Unit 2C
Characteristics of Linear Functions and Arithmetic
Sequences
Table of Contents
Title
Page #
Glossary.……..………………………………………………………………… 2
Lesson 2C-1 Domain and Range …………….…………………………… 3
Lesson 2C-2 Function Notation ……...…………………………………. 7
Lesson 2C-3 Slope and Rate of Change..……………………………… 12
Lesson 2C-4 Other Characteristics of Linear Functions……...……… 17
Lesson 2C-5 Arithmetic Sequences.…………………………………… 23
This packet belongs to:
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Unit 2C: Characteristics of Linear Functions and Arithmetic Sequences
1
Glossary
Arithmetic Sequence:
Continuous:
Constant Rate of Change:
Discrete:
Domain:
End Behavior:
Explicit Formula
Interval Notation:
Linear Function:
Range:
Recursive Formula:
Slope:
X-Intercept:
Y-Intercept:
Unit 2C: Characteristics of Linear Functions and Arithmetic Sequences
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Lesson 2C – 1: Domain and Range
Learning Target:
 I can identify the domain and range of a function.
F.IF.1
Guided Notes:
The relationship between two quantities can be shown by a set of
ordered pairs called a ______________.
The _____________of a relation is the set of first coordinates (or xvalues) of the ordered pairs. The ____________ of a relation is the set of
second coordinates (or y-values) of the ordered pairs.
Example 1:
Find the domain and range of
the following relation.
{(−1,4), (2,6), (0, −4), (−7, 10)}
Example 2:
Find the domain and range of the
following relation.
2
-3
Domain:
Domain:
Range:
3
-2
4
5
5
7
Range:
9
We have two different notations that we can use to describe domain and
range of continuous functions.
The first is called ________________ and uses inequality symbols like
<,>,≤, and ≥.
When a boundary is included in the domain or range, we use either the
___________ symbol since our domain or range can equal that
boundary.
If the boundary is not included in the domain or range, we use either the
Unit 2C: Characteristics of Linear Functions and Arithmetic Sequences
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_______ symbol since the domain or range cannot equal that
boundary. When either ___________________ is our bounds, we
always use
< or > since you can never actually reach infinity.
Another notation that we use to describe domain and range is called
______________ With interval notation, we use parentheses and
brackets instead of inequality symbols.
We use brackets when the boundary is ________________.
We use parentheses when the boundary is ___________ We always use
parentheses when a boundary is ___________________ since you can
never actually reach either one.
Set notation looks at the boundaries on the interval.
For example:
3≤x<7 would be written as ______________in interval notation.
−∞ < 𝑥 ≤ 9 would be written as (−∞, 9].
Example 3:
Give the domain and range of each graph below.
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Domain:
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Domain:
Range:
Range:
Domain:
Range:
In a non-restricted linear function, the domain and range will both always
be _______________. This can be expressed with the symbol ________.
Sometimes, your domain and range might be restricted based on the real
world situation of a function.
Example 4
The amount owed on a car loan can be modeled by the function 𝑓(𝑥) =
15,000 − 500𝑥, where x is the number of months after purchase. What is
the reasonable domain and range of this function?
Practice 2C-1: Domain & Range
Find the domain and range for each relation.
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5.
6.
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D
7.
10.
Domain:_______________
Domain:_______________
Range:________________
Range:________________
8.
11.
9.
12.
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1
3
-10
6
11
14.
11
14
14
13
13
Domain:________________
Domain:________________
Range:_________________
Range:_________________
1
2
-4
6
Application 2C-1
Pick up Application 2C-1 Domain And Range Scenarios from the Algebra
Embassy.
Lesson 2C – 2: Function Notation
Learning Target: I can read and write functions in function notation.
F.IF.1, F.IF.2
Guided Notes:
Relations can be expressed in several different ways.
The relation {(2, 3), (4, 7), (6,8)} can be displayed in the following ways
An x/y table
A graph
Using Mapping
X
Y
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A _______ is a special type of relation that pairs each domain value with
____________ range value.
Example 1:
Give the domain and range of the relation and tell whether the relation is
a function. Justify your answer.
{(3,–2), (1, –1), (4, 0), (1,1)}
Example 2:
Give the domain and range of the relation and tell whether the relation is
a function. Justify your answer.
{(-1,4), (3, –9), (0, 4), (8,7)}
The Vertical Line Test
If any ___________ line passes through more than one point of the
graph, then the relation is not a function.
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The input of a function is also called the ________________. The output
of a function is the ________________ The value of the dependent
variable ____________ on, or is a function of, the value of the independent
variable.
When writing equations for functions, we use a specific format called
_______________.
The ______________ is a function of the __________________.
