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Foundations of Algebra Unit 2B: Linear Functions Notes
1
Unit 2B: Linear Functions
In this unit, you will learn how to do the following:
Learning Target #1: Creating and Evaluating Functions
Determine if a relation is a function
Identify the domain and range of a function
Evaluate a function
Create an input and output table
Create a rule to describe a table, graph, or context
Learning Target #2: Graphs and Characteristics of Linear Functions
Graph a function in slope intercept or standard form
Convert between standard and slope intercept forms
Calculate the slope in multiple representations
Identify the y-intercept from multiple representations
Identify the domain and range, x and y intercepts, intervals of increase and decrease,
maximums and minimums, end behavior, and positive and negative areas from a graph
Learning Target #3: Applications of Linear Functions
Interpret linear functions in context
Write an equation of a line given a point and slope or two points
Analyze linear functions using different representations
Find and interpret appropriate domains and ranges for authentic linear functions
Calculate and interpret the average rate of change
Learning Target #4: Arithmetic Sequences
Explain why sequences are functions
Write recursive and explicit formulas for arithmetic sequences
Unit 2b Timeline
Monday Tuesday Wednesday Thursday Friday
January 7th
Day 1:
Intro to
Functions/Evaluating
Functions
8th
Day 2:
Creating
Function Rules
9th
Day 3:
Slopes & Y-
Intercepts
10th
Day 4:
Converting
Between Slope
Intercept &
Standard Form
11th
Day 5:
Characteristics of
Linear Functions -
1
14th
Day 6:
Quiz 1
Over Days 1 – 5
Characteristics of
Linear Functions - 2
15th
Day 7:
Characteristics in
a Real World
Context
16th
Day 8:
Writing Equations
of Lines given
Point & Slope, &
2 Points/ SMI
17th
Day 9:
Standard Form of
Linear Equation
18th
Day 10:
Comparing
Linear Functions
21st
MLK Jr. Holiday
22nd
Day 11:
Arithmetic
Sequences - 1
23rd
Day 12:
Arithmetic
Sequences – 2
24th
Day 13:
Unit 2b Review
25th
Day 14:
Unit 2b Test
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 1 – Functions
Relations
A relation can be represented as: _______________, ______________, ______________ or _________________ .
Functions
Map each ________ to one and _________ one _____________
No input has more than one output (No x-values going to two different y-values
Domain and Range
The first coordinate of an ordered pair in a relation in the input, and the second coordinate is the output.
We refer to the set of all inputs as the domain and the set of all outputs as the range.
Determine if the following are functions. Then state the domain and range:
a. b. {(3, 4), (9, 8), (3, 7), (4, 20)} c. {(15, -10), (10, -5), (5, 2), (10, 5), (15, 10)}
Function or Not a Function Function or Not a Function Function or Not a Function
Reason: Reason: Reason:
Domain: Domain: Domain:
Range: Range: Range:
d. e. f.
Function or Not a Function Function or Not a Function Function or Not a Function
Explain: Explain: Explain:
Domain: Domain: Domain:
Range: Range: Range:
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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g. (telephone number, person) h. (person, car) i. (shirt color, student)
Function or Not a Function Function or Not a Function Function or Not a Function
Different Meanings of Domain and Range Organizer
D
R
I
O
X
Y
I
D
Function Notation
The following problems are written in function notation.
What do you think function notation means?
If x is the independent variable and y is the dependent variable, then function notation for y is f(x), which is
read “f of x,” where f names the function. When an equation is in two variables and it describes a function, you
can use function notation to write it:
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Ex. Convert the following equations into function notation.
a. y = 5x + 7 b. g = 8h – 2 c. b = -4d
Understanding Function Notation
While visiting her grandmother, Fiona found markings on the inside of a closet door showing the heights of her
mother, Julia, and Julia’s brothers and sisters on their birthdays growing up. From the markings in the closet,
Fiona wrote down her mother’s height each year from ages 2 to 16. Her grandmother found the
measurements at birth and one year by looking in her mother’s baby book. The data is provided in the table
below, with heights rounded to the nearest inch.
Age (yrs.) x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Height (in.) y 21 30 35 39 43 46 48 51 53 55 59 62 64 65 65 66 66
1. Which variable is the independent variable, and which is the dependent variable? Explain your choice.
2. What is the value of h(11)? What does this mean in context?
3. When x is 3, what is the value of y? Express this fact using function notation.
4. Find an x such that h(x) = 53. What does your answer mean in context?
5. Find an x such that h(x) = 65. What does your answer mean in context?
6. Describe what happens to h(x) as x increases from 0 to 16. What can you say about h(x) for x greater than
16?
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Evaluating Functions
When you want to know the output of a function, you can use your input values by substituting them into your
function for the independent variable.
