive correlation because as the
temperature increases the sales in ice
cream increases as well.
Partner Exploration
Finding the Line of Best Fit
by Hand
-Find a balance point….means to find the average of the
inputs 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 and the average of the outputs
𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠 …this is called point 𝑥, 𝑦
-Find the average of the ∆′ 𝑠 (this will be your average
slope)
-Plug information into the point-slope form using 𝑥, 𝑦
as your point and your average of the ∆′ 𝑠 as your slope
Letting the Calculator do
the Work
-Make sure Diagnostics are turned on
𝑆𝑇𝐴𝑇 → 𝐸𝐷𝐼𝑇 → 1: 𝐸𝐷𝐼𝑇 … → 𝐸𝑁𝑇𝐸𝑅 → 𝐿1
CLEAR L1:
Push the up arrow→ 𝐶𝐿𝐸𝐴𝑅 → 𝑑𝑜𝑤𝑛 𝑎𝑟𝑟𝑜𝑤
Enter your rolls in L1
Clear L2
Enter Totals in L2
2nd Quit
Continued….
𝑆𝑇𝐴𝑇 → 𝐶𝐴𝐿𝐶 → 4: 𝐿𝑖𝑛𝑅𝑒𝑔 𝑎𝑥 + 𝑏 → 𝐸𝑁𝑇𝐸𝑅 → 𝐿1 , 𝐿2 → 𝐸𝑁𝑇𝐸𝑅
𝐿𝑖𝑛𝑅𝑒𝑔
𝑦 = 𝑎𝑥 + 𝑏
𝑎 = Slope
𝑏 = y-intercept
𝑟 2 = Coefficient of Determination
𝑟 = Correlation Coefficient
-the 𝑟 value appears only with linear functions. It tells you if the
correlation is positive or negative and how strong…..closer to +1 the
more stronger positive correlation….closer to -1 the more stronger
negative correlation
-the 𝑟 2 value tells you how closely the line matches the data – called
the coefficient of determination, it is a percentage of points that
are explained by the equation
Graphing Investigation with
Calculator(Dice Roll)
Make sure Scatter Plot is on
𝑌 =→ 𝑢𝑝 𝑎𝑟𝑟𝑜𝑤 → 𝑃𝐿𝑂𝑇 1 → 𝐸𝑁𝑇𝐸𝑅 → 𝑑𝑜𝑤𝑛 𝑎𝑟𝑟𝑜𝑤
Window
 Xmin:
 Xmax:
 Xscal:
 Ymin:
 Ymax:
 Yscal:
0
20
1
0
120
5
Now Let the Computer Do
the Work
Assignment
Point-Slope WS
Fitting Lines to Data
(Olympics)
• Look at the table and scatter plot on page 46
• What does the table of data display?
• What type of function does the scatter plot most closely
represent?
• What type of correlation does the data have?
Positive/Negative? How can you tell?
Letting the Calculator do
the Work
-Make sure Diagnostics are turned on
𝑆𝑇𝐴𝑇 → 𝐸𝐷𝐼𝑇 → 1: 𝐸𝐷𝐼𝑇 … → 𝐸𝑁𝑇𝐸𝑅 → 𝐿1
CLEAR L1:
Push the up arrow→ 𝐶𝐿𝐸𝐴𝑅 → 𝑑𝑜𝑤𝑛 𝑎𝑟𝑟𝑜𝑤
Enter the years of the Olympics in L1
Clear L2
Enter the times into L2
2nd Quit
Continued….
𝑆𝑇𝐴𝑇 → 𝐶𝐴𝐿𝐶 → 4: 𝐿𝑖𝑛𝑅𝑒𝑔 𝑎𝑥 + 𝑏 → 𝐸𝑁𝑇𝐸𝑅 → 𝐿1 , 𝐿2 → 𝐸𝑁𝑇𝐸𝑅
𝐿𝑖𝑛𝑅𝑒𝑔
𝑦 = 𝑎𝑥 + 𝑏
𝑎 = Slope (-.3939854136)
𝑏 = y-intercept (996.023809)
𝑟 2 = Coefficient of Determination (.794626912)
𝑟 = Correlation Coefficient (-.8914184831)
Line of Best Fit:
𝑦 = −.3939854136𝑥 + 996.023809
-the 𝑟 value appears only with linear functions. It tells you if the correlation is
positive or negative and how strong…..closer to +1 the more stronger positive
correlation….closer to -1 the more stronger negative correlation
-the 𝑟 2 value tells you how closely the line matches the data – called the
coefficient of determination, it is a percentage of points that are explained by
the equation
Questions?
Why are there two breaks in years for the Olympics? (19121920 & 1936-1948)
Why do you think there was a huge drop from 1896 to 1900?
Why do you think we keep getting faster and faster?
Could we use this line of best fit (function) to predict what
times would be in the future?
Why? or Why Not?
What would be the time in 2020 if we keep going at this
trend?
Graphing Investigation with
Calculator (Olympics)
Make sure Scatter Plot is on
𝑌 =→ 𝑢𝑝 𝑎𝑟𝑟𝑜𝑤 → 𝑃𝐿𝑂𝑇 1 → 𝐸𝑁𝑇𝐸𝑅 → 𝑑𝑜𝑤𝑛 𝑎𝑟𝑟𝑜𝑤
Window
 Xmin:
 Xmax:
 Xscal:
 Ymin:
 Ymax:
 Yscal:
1850
2030
1
150
300
100
Modeling Motion
Questions?
What is the path of ball as it travels through the air?
What function does this most closely resemble?
What information do we need to have to model this
function?
Input
Output
80
0
1
70
18
33
60
36
54
50
54
68
40
67
72
30
Series1
71
73
20
76
73
10
80
73
0
93
69
-10
108
61
-20
121
48
-30
135
32
149
10
163
-16
0
50
100
150
200
Let the Calculator do the
Work
𝑆𝑇𝐴𝑇 → 𝐶𝐴𝐿𝐶 → 5: 𝑄𝑢𝑎𝑑𝑅𝑒𝑔 → 𝐿1 , 𝐿2 → 𝐸𝑁𝑇𝐸𝑅
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑎 = −.0120712664
𝑏 = 1.849039066
𝑐 = 2.464304234
𝑟 2 = .9993252846
QuadReg:
𝑦 = −.0120712664𝑥 2 + 1.849039066𝑥 + 2.464304234
Predict how high up the ball would be if it was 20 units away
from you.
What Function are You?
Unit1 Project