Download Chapter 4A: Linear Functions Index:

Chapter 4A: Linear Functions
Index:
1: Proportional Relationships – U4L1
Pg. 2
2: Conversions – U4L2
Pg. 11
3: Non-proportional Linear Relationships – U4L3
Pg. 17
4: Graphing Linear Equations Introduction
Pg. 24
5: More Work Graphing Linear Functions – U4L4
Pg. 31
F: Graphing Linear Equations (Day 2)
Pg. 37
6: Writing Equations in Slope-Intercept Form – U4L5
Pg. 45
7: Modeling with Linear Functions – U4L6
Pg. 52
8: More Linear Modeling – U4L7
Pg. 57
9: Strange Lines – Horizontal and Vertical – U4L8
Pg. 63
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10: Absolute Value and Step Functions – U4L9
Pg. 68
11: The Truth About Graphs – U4L10
Pg. 73
12: Graphs of Linear Inequalities – U4L11
Pg. 78
13: Introduction to Sequences – U4L12
Pg. 83
14: Arithmetic Sequences – U4L13
Pg. 88
2
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Proportional Relationships
Lesson 1:
Proportional Relationships
You have studied proportional relationships in previous courses, but they are the basis of all linear
functions, so we will take a lesson to recall their particulars
Two variables have a proportional relationship if their respective values are always in the same ratio
(They have the same relative size to one another).
For Example:
A rope’s length and weight are in proportion. When 20meters of rope weights 1kilogram, then:
- Now using the given information lets establish a ratio:
-
You can now use this ratio in a proportion, which can help you solve for missing information.
How much would 40 meters of rope weight?
o
Cross Multiply
Isolate the variable
Therefore 40 meters of that rope will weight 2 kilograms
3
Exercise #1: At a local farm stand, six apples can be bought for four dollars. Determine how much it would
cost to buy the following amounts of apples. Round to the nearest cent, as necessary.
(a) Identify the given ratio
(b) 12 apples
(c) 20 apples
(d) If c is the total cost of apples and n is the
number of apples bought, write a proportional
relationship between c and n. Solve this
equation for the variable c.
(f) According to the graph,
terms of apples and money spent?
(e) Use the equation from part d and graph the
relationship on the grid below.
. Illustrate this on your graph. How do you interpret
in
4
Exercise #2: Erika is driving at a constant rate. She travels 120 miles in the span of 2 hours.
(a) If Erika travels at the same rate, how far
(b) Write a proportional relationship between the distance D
will she travel?
that Erika will drive over the time t that she travels,
assuming she continues at this same rate. Solve the
proportion for D as a function of t.
(c) What is the value of the proportionality constant? What are its units?
(d) How much time will it take for Erika to travel 150 miles?
(e) Graph D as a function of t on the axes to the right.
(f) What does the constant of proportionality from part (c) represent about this graph. Explain your thinking.
5
Exercise #3: If Jenny can run 5 meters in 2 seconds, then which of the following gives the distance, d, she
can run over a span of t-seconds going at the same constant rate of meters per second?
(1)
(2)
(3)
(4)
6
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Proportional Relationships
4A HW
Lesson 1:
Remember – Proportional Relationships are just ratios represented as fractions.
Steps: Set up your proportion to compare the two sets of ratios.
Cross Multiply to begin solving your proportion
Divide to fully isolate the variable and solve your problem
Directions: Solve the basic proportions to solve for the value of x.
1.)
2.)
3.)
4.)
6.)
5.)
6.) A nutrition company is marketing a low-calorie sack brownie. A serving size of the snack is
3 brownies and has a total of 50 calories.
(a) Determine how many calories
(b) Determine how many calories 21 brownies
6 brownies would have.
would have.
(c) Determine how many calories 14 brownies
would have. Round to the nearest calorie.
(d) If c represents the number of calories and b
represents the number of brownies, write a
proportional relationship involving c and b and then
solve it for c.
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(e) Graph the proportional relationship you
found in part (d) on the grid shown.
(f) Using the graph, what is the smallest whole number of brownies that a person would need to eat in order
to consume 125 calories? Illustrate this on your graph.
(g) Algebraically find the number of brownies that a person would eat in order to consume 300 calories.
7.) A local animal feed company makes its feed by the ton, which is 2,000 pounds. They want to include a
medication in the feed. Each cow needs 300 milligrams (mg) of this medication a day and each cow
consumes 15 pounds of the feed per dat. If there are 1,000 milligrams in a gram, how many grams of the
medication should the feed company add for each ton of feed they produce?