Ex. Evaluate f(x) = 3x when x = 2 and x = -8
Ex. Evaluate g(x) = ½x – 3 when x = -4 and x = 8
Evaluating a Function from a Graph
Given this graph of f(x), evaluate the following:
a. f(-4) = b. f(0) = c. f(-5) =
d. f(____) = -2 e. f(____) = 0 f. f(____) = 4
F(x) = x + 1
F(2) = 2 + 1
F(2) = 3
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Input and Output Tables and Graphing Functions
You can also evaluate functions to create input and output tables that can be used to graph the function.
Ex. Using the values of -2, -1, 0, 1, and 2, complete the input/output table and graph.
Input f(x) = -2x - 3 Output
Testing if a Function is a Function (Vertical Line Test)
Another way to tell if a relation is a function is the Vertical Line Test. The Vertical Line Test is used with graphs of
relations. To use the Vertical Line Test, consider all of the vertical lines that could be drawn on the graph of the
relation. If any of the vertical lines intersect the graph of the relation at more than one point, then the relation is
not a function.
Ex. Use the Vertical Line Test to determine if the graphs of the relations are functions.
A. B. C.
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 2 – Creating Function Rules
Scenario: Consider the following situations…
The number of hours worked and the money earned
Your grade on a test and the number of hours you studied
The number of people working on a particular job and the time it takes to complete a job
The total cost of a pizza delivery and the number of pizzas ordered
The speed of a car and how far the drives pushes down on the gas pedal
There are two quantities changing in each situation. When one quantity depends on the other in a problem
situation, it is said to be the dependent quantity. The quantity that the dependent quantity depends on is called
the independent quantity. When you have a function, the input value that represents the independent
quantity is considered the independent variable and the output value that represents the dependent quantity
is considered the dependent variable.
Independent Quantities/Variables Dependent Quantities/Variables
Input values Output values
Not changed by other quantities Changes due to independent
quantity
Located on x-axis Located on y-axis
In the scenarios listed above, circle the independent quantity and underline the dependent quantity. Then
name a variable to represent the independent and dependent quantities.
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Creating Function Rules from a Context
Creating functions is very similar to creating equations. You will want to define a variable, identify the changing
value, and the constant value. An algebraic expression that defines a function is a function rule.
Scenario: An art teacher has $500 for supplies and plans to spend $25 per week.
A. Name the independent and dependent quantities.
B. Create a function rule that relates the independent and dependent quantities.
C. How much money will be remaining after 4 weeks? D. After 6 weeks? E. After 8 weeks?
F. How many weeks did it take to have $100 remaining? G. How long did it take to spend all the money?
H. Create an input-output table and then graph your points.
I. What is a reasonable domain? What’s a reasonable range?
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Creating Function Rules
Ex. Create a function rule for the tables below:
A. B.
C. {(1, 3), (2, 6), (3, 9), (4, 12)} D. {(1, -6), (2, -5), (3, -4), (4, -3)}
E. A hot air balloon cruising at 1000 feet begins to ascend. It ascends at a rate of 200 feet per minute. Create
a function f to represent the height of the balloon for m minutes. How many minutes does it take to reach 1400
feet?
F. A fish tank filled with 12 gallons of water is drained. The water drains at a rate of 1.5 gallons per minute.
Create a function f to represent the number of gallons remaining after m minutes. How long does it take for the
tank to have 3 gallons remaining?
Ex. Create a function rule for each person
Maya runs 7 miles per week and increases her distance by 1 mile each week. Matthew runs 4 miles per week
and increases his distance by 2 miles each week.
a. Maya’s Function Rule: b. Matthew’s Function Rule:
c. Who has run farther after 4 weeks?
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 3 – Slope & Y-intercepts
Scenario: The graph below shows a model of a skier’s elevation, over time, while skiing down a hill. Answer the
questions below the graph.
A. What does point A represent? B. At what elevation did the skier start? Label
that point B.
C. Label the point (24, 200) with C. What does it represent? D. How long would it take the skier to reach the
bottom? Draw a line to where the skier finished.
Label that point D.
E. How many feet did the skier descend down the hill each second? Use the following points to determine:
a. Points B and D B. Points A and B C. Points A and C
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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What did you notice?_______________________________________________________________________________
What you just calculated was the slope of the line. Slope can be described in several ways:
Steepness of a line
Rate of change – rate of increase or decrease
Rise
Run
Change (difference) in y over change (difference) in x
Slope from a Graph
Slope can be calculated in several different ways: graphs, tables, formulas, word problems, and equations.
Ex. Calculate the slope of each of the graphs.