We are used to writing equations like this:
y
is a function of
x
In function notation, we write equations like this:
f(x)
is a function of
x
Really, the only difference between function notation and what we are used
to writing is that we use f(x) instead of y.
So if we had the equation 𝑦 = 5𝑥 + 9, in function notation we would write
this as ___________________.
Often times, we are asked to evaluate
functions at a specific point using
function notation. Let’s take a look at an
example.
Example 3:
If 𝑓(𝑥) = 7𝑥 + 2, find f(2).
Sometimes you may be asked to find x from an equation written in function
notation.
Example 4:
If 𝑔(𝑥) = 3𝑥 + 3 and 𝑔(𝑥) = 12, find x.
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We can also write ordered pairs in function notation.
Example 5a:
Write the following order pair in function notation: (9,-7).
Example 5b: Translate the following statement into an ordered pair:
ℎ(3) = 12.
Practice 2C-2: Function Notation
Determine whether each of the following represents a function. Explain your answer.
1) Input: State
Output: Capital of that state
2) (3,2); (-1,4); (5,4); (8,-1)
3) Input: State
Output: City in that state
4) (4,1); (5,-1); (-2,0); (4,3)
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5)
Input
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Output
1
3
-10
6
6)
11
14
13
Use the functions below to answer questions 7 – 14.
𝒇(𝒙) = 𝒙 − 𝟓
𝒈(𝒙) = 𝒙𝟑 − 𝟒
𝒉(𝒙) = 𝟑𝒙 − 𝟔
𝒋(𝒙) = 𝒙𝟐 + 𝟑𝒙 + 𝟐
𝒌(𝒙) = (𝒙 + 𝟐)𝟐
𝒎(𝒙) = √𝒙 − 𝟐
𝒒(𝒙) = 𝒙′ 𝒔 𝒃𝒊𝒓𝒕𝒉𝒅𝒂𝒚
𝒑(𝒙) = 𝒕𝒉𝒆 𝒅𝒂𝒚 𝒃𝒆𝒇𝒐𝒓𝒆 𝒙
7) Find 𝑓(1).
8) Find 𝑔(−2).
9) Find ℎ(−3).
10) Find 𝑗(−4).
11) Find 𝑘(−5).
12) Find 𝑚(18).
13) Find 𝑝(𝑀𝑎𝑟𝑐ℎ 20, 1969).
14) Find 𝑞(𝑦𝑜𝑢).
Use the functions below to answer questions 15 – 22.
𝒈(𝒙) = −𝟑𝒙 + 𝟏
𝒇(𝒙) = 𝒙 + 𝟕
𝒉(𝒙) =
𝟏𝟐
𝒙
𝒋(𝒙) = 𝟐𝒙 + 𝟗
15) Find x if 𝑔(𝑥) = 16.
16)Find x if ℎ(𝑥) = −2.
17) Find x if 𝑓(𝑥) = 23.
18) Find 𝑔(10).
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19) Find ℎ(−2).
20) 𝑗(𝑘).
Translate the following statements into ordered pairs.
21) 𝑓(−1) = 1
22) ℎ(2) = 7
23) −1 = 𝑔(1)
24) 𝑗(0) = 0
25) What values of x would make this relation not be a function?
(3,2); (x,4); (5,7); (8,-1)
Application 2C-2
Collect 2C-2 Function Notation Application sheet from the Algebra Embassy.
Lesson 2C – 3: Slope and Rate of Change
Learning Target: I can find and interpret the rate of change of a linear
function.
F.IF.5
Guided Notes:
The _____________, or the slope, is a ratio describing how one quantity changes
as another quantity changes. In algebra, we are typically comparing how the
output changes to how the input changes.
Linear functions have a ___________ rate of change, meaning values increase or
decrease at the same rate over a period of time.
If the rate of change ____________ over time, we say that the rate of change is
______________.
If the rate of change _____________ over time, we say that the rate of change is
negative.
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One way that we can find rate of change is from a graph.
Example 1
We can also find the slope from an equation.
Example 2
Find the slope from the following equations:
4
𝑦 =− 𝑥+3
3
3𝑥 − 2𝑦 = 8
We can also find the rate of change when we are given ordered pairs.
To be able to do this, however, we need to remember our slope formula from our
middle school years.
𝑦 −𝑦
The formula for slope is 𝑚 = 𝑥2−𝑥1. This can be found on the Algebra I formula
2
1
sheet.
Example 3
Find the slope of the line that goes through the point (4,2) and (6,4).
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Example 4
We can also find the rate of change from a table.
The average price for a ticket to a movie theater in North America for selected
years is shown in the table below.
