foundations of algebra unit 2b: linear functions notes unit 2b: … · 2019. 1. 9. · slope: _____...

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


2

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.


Explain: Explain: Explain:



Standard(s): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


3

g. (telephone number, person) h. (person, car) i. (shirt color, student)


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:


4

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?


5

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


6

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


7

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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


8

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?


9

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?


10

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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


11

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: ___________________


12

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


13

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


14

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:


15

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?


16

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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


17

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


18

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


19

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.



Linear Examples:

1. 2.

Domain: Domain:

Range: Range:

Standard(s): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


20

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:


21

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:


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:


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:


22

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:


23

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: _________________________


24

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) _____


25

Practice

Practice Example 1

Practice Example 2

Domain:

Range:

Domain:

Range:

Y-intercept:

X-intercept:

Zero:

Y-intercept:

X-intercept:

Zero:







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) _____


26

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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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?


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?


Domain: Range:

Example 4: Jason goes to an amusement park where he pays $8 admission and $2 per ride. He has $30 to

spend.


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.


Domain: Range:


28

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?


29

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.


30

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.


of m and the newly

founded b into

y = mx + b.


(y – y1) = m(x – x1).




point in for x1 and y1.


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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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?


Dependent Quantity:

Slope:

Y-intercept:

Equation:

m = __________ b = ___________

Equation: ________________________


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?


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?


Dependent Quantity:

Slope:

Y-intercept:

Equation:


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.


y = mx + b.




point in for x and y.


for b.


of m and the newly

founded b into

y = mx + b.

1. Calculate the slope

using the slope

formula.


(y – y1) = m(x – x1).




point in for x1 and y1.


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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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?


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?


Depending Quantity:

Slope:

Y-intercept:

Equation:

Point 1 Point 2

m = __________ b = ___________

Equation: ________________________


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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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


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


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.


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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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.


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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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:


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


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


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): ________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________

____________________________________________________________________________________________________


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

foundations of algebra unit 2b: linear functions notes unit 2b: … · 2019. 1. 9. · slope: _____...

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