A. Slope: _______ y-intercept: _______ B. Slope: _______ y-intercept: _______
Equation: ___________________ Equation: ___________________
C. Slope: _______ y-intercept: _______ D. Slope: _______ y-intercept: _______
Equation: ___________________ Equation: ___________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Zero Slope
4 Types of Slope
Slope from a Table
Calculate the slope using points in the table from our scenario at the
beginning of the lesson. (Remember slope is the change in y divided
the change in x.)
a. b.
Time Elevation
0 320
20 220
24 200
64 0
Un
def
ined
Slo
pe
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Slope from a Formula
In the previous problems with the table, you had to calculate the difference in two y-values first before you
calculated the difference in two x-values. This leads us to the slope formula which can be used to calculate
the slope of any two points.
Ex. Calculate the slope of two points using the slope formula.
A. (9, 3), (19, -17) B. (1, -19), (-2, -7)
Y-intercepts
A y-intercept is the point where the graph crosses the y-axis. Its coordinate will always be the point (0, b),
where b stands for the number on the y-axis where the graph crosses and the value of the x-coordinate will
always be 0.
Ex. Identify the y-intercept in the following representations:
A. B.
C. D.
Slope Formula
𝒎 =𝒚𝟐 − 𝒚𝟏𝒙𝟐 − 𝒙𝟏
where (x1, y1) & (x2, y2) are coordinate points
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Real World Y-Intercepts
In a real world situation, the y-intercept represents the starting value or starting point. Determine the y-intercept
for the following table:
A. How many pills were in the bottle to start? B. How much was admission to the carnival?
c. Alberto is saving for a new video game. After adding two weeks of his allowance to a savings account, he
has $105. After adding three more weeks of his allowance, his savings is now at $150. Determine the y-
intercept and explain what the y-intercept means in terms of the problem.
Real World Slopes
If a graph, table, equation, or context represents a real world situation, the slope has a meaning that can be
interpreted as a rate of change. For the following representations, calculate the slope and interpret it as a rate
of change.
a. b.
Slope/Rate of Change: Slope/Rate of Change:
Unit Rate of Change: Unit Rate of Change:
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Foundations of Algebra Unit 2B: Linear Functions Notes
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c.
Slope/Rate of Change:
Unit Rate of Change:
d. Bella’s Pizza Shop charges $4.50 for a small pizza, $7 for a medium pizza, and $9 for a large pizza. Toppings
cost extra depending on the size of the pizza ordered. Grayson ordered a large pizza with three toppings that
cost of a total of $12.60. What is the unit rate of cost per number of toppings for a large pizza?
e. A maintenance crew is paving a road. They are able to pave one eighth of a mile of a road during each
working shift. A working shift is 7 hours. What is the unit rate of yards of road paved per hour?
f. One hundred twenty teenagers attended the community center’s dance. Each ticket costs $5. The
community center’s expenses for the dance are $140 for the DJ and $60 for other expenses? What was the
profit that center made? What is the profit made in dollars for each ticket sold?
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 4 – Converting Between Slope Intercept & Standard Form
In the last unit, you reviewed how to solve for y. When you graph linear functions, it is much easier to graph in
slope intercept form than standard form.
Standard Form Slope Intercept Form Ax + By = C
a, b, and c are constants
y = mx + b
m = slope
b = y-intercept
Solve the equations for y. Then name the slope and y-intercept.
A. 3x – 2y = -16 B. 5x – y = 10
Slope: _______ y-intercept: _______ Slope: _______ y-intercept: ______
C. 4x – y = -3 D. 5y + 2x = 20
m = _______ b = _______ m = _______ b = _______
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Converting from Slope Intercept to Standard Form
When converting from slope intercept form to standard form, you want to move your equation around so that
the variables x and y are on the same side and the constant is on the other side. Additionally, the standard
form of an equation should not have the ‘x’ term be negative so you might have to multiply the entire equation
(both sides) by -1.
Convert the following equations to slope intercept form:
a. y = -3x + 2 b. y = 5x + 4 c. y = 7x – 3
d. y = −2
3𝑥 + 4 e. y =
5
3𝑥 − 3 f. y =
1
2𝑥 − 6
Things to Remember about Standard Form
Ax + By = C A, B, and C are integers
No fractions
A should be positive
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Finding x & y intercepts
Practice: Find the x and y intercepts of each equation. Then graph.
a. 2x – 5y = 10
x-intercept: y-intercept:
b. 3x + 6y = -18
x-intercept: y-intercept:
X –intercepts
Written as (a, 0)
The value of the y-coordinate is always 0.
Y-intercepts
Written as (0, b)
The value of the x-coordinate is always 0.
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 5 – Characteristics of Linear Functions
One key component to fully understanding linear functions is to be able to describe characteristics of the
graph and its equation. Important: If a graph is a line (arrows), we need to assume that it goes on forever.
Domain and Range
Domain Define:
All possible values of x
Think:
How far left to right does the
graph go?