Find the rate of change from 2010 to 2015. Then find the rate of change from 1995
to 2000? How do the rates of change compare?
Year
1985
Price ($) 3.90
1990
4.20
1995
4.50
2000
4.80
2005
5.10
2010
5.40
2015
5.70
Finally, we might be asked to find the rate of change over an interval on a graph
given function notation.
Example 5
Find the rate of change over the interval 𝑓(0) 𝑡𝑜 𝑓(2)
We also have two special cases when dealing with slope.
A horizontal line has slope of _________. These are equations like y=4. When
plugging points into the slope formula, you will have a zero in the ____________.
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A vertical line has a
slope that is _________
These are equations like
x=2. When plugging points
into the slope formula, you
will have a zero in the
________________.
We can use the mnemonic
HOY VUX to help us
remember our horizontal
and vertical slopes.
Practice 2C-3 Rate of Change
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Find the rate of change (slope) through each pair of points.
5. (-14, -8), (-10, -7)
6. (5, 4), (8, 5)
7. (-18, -17), (-17, 16)
8. (13, 7), (13, 10)
9. (14, -4), (-6, 6)
10. (9, 12), (14, 5)
Find the rate of change (slope) of each line.
11. 𝑦 = −5𝑥 + 5
12. 𝑦 = 5𝑥 + 5
15. 5𝑥 + 2𝑦 = 4
16. 4𝑥 − 𝑦 = 5
17. 3𝑥 + 𝑦 = 0
18. 𝑥 = 4
19. Find the rate of change of the cost per minute.
Minutes (x)
Cost in Dollars (f(x))
0
110.00
30
120.50
60
131.00
90
141.50
120
152.00
20. Find the rate of change of revolutions per second.
Seconds (x)
0
2
4
10
15
Revolutions (f(x))
0
6
12
30
45
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Find the rate of change over the given interval.
f(-1) to f(3)
f(-2) to f(1)
Application 2C-3
Collect 2C-3 Application Rate of Change Word Problems sheet from the
Algebra Embassy.
Lesson 2C – 4: Other Characteristics
Learning Target: I can identify and interpret characteristics of a linear
function.
F.IF.4
There are several other characteristics that we can use to describe linear
functions.
We have already touched on intercepts earlier in the semester.
Remember, the y-intercept is where the graph crosses the ____. Its
__________ is always zero.
The x-intercept is where the graph crosses the __________. Its _________
is always zero.
Example 1
Find the x- and y-intercept for the equation 3x+4y=12.
We also can discuss where the function is increasing or decreasing.
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If the rate of change is ___________, we say that the function is _________. It
is increasing on the interval from −∞ < 𝑥 < ∞ or in interval notation (−∞, ∞).
If the rate of change is ___________, we say that the function is __________. It
is decreasing on the interval from −∞ < 𝑥 < ∞ or in interval notation (−∞, ∞).
Example 2:
Are the graphs below increasing or decreasing?
Another characteristic that we can use to describe a function is end behavior. End
behavior describes what happens to the _________ as the _________
approaches infinity or negative infinity.
We typically write end behavior as:
As x−∞, 𝑦 ___________ (Read as x approaches negative infinity, y
approaches____?)
As x∞, 𝑦 ___________ (Read as x approaches positive infinity, y
approaches____?)
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Example 3
Describe the end behavior of the functions below.
Finally, we look at where on the graph the function has a positive y-value and
where it has a negative y value. We can use either set notation or interval
notation to describe the intervals of positive and negative.
Example 4:
Give the interval for which the graph below is positive and the interval for
which the graph is negative.
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Practice 2C-4 Other Characteristics:
1.
a.) rate of change over the interval
[-1, 2]
2.
a.) rate of change over the interval
[0, 3]
b.) domain
b.) domain
c.) range
c.) range
d.) x-intercept
d.) x-intercept
e.) y-intercept
e.) y-intercept
f.) interval of increase
f.) interval of increase
g.) interval of decrease
g.) interval of decrease
h.) End Behavior:
h.) End Behavior:
i.) interval of positive
i.) interval of positive
j.) interval of negative
j.) interval of negative
as 𝑥 → −∞, 𝑦 →_______
as 𝑥 → ∞, 𝑦 →_______
as 𝑥 → −∞, 𝑦 →_______
as 𝑥 → ∞, 𝑦 →_______
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Graph, “Sara started with a balance
of –$4 and for every 2 days, she
deposited $5.
a.) rate of change over the interval
[0, 2]
4.