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8.) Ken is driving at a constant speed. After
hours he has driven a total distance of 90 miles.
(a) How far will Ken drive in 2 hours at this rate?
(b) If D represents the distance Ken has driven
in miles and t represents the time he has been
driving, in hours, then write an equation for
D in terms of t.
(c) Use this equation to determine how far Ken
drives in 15 minutes.
(d) Ken is driving a total of 234 miles. How
long will his trip take him, to the nearest tenth
of an hour, assuming that he travels at this
constant rate? Use proper units.
Reasoning Section:
Unit rates are proportions where we compare the change in one variable to a change of one unit in the
other variable. When we typically report speeds in miles per hour, that is a unit rate. A speed of 65 miles
per hour should be interpreted as 65 miles traveled per 1 hour of time. When we say that fat has 9
calories per gram, that is a unit rate because we are comparing 9 calories to 1 gram.
9.) Convert each of these into unit rates.
(a) 24 feet per 3 seconds
(b) 30 pounds per 8 boxes
(c) 50 calories per 20 chips
9
Review Section:
10.) If there are 4 quarts in a gallon, 2 pints in a quart, and 2 cups in a pint, then how many cups are
in a gallon? Show your calculations or explain how you arrived at your answer.
11.) The inequality
is equivalent to:
[1]
[2]
[3]
[4]
12.) When solving for the value of x in the equation 4(x – 1) + 3 = 18, Aaron wrote the following on the
board. Which property was used incorrectly when going from line 2 to line 3?
[1] Distributive
[line 1] 4(x – 1) + 3 = 18
[2] Associative
[line 2] 4(x – 1) = 15
[3] Commutative
[line 3] 4x – 1 = 15
[4] Multiplicative Inverse
[line 4] 4x = 16
[line 5] x = 4
10
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Unit Conversions
4B
Lesson 2:
Units are amazingly important in mathematics, science and
engineering. They are how we decide on what constitutes the
number 1 (i.e. 1 gallon, 1 pound, etcetera.) We often need to
convert from one unit to another in practical problems. In this
situation we can almost always use proportional reasoning
to do the job.
In June, on the NYS CC Regents Exam you will be
given a reference sheet. The top portion of the reference
sheet is pictured to the right. This will help with any
conversions that you may struggle with.
Example: Amanda has traveled a total of 6.3 miles. If there are 1760 yards in each mile, how many yards
did Amanda travel? Set up and solve a proportion for the problem.
Exercise #1: John has traveled a total of 4.5 miles. If there are 5,280 feet in each mile, how many feet did john travel?
Set up and solve a proportion for the problem.
Exercise #2: If there are exactly 2.54 centimeters in each inch, how many centimeter are there is on foot? Show the
work that leads you to your answer.
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Exercise #3: On a certain day in Toronto Canada, the temperature was
elcius (C). Using the formula
, Peter converts his temperature to degrees Fahrenheit (F). Which temperature represents
in degrees Fahrenheit?
[1]
[2]
[3]
[4]
Exercise #4: A parking lot is 100 yards long. What is the length of of the parking lot, in feet?
[1] 300
[3] 75
[2] 225
[4]25
Exercise #5: A bathtub contains 14.5 cubic feet of water. If water drains out of the bathtub at a rate of 4 gallons per
minute, then how long will it take, to the nearest minute, to drain the bathtub? There are 7.5 gallons of water per
cubic foot. Show the work that leads to your answer.
Exercise #6: The mile and the kilometer are relative close in size. can you convert 1 mile into an equivalent in
kilometers? Here’s what I’ll give you. There are 2.54 centimeters in an inch, 5,280 feet in a mile, 100 centimeters in a
meter, and 1,000 meters in a kilometer. All else you should be able to do for yourself. Round your answer to the
nearest tenth of a kilometer. This takes quite a string of multiplications, but you can do it!!
12
We can also convert the ratio of two quantities, or rates, into different units if need be.
Exercise #7: One interesting conversion is from a speed expressed in feet per second to a speed in miles per
hour. We sometimes think better in miles per hour because that is how the speeds of our cars are measured.
(a) Convert a speed of 45 miles per hour into feet per second given that there are 5,280 feet in a mile.
(b) The current fastest human is Usain Bolt, from Jamaica. In 2009, Usain ran 100 meters in a blazing 32.2 feet
per second average speed. How does this compare to a typical car driving speed?