Write:
Smallest x ≤ x ≤ Biggest x
*use < if the circles are open*
Range Define:
All possible values of y
Think:
How far down to how far up
does the graph go?
Write:
Smallest y ≤ y ≤ Biggest y
*use < if the circles are open*
Non Linear Examples:
1. 2. 3.
Domain: Domain: Domain:
Range: Range: Range:
Linear Examples:
1. 2.
Domain: Domain:
Range: Range:
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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X and Y intercepts (including zeros)
Y-Intercept
Define:
Point where the graph crosses
the y-axis
Think:
At what coordinate point does
the graph cross the y-axis?
Write:
(0, b)
X-Intercept Define:
Point where the graph crosses
the x-axis
Think:
At what coordinate point does
the graph cross the x-axis?
Write:
(a, 0)
Zero
Define:
Where the function (y-value)
equals 0
Think:
At what x-value does the graph
cross the x-axis?
Write:
x = ____
Linear Examples:
1. 2.
Y-intercept: Y-intercept:
X-intercept X-intercept:
Zero: Zero:
3. 4.
Y-intercept: Y-intercept:
X-intercept X-intercept:
Zero: Zero:
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Interval of Increase and Decrease
Interval of Increase Define:
The part of the
graph that is rising
as you read left to
right.
Think:
From left to right, is
my graph going
up?
Write:
x value where it starts increasing
< x <
x value where it stops increasing
Interval of Decrease Define:
The part of the
graph that is
falling as you read
from left to right.
Think:
From left to right, is
my graph going
down?
Write:
x value where it starts decreasing
< x <
x value where it stops decreasing
Interval of Constant Define:
The part of the
graph that is a
horizontal line as
you read from left
to right.
Think:
From left to right, is
my graph a flat
line?
Write:
x value where it starts flat-lining
< x <
x value where it stops flat-lining
Non Linear Example:
Interval of Increase:
Interval of Decrease:
Interval of Constant:
Linear Examples:
1. 2.
Interval of Increase: Interval of Increase:
Interval of Decrease: Interval of Decrease:
Interval of Constant: Interval of Constant:
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Maximum and Minimum (Extrema)
Maximum Define:
Highest point or
peak of a function.
Think:
What is my highest
point or value on
my graph?
Write:
If none, write none
Otherwise,
y = biggest y-value
Minimum Define:
Lowest point or
valley of a function.
Think:
What is the lowest
point or value on
my graph?
Write:
If none, write none
Otherwise,
y = smallest y-value
Non Linear Examples:
1. 2. 3.
Maximum: Maximum: Maximum:
Minimum: Minimum: Minimum:
Linear Examples:
1. 2.
Maximum: Maximum:
Minimum: Minimum:
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 6 – Characteristics of Linear Functions (cont’d)
Positive and Negative Regions on a Graph
Positive Define:
The part of the
function that is
above the x-axis.
Think:
Which part of
the function is in
the positive
region and
where?
Write:
Inequality using
zero value (x)
Negative Define:
The part of the
function that is
below the x-axis.
Think:
Which part of
the function is in
the negative
region and
where?
Write:
Inequality using
zero value (x)
1. 2.
Positive: __________________________ Positive: __________________________
Negative: _________________________ Negative: _________________________
3. 4.
Positive: __________________________ Positive: __________________________
Negative: _________________________ Negative: _________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
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End Behavior
End Behavior Define:
Behavior of the ends of the function (what happens to the
y-values or f(x)) as x approaches positive or negative
infinity. The arrows indicate the function goes on forever so
we want to know where those ends go.
Think:
As x goes to the left (negative
infinity), what direction does
the left arrow go?
Write:
As x -∞, f(x) _____
Think:
As x goes to the right (positive
infinity), what direction does
the right arrow go?
Write:
As x ∞, f(x) _____
1. 2.
As x -∞, f(x) _____ As x -∞, f(x) _____
As x ∞, f(x) _____ As x ∞, f(x) _____
3. 4.
As x -∞, f(x) _____ As x -∞, f(x) _____
As x ∞, f(x) _____ As x ∞, f(x) _____
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Practice
Practice Example 1
Practice Example 2
Domain:
Range:
Domain:
Range:
Y-intercept:
X-intercept:
Zero:
Y-intercept:
X-intercept:
Zero:
Interval of Increase:
Interval of Decrease:
Interval of Constant:
Interval of Increase:
Interval of Decrease:
Interval of Constant:
Maximum:
Minimum:
Maximum:
Minimum:
Positive:
Negative:
Positive:
Negative:
End Behavior:
As x -∞, f(x) _____
As x ∞, f(x) _____
End Behavior:
As x -∞, f(x) _____
As x ∞, f(x) _____
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 7 – Characteristics of Linear Functions (Real World)
Now that you have learned all the characteristics that apply to linear functions, we are going to focus on a few
characteristics that have very real world applications to them – slope, domain & range, and intercepts.
