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Name:____________________________ Period:______________Date:_________

Proctor Hug HS Algebra 1 S1 Unit #3- Linear Functions

Learning Objectives

I can…

Self-Rating 0 – I have no idea.

1 – I can solve problems but do not know why the math works.

2 – I understand why the math works

and can solve most problems but still make mistakes.

3 – I understand why the math works

and can accurately solve problems.

Evidence Cited

The evidence cited here must back-up your

self-rating claim.

PRE POST

Skill 1. I can understand whether a

relation is a function.

Skill 2. I can identify, evaluate and

graph, and write linear functions.

Skill 3. I can graph transformations of

linear functions.

Skill 4. I can write arithmetic and

geometric sequences.

Skill 5. I can utilize a scatter plot and

interpret line of best fit.

Monday Tuesday Wednesday Thursday Friday 10 - Sept 11 – Sept 12 - Sept 13 - Sept 14 - Sept

Unit 2 Unit 2 Quiz Relations and

Functions Relations and

Functions

17 - Sept 18 - Sept 19 - Sept 20 – Sept 21 - Sept

Linear Functions Linear Functions Quiz

Transformations Transformations

24 – Sept 25 – Sept 26 – Sept 27 – Sept 28 - Sept

Arithmetic Sequence

Arithmetic Sequence

Quiz

Scatter Plots/Lines of Best Fit

Scatter Plots/Lines

of Best fit

1 - Oct 2 - Oct 3 - Oct 4 - Oct 5 - Oct

FALL BREAK FALL BREAK FALL BREAK FALL BREAK

FALL BREAK

8 - Oct 9 - Oct 10 - Oct 11 - Oct 12 - Oct

Review Review Review/SLO Pretest Unit 3

Unit 3

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Skill #1 I can understand whether a relation is a function.

Warm –Up

#1) identify the point and the slope

𝑦 − 5 = −2

3(𝑥 + 4)

#2) identify the intercepts

𝟐𝒙 − 𝟓𝒚 = 𝟐𝟎

#3) solve for the variable

𝟒𝒙 + 𝟕 = 𝟏𝟓

Guided Notes

Essential Questions: What is a function? Why is domain and range important in defining

a function?

Let’s do some critical thinking…

The desks in a study hall are arranged in rows like the horizontal ones in the picture.

A. What is a reasonable number of rows for the study hall? What is a reasonable

number of desks?

B. What number of rows would be impossible? What number of desks would be

impossible? Explain

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Domain and Range

_______________ is a set of ordered pairs. A _______________ is a relation in which

each input is assigned to exactly one output. The __________________ of a function is

the set of inputs. The _______________ of a function is the set of outputs.

Range Function Relation Domain

____________ - x-values

____________ - y-values

Example 1

What are the domain and the range of the function?

Now you try!

Identify the domain and range of each function.

Domain-

Range-

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Example 2

B. Is the domain for each situation continuous or discrete?

The domain of a function is _________________ when it includes all real numbers. The

graph of the function is a line or curve.

The domain of a function is _________________ when it consists of just whole numbers

or integers. The graph of the function is a series of data points.

Draw an example of a continuous graph. Draw an example of a discrete graph.

Continuous Discrete

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Practice

For the set of ordered pairs shown, identify the domain and range. Does this relation

represent a function?

{(1,8), (5,3), (7,6), (2,2), (8,4), (3,9), (5,7)}

Identify the domain and range of each function.

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A student was asked to name all values of n that make the relation a function. Correct

the error.

Analyze each situation. Identify a reasonable domain and range for each situation.

a) An airplane travels at 565 mph.

b) Tickets to a sporting event cost $125 each.

Determine whether each relation is a function. If yes, classify the function as one-to-one

or not.

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Using the names of the emoticons as the domain and the shapes of the emoticons’

mouths as the range, make a list of 5 emoticons that make a function.

Ticket & Self Reflection

Please answer the question that is posed to all students prior to class ending. This is a formative assessment

technique engages all students and provides the all-important evidence of student learning for the teacher.