Exercise #8: Elizabeth is baking chocolate chip cookies. A single batch uses of a teaspoon of vanilla, if
Elizabeth is mixing the ingredients for five batches at the same time, how many tablespoons of vanilla will
she use?
[1]
[2]
[3]
[4]
Exercise # 9: Roberta needs ribbon for a craft project. The ribbon sells for $3.75 per yard. Find the cost, in
dollars, for 48 inches of the ribbon.
13
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Unit Conversions
4B HW
Lesson 2 HW:
1.) The formula for changing Celsius (C) temperature to Fahrenheit (F) temperature is
Calculate to the nearest degree, the Fahrenheit temperature when the Celsius temperature is
.
.
2.) Peter walked 8,900 feet from home to school. How far, to the nearest tenth of a mile, did he walk?
[1] 0.5
[3] 1.6
[2] 0.6
[4] 1.7
3.) A total of 1680 ounces of pet food have to be packed in 5-pound bags. How may 5-pound bags of pet
food can be packed?
[1] 21
[3] 105
[2] 28
[4] 336
4.) Which expression can be used to change 75 kilometers per hour to meters per minute?
[1]
[2]
[3]
[4]
5.) A jogger ran at a rate of 5.4 miles per hour. Find the jogger’s exact rate, in feet per minute.
14
6.) How many centimeters are there in 1 yard if there are 2.54 centimeters per inch? Show your work
and express your answer without rounding.
7.) How close are a meter and a yard? Convert 1 meter into yards by using the fact that there are 100
centimeters in a meter, 2.54 centimeters in an inch, 12 inches in a foot, and 3 feet in a yard. Round
your answer to the nearest tenth of a yard.
8.) If there are 1000 grams in a kilogram and 454 grams in a pound, how many pounds are there per
kilogram? Round to the nearest tenth of a pound.
9.) A high school track athlete sprints 100 yards in 15 seconds.
(a) Determine the number of feet per second the runner is traveling at. Show your work.
(b) If there are 5280 feet in a mile and 3600 seconds in an hour, determine the runner’s
speed in miles per hour. Round to the nearest tenth.
15
10.) Mrs. Chen owns two pieces of property. The areas of the properties are 77,120 square feet and 33,500
square feet. Find the total number of acres Mrs. Chen owns, to the nearest hundredth of an acre?
11.) If a United States dollar is worth $1.41 in Canadian money, how much is $100 in Canadian money
worth in United States money, to the nearest cent?
Review Section:
12.) If
, then
[1]
[3] 6
13.) The statement
[1] associate
[3] distributive
equals:
[2]
[4] 18
is an example of the use of which property of real numbers?
[2] additive identiy
[4] additive inverse
14.) The length of a rectangular window is 5 feet more than its width . The area of the window is 36 square feet.
Which equation could be used to find the dimensions of the window?
[1]
[2]
[3]
[4]
16
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Non-Proportional Linear Relationships
4C
Lesson 3:
In this unit’s first lesson, we saw the simplest type of linear relationship, one where the two variables are
proportional to one another. In that case, recall:
PROPORTIONAL RELATIONSHIPS
The variables x and y are proportional if:
In other words, one variable is always a constant.
But, there are lots of linear relationships (ones that when graphed would form a line) that are not
proportional. How can we relate them with an equation?
Example: Consider the linear function graphed
.
a) Evaluate
and
. What two coordinate
points do these function values correspond to?
b) Calculate the average rate of change from from
to
. This is also known as what quantity for this line?
c) Identify the coordinate of the y-intercept.
Exercise #1: Consider the linear function graphed
.
a) Evaluate
and
. What two coordinate
points do these function values correspond to?
b) Calculate the average rate of change from from
to
. This is also known as what quantity for this line?
c) Identify the coordinate of the y-intercept.
17
Exercise #2: Consider the linear function
shown below.
(a) Evaluate
and
. What two coordinate points do
these function values correspond to?
(b) Calculate the average rate of change of f from
This is also known as what quantity for this line?
to
(c) Is there a proportional relationship between x and y? How can you check?
(d) Based on your 8th grade coursework, what relationship does exist between the two variables?
Write this equation and check it for the points from (a).
In general, what is always proportional on a linear function is the change in y to the change in x, also known
as the line’s slope. this gives rise to what is known as the slope – intercept form of a line.
The Slope Intercept Form of a Linear Function
Given a linear function
it can be expressed in equation form by:
OR
where m is the average rate of change or slope or
and b is the y-intercept of the line.