The Real Number System
When we apply domain and range to real world situations, we need to consider what types of numbers are
suitable for a domain and range. Typically, we describe domain and range using one of the types of number
classifications.
Types of Numbers Example
Counting Numbers 1, 2, 3, 4… (Zero is not included)
Whole Numbers 0, 1, 2, 3… (Also called non-negative integers)
Integers …-3, -2, -1, 0, 1, 2, 3, …
Rational Numbers Everything above plus decimals & fractions
Real Numbers Everything above plus irrational numbers
Most of the real world applications of domain and range do not include rational numbers (you can’t have a
fractional piece of an item or person) or non-negative numbers (such as time).
Domain & Range
When determining appropriate domains and ranges for a function, think about what the independent and
dependent quantities are and what type of numbers are appropriate and which are not appropriate.
Example 1: A plumber charges $96 an hour for making house calls to do plumbing work. What would be an
appropriate domain and range? Assume he charges by hour.
Independent Quantity: Dependent Quantity:
Domain: Range:
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
27
Example 2: Laura is selling cookies to raise funds for a school club. Each cookie costs $0.50. What would be an
appropriate domain and range?
Independent Quantity: Dependent Quantity:
Domain: Range:
Example 3: Rentals cars at ABC Rental Car Company cost $100 to rent, plus $1 per mile. What would be an
appropriate domain and range?
Independent Quantity: Dependent Quantity:
Domain: Range:
Example 4: Jason goes to an amusement park where he pays $8 admission and $2 per ride. He has $30 to
spend.
Independent Quantity: Dependent Quantity:
Domain: Range:
Example 5: Hunter is shopping for pencils. He has $5.00 from his allowance and he finds the pencils he wants
cost $0.65 each.
Independent Quantity: Dependent Quantity:
Domain: Range:
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Intercepts
1. A car owner recorded the number of gallons of gas remaining in the car's gas tank after driving a number of
miles. Use the graph below to answer the following questions.
a. What does x-intercept represent on the graph?
b. What does the y-intercept represent on the graph?
c. What does the point (200, 12) represent on the graph?
2. The graph below shows the relationship between the number of mid-sized cars in a car dealer's inventory
and the number of days after the start of a sale.
a. What does x-intercept represent on the graph?
b. What does the y-intercept represent on the graph?
c. What does the point (10, 50) represent on the graph?
Is the point a solution of the graph?
d. What does the point (5, 125) represent on the graph?
Is the point a solution of the graph?
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Slope/Average Rate of Change
Example 1: The graph shows the altitude of a plane.
a. Find the plane’s rate of change during the first hour.
b. Find the plane’s rate of change during the second
hour.
Example 2: An industrial-safety study finds there is a relationship between the number of industrial accidents
and the number of hours of safety training for employees. This relationship is shown in the graph below.
a. Find the rate of change.
b. Explain what it represents.
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Foundations of Algebra Unit 2B: Linear Functions Notes
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Day 8 – Writing Equations of Lines Given Point & Slope
So far, you have been able to determine the y-intercept from either a graph or an equation in slope intercept
form. How will you find the y-intercept or equation of a line without a graph or equation? You can use the
slope intercept form to find the y-intercept or equation of a line if you know the slope and a point on the line.
Writing Equations Using Slope Intercept Form
y = mx + b
Writing Equations Using Point Slope Form
(y – y1) = m(x – x1)
1. Write the formula
y = mx + b.
2. Substitute the value
of the slope in for m
and the value of the
point in for x and y.
3. Solve the equation
for b.
4. Substitute the value
of m and the newly
founded b into
y = mx + b.
1. Write the formula
(y – y1) = m(x – x1).
2. Substitute the value
of the slope in for m
and the value of the
point in for x1 and y1.
3. Solve the equation
for y
Ex 1: Write the equation of a line with a slope of -3 and y-intercept of 2.
Ex 2: Write the equation of a line if m = 9 and passes through the point (2, 11).
Ex 3: Write the equation of a line with m = -8 and passes through the point (3, 12).
m = __________ b = ___________
Equation: ________________________
m = __________ b = ___________
Equation: ________________________
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
31
Ex 4: Write the equation of a line with m = 4 and passing through the point (2, 5).
Applications of Slope Intercept Form
Y = M X + B Output Slope Input
Y-intercept (0, b)
Dependent
Variable Rate
Independent
Variable
Starting Amount
One Time Fee
Range
changeiny
changeinx Domain
When a problem involves a constant rate or speed and a beginning amount, it can be written using slope
intercept form. You need to recognize which value is the slope and which is the y-intercept.
