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Name:____________________________ Period:______________Date:_________
Proctor Hug HS Algebra 1 S1 Unit #3- Linear Functions
Learning Objectives
I canβ¦
Self-Rating 0 β I have no idea.
1 β I can solve problems but do not know why the math works.
2 β I understand why the math works
and can solve most problems but still make mistakes.
3 β 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 β 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 β Sept 21 - Sept
Linear Functions Linear Functions Quiz
Transformations Transformations
24 β Sept 25 β Sept 26 β Sept 27 β 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 βUp
#1) identify the point and the slope
π¦ β 5 = β2
3(π₯ + 4)
#2) identify the intercepts
ππ β ππ = ππ
#3) solve for the variable
ππ + π = ππ
Guided Notes
Essential Questions: What is a function? Why is domain and range important in defining
a function?
Letβs do some critical thinkingβ¦
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 emoticonsβ
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 Β½ hour, it increases its speed and travels at a
constant rate for 1 Β½ 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 βUp
Simplify Write the domain and range Is the previous question
ππ + ππ β π + ππ β ππ (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 β 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) β 1 5) f(h) = βπ
ππ
6) f(x)= -2(x+1) 7) f(m) = 1 - 4(π
π) 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 2ΛF 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 β x 4) f(x) = Β½ (x β 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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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.
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 βUp
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 Averyβs statement.
b. Give an argument to support Cindyβs 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
Frankβs starting point.
What function represents each turtleβs 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 β 4 and g(x) = 2(x +5) β 4
Make a table:
X f(x) =
2x β 4
g(x) =
2(x +5) β 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 β 2 Graph g(x) = Β½ (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 < (π ππ π) < 1
a Same Shape as parent
function
Stretches vertically Multiply the vertical distance by
βaβ for each point
Compresses vertically Multiply the vertical distance by
βbβ for each point
b Same Shape as parent
function
Compresses horizontally Multiply the horizontal distance
by βbβ for each point
Stretches horizontally Multiply the horizontal distance
by βbβ 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 β 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.
π¦ = Β±π(ππ₯ β β) + π
Establish a Shape 1 Graph the Starting Point 2
Graph the Stretch Point 3 Sketch the Function 4
(π₯ β β) (π₯ + β) Shifts Right Shifts
Left
+π βπ Shifts up Shifts
Down
Β± (+ 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 β
intercept of the graph off ?
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.
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 βUp
Find the slope and graph Solve and Graph
ππ + ππ = ππ ππ β π < ππ + π
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:
ππ = ππβπ + π
an β nth term of the sequence
a1 β 1st term of the sequence
an-1 β 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?
ππ = ππβπ + π
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.
ππ = ππ + (π β π)π
an β nth term of the sequence
a1 β 1st term of the sequence
n β 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?
ππ = ππ + (π β π)π
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Write an explicit formula from a recursive formula
ππ = ππβπ β π ; ππ = ππ ππ = ππβπ + π. π; ππ = βπ
ππ = ππβπ + ππ ; ππ = π ππ = ππβπ β ππ; ππ = ππ
Write a recursive formula from an explicit formula
ππ = π +π
ππ ππ = π + ππ
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, β¦
c) 37, 34, 31, 29, 26, β¦ d) 93, 86, 79, 72, 65, β¦
Write a recursive formula and an explicit formula for each sequence
a) 12, 19, 26, 33, 40, β¦ b) -4, 5, 14, 23, 32, β¦
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c) 62, 57, 52, 47, 42, β¦ d) -15, -6, 3, 12, 21, β¦
Write a recursive formula for each explicit formula and find the first term of the
sequence.
ππ = ππ + πππ ππ =π
πβ ππ
Describe and correct the error a student made in identifying the common difference of
the following sequence: 29, 22, 15, 8, 1, β¦
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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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.
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 βUp
#1) Is this an arithmetic sequence?
2, 4, 6, 8, 12β¦β¦
#2) Solve
π(π + π)π β π Γ· π
#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
______________ _____________________.
What is the relationship between hours after sunrise, x, and the temperature, y, shown in the scatter
plot?
When y-values tend to decrease as x-values increase, the two data sets have a
________________ ________________________.
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What is the relationship between hours after sunrise, x, and the temperature, y, shown in the scatter
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.
What is the relationship between hours after sunrise, x, and the temperature, y, shown in the scatter
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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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.
A student is tracking the growth of some plants. What type of association do you think
the data would show? Explain.