Example: Given the function
, identify the slope and the coordinates of the y-intercept.
18
Exercise #3: Given
Exercise #4: Given
, identify both the slope and the coordinates of the y-intercept.
, identify both the slope and the coordinates of the y-intercept.
Exercise #5: Given the linear function
(a) Complete the following table.
, do the following:
(b) Create a graph of the function on the axes below.
(c) Illustrate the slope of the function graphically
(d) Circle the graphs y-intercept
19
Exercise #6: Given the linear function
(a) Complete the following table.
, do the following:
(b) Create a graph of the function on the axes below.
(c) Illustrate the slope of the function algebraically using
two points from your graph.
(d) Identify the coordinates of the y-intercept.
20
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Non-Proportional Linear Relationships 4C HW
Lesson 3 HW:
1.) For the linear function
which of the following is true?
(1) It has a slope of 7 and a y-intercept of 2.
(2) It has a slope of 2 and a y-intercept of 7.
(3) It has a slope of 7x and a y-intercept of 2.
(4) It has a slope of 2 and a y-intercept of 7x .
2.) What is the equation of the line shown in the graph below?
(1)
(2)
(3)
(4)
3.) Which of the following represents the average rate of change of the function
over the interval
?
(1)
(2)
(3)
(4)
4.) Given the function
through the points
and
, does the function pass
Use of the grid is optional.
21
5.) Graph the function
6.) Graph the function
using the provided table.
using the provided table.
7.) What is the slope of a line passing through points (−7, 5) and (5, −3)?
[1]
[2]
[3]
[4]
8.) What is the slope of line in the accompanying diagram?
22
9.) What is the average rate of change between point A and point B?
Review Section:
10.) What is the value of x in the equation
[1]
[2] 16
[3]
[4] 4
11.) In a baseball game, the ball traveled 350.7 feet in 4.2 seconds. What is the average rate of speed of the
ball in feet per second?
[1] 83.5
[2] 177.5
[3] 354.9
[4] 1,4729
12.) The width of a rectangle is 4 less than half the length. If represents the length, write an equation
could be used to find the width,
23
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Graphing Linear Functions
4D
Lesson 4:
Graphing Linear Equations
To graph a line (linear equation), we first want to make sure the equation is in slope intercept form (y=mx+b).
We will then use the slope and the y-intercept to graph the line.
Slope (m): Measures the steepness of a non-vertical line. It is sometimes referred to as the
or
. It’s how fast and in what direction y changes compared to x.
y-intercept(b): The y-intercept is where a line passes through the y axis. It is always stated as an ordered
pair (x,y). The x coordinate is always zero. The y coordinate can be taken from the “b” in
.
Graphing The Linear Equation:
1) Find the slope:
 =
–
=
2) Find the y-intercept:

(Start Point)
3) Plot the y-intercept
4) Use slope to find the next point:
Start at
 up 3 on the y-axis
 right 1 on the x-axis
5) To plot to the left side of the y-axis, go to y-int.
and do the opposite(Down 3, left 1)
6) Connect the dots, using a straight edge.
7) Draw arrows at both ends of the line.
MANDATORY WORK:
8) Label the line with the equation
1st pt =
24
Directions: Graph each of the following equations, showing all mandatory work!
Exercise #1:
Exercise #2:
Exercise #3:
25
Identifying Points on Linear Equations
When graphing lines we use the slope intercept form
. The represents the slope of the line,
the represents the y-intercept of the line and the and the are the coordinates
of points on the line.
Example: Is the point
on the line represented by
?
No the point is not on the line.
Exercise #4: Does the function
pass through the point
Exercise #5: Which of the following is the equation of a line whose slope is 3 and which passes through the
point (2,7)?
[1]
[2]
[3]
[4]
Exercise #6: Which of the following is the equation of a line whose slope is and which passes through the
point (12,13)?
[1]
[2]
[3]
[4]
26
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Graphing Linear Functions
4D HW
Lesson 4 HW:
1.) What are the coordinates of the y-intercept of the line:
[1] (0, 5)
[3] (0, -11)
[2] (5, 0)
[4] (-11, 0)
2.) What is an equation of the line that has a slope of
[1]
[2]
[3]
[4]
and passes through
?