Example 1: An airplane 30,000 feet above the ground begins descending at a rate of 2000 feet per minute.
Assume the plane continues at the same rate of descent. The plan’s height and minutes above the ground are
related to each. What is the altitude after 5 minutes?
Independent Quantity:
Dependent Quantity:
Slope:
Y-intercept:
Equation:
Example 2: Suppose you receive $100 for a graduation present, and you deposit it into a savings account.
Then each week after that, you add $20 to your savings account. When will you have $460?
Independent Quantity:
Dependent Quantity:
Slope:
Y-intercept:
Equation:
m = __________ b = ___________
Equation: ________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
32
When a word problem involves a constant rate or speed and gives a relationship at some point in time between
each variable, you need to use y = mx + b to find the b value/y-intercept to create an equation to model the
relationship.
Example 3: Marty is spending money at an average rate of $3 per day. After 14 days, he has $68 left. How
much money did he begin with? After 6 days, how much money does he have remaining?
Independent Quantity:
Dependent Quantity:
Slope:
Y-intercept:
Equation:
Example 4: Diane knows a phone call to a friend costs 25 cents for the first 3 minutes and 10 cents for each
additional minute. How much will a 30 minute phone call cost?
Independent Quantity:
Dependent Quantity:
Slope:
Y-intercept:
Equation:
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Foundations of Algebra Unit 2B: Linear Functions Notes
33
Day 8: Writing Equations of Lines Given Two Points
Writing Equations Using Slope Intercept Form
y = mx + b
Writing Equations Using Point Slope Form
(y – y1) = m(x – x1)
1. Calculate the slope
using the slope
formula.
2. Write the formula
y = mx + b.
3. Substitute the value
of the slope in for m
and the value of the
point in for x and y.
4. Solve the equation
for b.
5. Substitute the value
of m and the newly
founded b into
y = mx + b.
1. Calculate the slope
using the slope
formula.
2. Write the formula
(y – y1) = m(x – x1).
3. Substitute the value
of the slope in for m
and the value of the
point in for x1 and y1.
4. Solve the equation
for y
Ex 1: Write the equation of a line given points (15, -13) and (5, 27).
Ex 2: Write the equation of a line given points (6, 19) and (0, -35).
m = __________ b = ___________
Equation: ________________________
m = __________ b = ___________
Equation: ________________________
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
34
Ex 3: Write the equation of a line given points (1, -4) and (3, 2).
Applications of Writing Equations Given Two Points
When a word problem gives two relationships at different points in time, they are giving you two points. You
must find the slope and y-intercept to write an equation.
Example 1: The math department sponsors Math Family Fun Night every year. In the first year, there were 35
participants. In the third year, there were 57 participants. Write an equation that can be used to predict the
amount of participants, y, for any given year, x (We are going to assume the relationship is linear). Based on
your equation, how many participated are predicted for the 6th year?
Independent Quantity:
Depending Quantity:
Slope:
Y-intercept:
Equation:
Point 1 Point 2
Example 2: Biologists have found that the number of chirps some crickets make per minute is related to
temperature. The relationship is very close to being linear. When crickets chirp 124 times a minute, it is about 68
degrees. When they chirp 172 times a minute, it is about 80 degrees. Find an equation for the line that models
this situation. How warm is it when the crickets are chirping 150 times a minute?
Independent Quantity:
Depending Quantity:
Slope:
Y-intercept:
Equation:
Point 1 Point 2
m = __________ b = ___________
Equation: ________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
35
Day 9 – Standard Form of Equations
Scenario: In the mid 1800’s, delivering mail and news across the American Great Plains was time consuming
and made for a long delay in getting vital information from side of the country to the other. At the time, most
mail and news traveled by stagecoach along the main stagecoach lines at about 8 miles per hour. The Pony
Express Riders averaged about 10.7 miles per hour. The long stretch of 782 miles from the two largest cities on
either side of the plains, St. Louis and Denver, was a very important part of this trail.
a. Use the variable x to write an
expression to represent the
distance the stagecoach was
driven in miles.
8x
b. Use the variable y to write an
expression to represent the
distance the Pony Express rode
in miles.
10.7y
c. Write an expression for the
distance that was traveled
using both of these methods on
one trip.
8x + 10.7y
d. Write an equation that represents using both methods to deliver mail from St. Louis to Denver.
8x + 10.7y = 782
a. If the Pony Express Riders rode for 20 hours from St. Louis before handing off the mail to a
stagecoach, how long would it take the stagecoach to get to Denver?
X Y
b. If the stagecoach rode for 50 hours from St. Louis before handing off the mail to a Pony Express
Rider, how long would it take the rider to get to Denver?
X Y
c. If mail was delivered by stagecoach only, how long would it take the stagecoach to get the
mail from St. Louis to Denver?