After a train has traveled for ½ hour, it increases its speed and travels at a

constant rate for 1 ½ hours.

a. What is the domain? What is the range?

b. How can you represent the relationship between time traveled and speed?

c. Why did you choose this representation?

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Unit 3 Skill #2 I can identify, evaluate, graph and write linear equations

Warm –Up

Simplify Write the domain and range Is the previous question

𝟒𝒂 + 𝟑𝒃 − 𝟕 + 𝟓𝒂 − 𝟐𝒃 (4,1), (3,5), (2,6), (-3,10) a function? Why or why

not?

Guided Notes

_________________ ____________________ is a method for writing variables as a

function of other variables.

Write the equation y = 5x + 1 using function notation

What is the value of f(x) = 5x + 1 when x = 3?

You try!

Evaluate each function for x = 4

1) g(x) = -2x – 3 2) h(x) = 7x +15

The function f is defined in

function notation by f(x) = 5x +1

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Find the value of f(5) for each function

3) f(x) = 6 + 3x 4) f(a)= 3(a+2) – 1 5) f(h) = −𝒉

𝟏𝟎

6) f(x)= -2(x+1) 7) f(m) = 1 - 4(𝒎

𝟐) 8) f(m) = 2(m - 3)

Writing a linear function rule

The cost to make 4 bracelets is shown in the table. How can you determine the cost to

make any number of bracelets?

1)Write a function using slope-intercept form for the rule 2) Find the value of b

f(x) = mx + b

Try it!

Write a linear function for the data in each table using function notation.

1) 2)

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3) 4)

Describe and correct the error Mr. Bishop made when finding the function rule for the

data in the table.

Analyze a linear function

A _________ ___________ is a function whose graph is a line.

Example

Tamika records the outside temperature at 6:00 am. The outside temperature increases

by 2˚F every hour for the next 6 hours. If the temperature continues to increase at the

same rate, what will the temperature be at 2:00 pm?

1) Write a function that models the situation.

2) Sketch a graph of the function.

3) Find the value of y when x = ____

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Try it! Sketch the graph of each function.

1) f(x) = -x +1 2) g(x) = 3x + 1

3) g(x) = 3 – x 4) f(x) = ½ (x – 1)

A chairlift starts 0.5 miles above the base of a mountain and travels up the mountain at a constant

speed. How far from the base of the mountain is the chairlift after 10 minutes?

Write a linear function to represent the distance the chairlift travels from the base of the mountain.

Find the distance.

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A snack bar at an outdoor fair is open from 10 am to 5:30 pm and has 465

bottles of water for sale. Sales average 1.3 bottles of water per minute.

a. Graph the number of bottles remaining each hour as a function of time in hours.

Find the domain and range.

b. At this rate, what time would they run out of water? How many bottles of water

are needed at the start of the next day? Explain.

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Unit 3 Skill #3 I can transform linear functions.

Warm –Up

Avery states that the graph of g is the same as the graph of f with every point shifted vertically. Cindy states that the graph

of g is the same as the graph of f with every point shifted horizontally.

Guided Notes

A ________________ of a function f maps each point of its graph to a new location.

One type of transformation is a _____________. A ____________ shifts each point of the

graph of a function the same distance. It may be horizontal or vertical.

a. Give an argument to support Avery’s statement.

b. Give an argument to support Cindy’s statement

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Vertical Translation

The positions of 2 baby sea turtles making their way to the water after hatching from

their eggs is recorded. They move at the same speed with Bryon starting 2 ft ahead of

Frank’s starting point.

What function represents each turtle’s position as they make their way to shore?

What is similar between the two turtles? What is different about the two turtles?

Horizontal Translation

Consider the graphs of f(x) = 2x – 4 and g(x) = 2(x +5) – 4

Make a table:

X f(x) =

2x – 4

g(x) =

2(x +5) – 4

-2

-1

0

1

2

Conclusions?

Find the speed of each turtle by finding

the slope of each line.