b.) domain
a.) rate of change over the interval
[1, 3]
c.) range
b.) domain
d.) x-intercept
c.) range
e.) y-intercept
d.) x-intercept
f.) interval of increase
e.) y-intercept
g.) interval of decrease
f.) interval of increase
h.) End Behavior:
g.) interval of decrease
as 𝑥 → −∞, 𝑦 →_______
as 𝑥 → ∞, 𝑦 →_______
i.) interval of positive
j.) interval of negative
h.) End Behavior:
as 𝑥 → −∞, 𝑦 →_______
as 𝑥 → ∞, 𝑦 →_______
i.) interval of positive
j.) interval of negative
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5. {(-4, 1), (-2, 4), (0, 7), (2, 10)}
a.) rate of change over the interval
[-4, 0]
b.) domain
c.) range
d.) x-intercept
e.) y-intercept
f.) interval of increase
g.) interval of decrease
6.
a.) rate of change over the interval
[-4, 1)?
rate of change over the interval
(1, ∞]?
b.) domain
c.) range
d.) x-intercept
e.) y-intercept
h.) End Behavior:
as 𝑥 → −∞, 𝑦 →_______
as 𝑥 → ∞, 𝑦 →_______
f.) interval of increase
i.) interval of positive
g.) interval of decrease
j.) interval of negative
h.) End Behavior:
as 𝑥 → −∞, 𝑦 →_______
as 𝑥 → ∞, 𝑦 →_______
i.) interval of positive
j.) interval of negative
Application 2C-4
Collect 2C-4 Other Characteristics Application sheet from the Algebra Embassy.
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Lesson 2C – 5: Arithmetic Sequences
Learning Target: I understand that sequences are a function with a
domain of integers.
F.IF.3
A ________ is an ordered list of numbers, pictures, letters, geometric figures, or
just about any object you like.
Each number, figure, or object is called a ________ in the sequence. For
convenience, the terms of sequences are often separated by commas.
We can also think of a sequence as a function whose domain consists of
___________________________.
With sequences, we use another new notation, called sequence notation.
In sequence notation, we refer the terms like this:
a1, a2, a3,…
a2 represents the _____________ term in the sequence.
an represents the _____________ in the sequence.
an-1 represents the ______________ in a sequence.
The subscript in each name is called the ______ and refers to the term number.
In an arithmetic sequence, a common number (__________________) is
__________ to each term to get the next term in the sequence.
For example, the sequence:
5, 10, 15, 20, 25….
is arithmetic because a common difference of 5 is added to each previous term
to get the next term.
Arithmetic sequences are linear functions.
We have two different ways to write formulas for sequences.
The first method is called a recursive formula.
Recall that instead of using x and y or f(x) and x, we are going to use a n and n in
our formulas. Note: Both formulas are on your Algebra 1 Formula Sheet.
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The recursive formula for an arithmetic sequence is as follows:
𝑎𝑛 = 𝑎𝑛−1 + 𝑑
where
an = the current term
an-1 = the previous term
d = the common difference.
Example 1:
Write the recursive formula for the following sequence and then use it to find a 7
3, 6, 9, 12, 15…..
It is important to note that to be able use a recursive formula, you MUST have the
previous term.
Using a recursive formula could be quite tedious for terms like the a25 or a100, as
we would have to find every single term that came before them. Wouldn’t it be
nice if we had another formula we could use that didn’t require the previous term?
Thankfully, such a formula does indeed exist.
The explicit formula for a sequence is as follows:
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
where
an = the current term
a1= the first term
n = the index (a.k.a the term number)
d= the common difference
Example 2
Write the explicit formula for the following
sequence and then use it to find a7 and
a20.
3, 6, 9, 12, 15…..
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Practice 2C-5
Practice 2C-5: Arithmetic Sequences
Sequence
1.
-5, -3, -1, 1,…
2.
0, 5, 10, 15, …
3.
10, 7, 4, 1, …
4.
3, 9, 15, 21, …
5.
1
6.
−1, −3, −5, −7, …
Recursive
Definition
Explicit
Definition
Find 𝑎12
3
, 1, 2 , 2, …
2
7. If 𝑎1 = −9 and 𝑎𝑛 = 𝑎𝑛−1 + 3
Find 𝑎3
9. If 𝑎𝑛 = 7 + (𝑛 − 1)4
Find 𝑎24
Explicit or Recursive?____________
Explicit or Recursive?________________
8. If 𝑎1 = 6 and 𝑎𝑛 = 𝑎𝑛−1 − 2
Find 𝑎3
Explicit or Recursive?____________
10. If 𝑎𝑛 = −5 + (𝑛 − 1)1
Find 𝑎24
Explicit or Recursive?_____________
Application 2C-5
Collect 2C-5 Arithmetic Application sheet from the Algebra Embassy.
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