X Y
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
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Foundations of Algebra Unit 2B: Linear Functions Notes
36
d. If mail was delivered by Pony Express Riders only, how long would it take a rider to get the mail
from St. Louis to Denver?
X Y
The Parts of the Pony Express Problem
The equation, 8x + 10.7y = 782 is in standard form of a linear equation, which is Ax + By = C. Below, describe
what each variable or expression represents in this equation.
X
Y
8x
10.7y
8x + 10.7y
782
x-intercept
y-intercept
Time the mail
was in a
Stagecoach
(hours)
Time the mail
was with the
Pony Express
(hours)
20
50
0
0
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Foundations of Algebra Unit 2B: Linear Functions Notes
37
Comparing Standard Form and Slope Intercept Form
Standard Form Slope Intercept Form
Form
Ax + By = C
a, b, and c are constants
y = mx + b
m = slope
b = y-intercept
Information
Gives x intercept (when substituting 0 for y)
Gives y-intercept (when substituting 0 for x)
Gives slope and y-intercept
Advantages
Easy to calculate x and y intercepts
Helpful when we solve systems of equations
(Unit 3) using elimination
Easily determine slope and y-intercept
Easiest and fastest to graph the line
Only form you can put in the graphing
calculator
Disadvantages
Do not know the slope unless you convert to
slope intercept form (solve for y)
A, B, and C do not stand for anything
obvious (like slope or y-intercept)
Harder to graph a line
Finding the x-intercept takes a little more
work
Not every linear equation can be written in
slope intercept form (like x = 5)
Context
Adding or subtracting two amounts and
setting equal to a total
Example: Tickets for the school play cost
$5.00 for students and $8.00 for adults. On
opening night $1600 was collected in ticket
sales.
5x + 8y = 1600
Multiplying a constant to a changing
amount and then adding or subtracting a
starting amount
Example: Carl has $200 in his bank account
and each week he withdraws $25 dollars.
y = 200 – 25x
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Foundations of Algebra Unit 2B: Linear Functions Notes
38
Practice with Standard and Slope Intercept Form in a Context
Practice: For each scenario, create an equation and solve for the missing variable.
a. A bookstore has mystery novels on sale for $2 each and sci-fi novels on sale for $3 each. Bailey has $30 to
spend on books. How many mystery novels can she buy if she buys 6 sci-fi novels?
b. Your little brother is having a party at the local zoo. The zoo charges a party fee of $50 plus $5 for each
guest. How many guests did he invite if the total cost was $115?
c. Alex’s goal is to sell $100 worth of tickets to the school play. The tickets are $4 for students and $10 for adults.
How many student tickets does he need to sell if he sells 6 adult tickets?
d. It costs $4 to order a chicken sandwich and $3 to order a cheeseburger form the local fast food restaurant
down the street for dinner for the math team before their competition. They have $60 to spend on food.
Calculate the x and y intercepts of this problem and interpret your answers in terms of the problem.
Page 39
Foundations of Algebra Unit 2B: Linear Functions Notes
39
Day 10 – Comparing Linear Functions
Linear Functions can come in many forms:
Context:
Graph:
Table:
Equation:
Now that you have studied linear functions and their characteristics for over two weeks, you need to be able to
compare and answer questions in whatever form is given to you.
Practice 1: Which function has the biggest y-intercept?
Function A: Function B: Function C:
Practice 2: Which function has the greatest rate of change?
Function A: Function B: Function C:
30x + 2y = -24
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
Page 40
Foundations of Algebra Unit 2B: Linear Functions Notes
40
Practice 3: Two airplanes are in flight. The function f(x) = 400x + 1200 represents the altitude, f(x), of Plane 1 after
x minutes. The graph below represents the altitude of the second airplane.
Plane 2
Compare the starting altitudes of the two planes.
Compare the rate of change of the two planes.
Practice 4: Your employer has offered two pay scales for you to choose from. The first option is to receive a
base salary of $250 a week plus 15% of the price of any merchandise you sell. The second option is represented
in the graph below.
Option 2
a. Create an equation to represent the first option for
one week’s worth of pay.
b. Create an equation to represent the second option
for one week’s worth of pay.
c. Which option has a higher base salary? Explain how you know.
d. Which option has a higher rate for selling merchandise? Explain how you know.
Page 41
Foundations of Algebra Unit 2B: Linear Functions Notes
41
Day 11 – Arithmetic Sequences (Explicit Formula)
For the following patterns, find the next two numbers. Then describe the rule you are applying each time.