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Stretches and Compressions of linear functions

Graph f(x) = x – 2 Graph g(x) = ½ (x - 2)

What are similar qualities between the two graphs? What are different qualities between

the two graphs?

Graph f(x) = x + 1 Graph g(x) = (3x) + 1

What are similar qualities between the two graphs? What are different qualities between

the two graphs?

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Forms a point on the line: (-h, k)

Equal to 1 Bigger than 1 Between 0 and 1

= 1 > 1 0 < (𝑎 𝑜𝑟 𝑏) < 1

a Same Shape as parent

function

Stretches vertically Multiply the vertical distance by

“a” for each point

Compresses vertically Multiply the vertical distance by

“b” for each point

b Same Shape as parent

function

Compresses horizontally Multiply the horizontal distance

by “b” for each point

Stretches horizontally Multiply the horizontal distance

by “b” for each point

Determine the type of transformation of the following

a) g(x) = (3x +5) + 8 b) g(x) = 2 (0.5x) + 3 c) g(x) = 3(0.1x) + 5

A student graphs f(x) = 3x – 2. On the same grid they graph the function g which is a

transformation of f made by subtracting 4 from the input of f. Describe and correct the

error they made when graphing g.

𝑦 = ±𝑎(𝑏𝑥 − ℎ) + 𝑘

Establish a Shape 1 Graph the Starting Point 2

Graph the Stretch Point 3 Sketch the Function 4

(𝑥 − ℎ) (𝑥 + ℎ) Shifts Right Shifts

Left

+𝑘 −𝑘 Shifts up Shifts

Down

± (+ or -): Reflects the graph over the x-axis (a) or y-axis (b)

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The cost of renting a landscaping tractor is a $100 security deposit plus the hourly rate.

b. How would the slope and y-intercept of the graph g compare to the slope and y –

intercept of the graph off ?




a. The function f represents the cost of renting the

tractor. The function g represents the cost if the

hourly rate were doubled. Write each function.

The graph of a linear function f has a negative slope. Describe the effect on the graph of the function

if the transformation has a value of k < 0.

a) Adding k to the outputs of f

b) adding k to the inputs of f

c) multiplying the outputs of f by k

d) multiplying the inputs of f by k

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Unit 3 Skill #4 I can identify and describe arithmetic sequences.

Warm –Up

Find the slope and graph Solve and Graph

𝟒𝒙 + 𝟑𝒚 = 𝟏𝟓 𝟑𝒙 − 𝟓 < 𝟐𝒙 + 𝟕

Guided Notes

A ________________ is an ordered list of numbers that often forms a pattern. Each number is a

_________________________. In an ________________________________, the difference between any

two consecutive terms is a constant called the __________________________.

Term of the sequence Common difference Sequence Arithmetic sequence

Is the ordered list 26, 39, 52, 65, 78 an arithmetic sequence?

How are sequences related to functions?

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Recursive formula for an arithmetic sequence is:

𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒅

an – nth term of the sequence

a1 – 1st term of the sequence

an-1 – previous term of the sequence

d- common difference

Recursive formula describes the pattern of a sequence that can be used to find the next

term in a sequence.

What is a recursive formula for the height above the ground of the nth step of the

pyramid shown?

𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒅

Find the height above the ground of the 3rd step.

Try it!

Write a recursive formula to represent the total height of the nth stair above the ground

if the height of each stair is 18 cm.

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Explicit formula expresses the nth term of a sequence in terms of n.

𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅

an – nth term of the sequence

a1 – 1st term of the sequence

n – term we are solving for in sequence

d- common difference

The cost of renting a bicycle is given in the table. How can you represent the rental cost

using an explicit formula?

What is the cost of renting the bicycle for 10 days?

Try it!

The cost to rent a bike is $28 for the first day plus $2 for each day after that. Write an

explicit formula for the rental cost for n days. What is the cost of renting the bike for 8

days?

𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅

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Write an explicit formula from a recursive formula

𝒂𝒏 = 𝒂𝒏−𝟏 − 𝟑 ; 𝒂𝟏 = 𝟏𝟎 𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟐. 𝟒; 𝒂𝟏 = −𝟏

𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟏𝟓 ; 𝒂𝟏 = 𝟖 𝒂𝒏 = 𝒂𝒏−𝟏 − 𝟐𝟏; 𝒂𝟏 = 𝟓𝟔

Write a recursive formula from an explicit formula

𝒂𝒏 = 𝟏 +𝟏

𝟐𝒏 𝒂𝒏 = 𝟖 + 𝟑𝒏

Practice

Tell whether or not each sequence is an arithmetic sequence. If it is, give the common

difference.

a) 1, 15, 29, 43, 57, ... b) 1, -2, 3, -4, 5, …

c) 37, 34, 31, 29, 26, … d) 93, 86, 79, 72, 65, …

Write a recursive formula and an explicit formula for each sequence

a) 12, 19, 26, 33, 40, … b) -4, 5, 14, 23, 32, …

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c) 62, 57, 52, 47, 42, … d) -15, -6, 3, 12, 21, …

Write a recursive formula for each explicit formula and find the first term of the

sequence.

𝒂𝒏 = 𝟑𝟓 + 𝟓𝟐𝒏 𝒂𝒏 =𝟕

𝟐− 𝟑𝒏

Describe and correct the error a student made in identifying the common difference of

the following sequence: 29, 22, 15, 8, 1, …

In a video game, you must score 5,500 points to complete level 1. To move through

each additional level, you must score an additional 3,250 points. What number would

you use as a1 when writing an arithmetic sequence to represent this situation? What

would n represent? Write an explicit formula to represent this situation. Write a

recursive formula to represent this situation.

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The graph of an arithmetic sequence is shown. Write a recursive formula for the arithmetic sequence if

the y-value of each point is increased by 3.

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Skill #5 I can use a scatter plot to describe the relationship between two data points

Warm –Up

#1) Is this an arithmetic sequence?

2, 4, 6, 8, 12……

#2) Solve

𝟐(𝟑 + 𝟐)𝟐 − 𝟔 ÷ 𝟑

#3)

Guided Notes

What is the relationship between hours after sunrise, x, and the temperature, y, shown in the scatter

plot?

When y-values tend to increase as x-values increase, the two data sets have a

______________ _____________________.


plot?

When y-values tend to decrease as x-values increase, the two data sets have a

________________ ________________________.

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plot?

When there is no general relationship between x-values and y-values, the two data sets

have _________ __________________.

Describe the type of association each scatter plot shows.


plot?

The scatter plot suggests a linear relationship. There is a

_______________ _____________________ between

hours after sunrise and the temperature.

When data with a negative association are modeled

with a line, there is a _______________

_____________________. If the data do not have an

association, they cannot be modeled with a linear

function.

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Try it!

How can the relationship between the hours after sunset, x, and the temperature, y, be

modeled? If the relationship is modeled with a linear function, describe the correlation

between the two data sets.

Equation of a Trend Line

A ______________ ___________ models the data in a scatter plot by showing the general

direction of the data. A trend line fits the data as closely as possible.

Try it!

Find the equation of the trend line of this graph.

Step 1: Sketch a trend line for the data.

A trend line approximates a balance of points

above and below the line. It does not mean it

will pass through any of the points

Step 2: Write the equation of this trend line.

Select two points on the trend line to find the

slope

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Interpret Trend Lines

The table shows the amount of time required to download a 100-megabyte file for

various Internet speeds. Assuming the trend continues, how long would it take to

download the 100-megabyte file if the Internet speed is 75 kilobytes per second?

Use the equation of the linear model to find the y-value that corresponds to x = 75.

Practice!

Describe the type of association between x and y for each set of data. Explain.

a) b)

Find the equation of the trend line.

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Make a scatter plot of the data. Describe the type of association that the scatter plot

shows. Draw a trend line and write its equation.

a) b)

Describe and correct the error Mr. Bishop made in describing the association of the data

in the table.

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A student is tracking the growth of some plants. What type of association do you think

the data would show? Explain.

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