Pattern Rule Common Diff.
a. -4, -2, 0, 2, ______, ______, … ____________________________________________________ _____________
b. -20, -16, -12, -8, --4, ______, ______,…____________________________________________________ _____________
c. 6.5, 5, 3.5, 2, ______, ______, … ____________________________________________________ _____________
d. 12, 18, 24, ______, ______, … ____________________________________________________ _____________
e. 50, 40, 30, ______, ______, … ____________________________________________________ _____________
f. 11, 9, 7, ______, ______, … ____________________________________________________ _____________
g. What did you notice about your patterns? _______________________________________________________________
h. What do you think the “…” means? ______________________________________________________________________
Sequences
A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other
objects. A sequence in which you get the next consecutive term by adding or subtracting a constant value is
called an arithmetic sequence. In other words, we just add or subtract the same value over and
over…infinitely. This constant value is called the common difference.
What you may not realize is when it comes to sequences, they are considered linear functions. The position of
each term is called the term number or term position. We can think of the term number or position as the input
(domain) and the actual term in the sequence as the output (range). Instead of using x for the input, we are
going to use n and instead of using y for the output, we are going to use an.
Pattern A: Pattern B:
Term Number (n)
Term (an) 12 18 24
Term Number (n)
Term (an) -4 -2 0 2
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
Page 42
Foundations of Algebra Unit 2B: Linear Functions Notes
42
Formula for Arithmetic Sequences
Why We Have a Formula for Sequences
Take a look at the following pattern: 4, 8, 12, 16, ….
What is the 3rd term? _________ What is the 5th term? _________ What is the 7th term? ________
What is the pattern? _________________________________________ What is the 1st term? ________
What is the 54th term? ________ (You don’t want to add ____ over and over 54 times?!?!?!?)
This is why the Explicit Formula was created – as long as you know your common difference and 1st term, you
can create a rule to describe any arithmetic sequence and use it to find any term you want.
Creating an Explicit Rule
1. Write down the Explicit Formula.
2. Substitute the first term in for a1 and common
difference in for d.
3. Simplify the right side of the equation so that you
have an equation that looks very similar to
y = mx + b (except it will look more like an = dn + c).
4. To find an nth term, substitute the term number you
are wishing to find into n.
Write an Explicit Rule for the following sequences:
a. 1, 8, 15,… b. 4, 0, -4,… c. -5, 3, 11,…
a1 = _______ a1 = _______ a1 = _______
d = _______ d = _______ d = _______
Explicit Formula:
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Foundations of Algebra Unit 2B: Linear Functions Notes
43
Finding the Nth Term
To find the nth term, particularly when the nth term is quite large, you want to create an Explicit Rule first and
then substitute that term number into the rule for n.
For the given sequences, create an explicit rule and then use the rule to find the following terms:
a. 5, 10, 15, 20, ….. Find 21st term b. 121, 110, 99, 88, …. Find a10
c. -30, -22, -14, -6, …. Find a30 d. 3, 8, 13, 18, … Find 17th term
Finding Terms Using an Explicit Rule
For the following sequences, find the first five terms:
a. 4 3( 1)na n b. ( 1)na n c. 9( 1) 13na n
Page 44
Foundations of Algebra Unit 2B: Linear Functions Notes
44
Graphing Sequences
For the following sequences, complete the following:
a. Create a table representing the term numbers and terms and then graph
b. Create an Explicit Rule to describe the sequence.
c. If you could use your graph to write the equation of the line, what would the equation be?
1. -8, -5, -2, 1, … 2. 6, 2, -2, -4, …
b. Explicit Rule: b. Explicit Rule:
c. Equation of the line: c. Equation of the line:
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
Page 45
Foundations of Algebra Unit 2B: Linear Functions Notes
45
Day 12 – Arithmetic Sequences (Recursive Formula)
There is a second formula for arithmetic sequences called the Recursive Formula. The recursive formula allows
you to find the next term in a sequence if you know the common difference and any term of the sequence.
Finding Terms Using a Recursive Formula
For the following recursive formulas, find the first five terms:
1. 1
1
4
4n n
a
a a
2.
1
1
7
6n n
a
a a
3.
1
1
3.5
9n n
a
a a
4. 1
1
99
100n n
a
a a
5.
1
1
17
28n n
a
a a
6.
1
1
2
4
n n
a
a a
Common Difference Previous Term Nth Term
Standard(s): ________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
Page 46
Foundations of Algebra Unit 2B: Linear Functions Notes
46
Creating a Recursive Rule
For the following sequences, create a recursive rule:
a. 1, 8, 15,… b. 4, 0, -4,… c. -5, 3, 11,…
d. 14, 3, -8,… e. 7, 10, 13,… f. -6, -13, -20…
Using Figures to Create Rules
a. Create an explicit rule for finding the number of Popsicle sticks.
b. Create an explicit rule for finding the perimeter.
a. Create an explicit rule for finding the number of dashes.
# of Popsicle
Sticks
Perimeter
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
# of Dashes
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6