3.) Which verbal expression can be represented by
[1] 5 less than 2 times x
[2] twice the difference of x and 5
[3] 2 multiplied by x less than 5
[4] the product of 2 and x, decreased by 5
4.) An example of an algebraic inequality is:
[1]
[2]
[3]
[4]
5.) When
[1] 0
[3] 8y
and
what is the value of
[2] 10x
[4] 8y
6.) Which equation has a slope of 2 and passes through the point
[1] y = 2x – 2
[2] y = 2x – 2
[3] y = 2x + 2
[4] y = 2x + 2
?
7.)
27
8.)
9.)
10.)
28
11..)
Review Section:
12.) Joey enlarged a 3-inch by 5-inch photograph on a copy machine. He enlarged it four times. The table
below shows the area of the photograph after each enlargement.
What is the average rate of change of the area from the original photograph to the fourth enlargement, to
the nearest tenth?
[1]4.3
[2] 5.4
[3]4.5
[4] 6.0
29
13.) If
and
[1]
[3]
, at which value of x is
[2]
[4]
14.) The distance a free falling object has traveled can be modeled by the equation
, where is
acceleration due to gravity and is the amount of time the object has fallen. What is in terms of and ?
[1]
[2]
[3]
[4]
30
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
More Work Graphing Linear Functions
4E
Lesson 5:
More Work Graphing Linear Functions
Exercise #1: Four lines are shown graphed. Place the number of the line next to the equation that most
appropriately models it and explain why you chose that line.
a)
[3] – Positive y-intercept & Negative Slope
b)
_________________________________________________
_________________________________________________
c)
_________________________________________________
_________________________________________________
d)
_________________________________________________
_________________________________________________
Recall that if a line is written in the form
, then it is relatively easy to graph, especially if
reasonably easy to work with. A quick review from the pervious lessons.
Exercise #2: On the grid, graph the equation
and are
First identify its slope and y-intercept.
31
Exercise #3: Write down two points this line passes through
and use them to calculate the average rate of change of this function.
Sometimes linear equations are not written in a form that makes it easy to determine the slope and the
intercept. It is important to be able to rearrange these formulas in order to quickly identify these linear
parameters.
Exercise #4: Consider the linear equation given by
(a) Steps are shown below that rearrange this equation. Justify each step with a property of equality or a
property of numbers.
(1) _______________________________________________________________
(2) _______________________________________________________________
(3) _______________________________________________________________
(4) _______________________________________________________________
(b) Identify the slope and the y-intercept of this line.
32
Exercise #5: Rearrange each of the following linear equations into
(a)
(b)
(c)
(d)
(e)
(f)
, then identify the slope and y-intercept.
33
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
More Work Graphing Linear Functions
4E HW
Lesson 5 HW:
1.) Which of the following is true about the linear function
[1] It has a slope of 2 and a y-intercept of 18
[2] It has a slope of 2 and a y-intercept of 9
[3] It has a slope of
and a y-intercept of 9
[4] It has a slope of and a y-intercept of 18
2.) For the line
, for every unit increase in x, which of the following is true?
[1] y decreases by 6
[2] y increases by 3
[3] y increases by 2
[4] y decreases by 10
3.) The two lines
and
are shown graphed below. The values of a,b,c and d are not given, but
properties of them can be inferred from the graph. Determine if the pair of values could be equal and explain your
answer.
[1] b and d
[2] a and d
[3] a and c
4.) Four lines are shown graphed. Place the number of the line next to the equation that most appropriately models it
and explain why you chose that line.
a)
_________________________________________________
_________________________________________________
b)
_________________________________________________
_________________________________________________
c)
_________________________________________________
_________________________________________________
d)
_________________________________________________
_________________________________________________
34
5.) Rewrite each of the following linear equations in equivalent slope-intercept form, then identify the slope and
the y-intercept.
(a)
(b)
(c)
(d)
(e)
(f)
35
Review Section:
6.) The value of the x-intercept for the graph of
[1] 10
[2]
[3]
7.) Let
is:
[4]
be a function such that
[1]
[2]
[3]
[4]
is defined on the domain
8.) Which graph shows a line where each value of
[1]
[2]
9.) What is the value of x in the equation
[1] 4
[2] 6
[3] 8
[4] 11
. The range of this function is:
is three more than half of x?
[3]
[4]
?
10.) Which situation could be modeled by using a linear function?
[1] a bank account balance that grows at a rate of 5% per year, compounded annually
[2] a population of bacteria that doubles every 4.5 hours
[3] the cost of cell phone service that charges a base amount plus 20 cents per minute
[4] the concentration of medicine in a person’s body that decays by a factor of one-third every hour
36
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Graphing Linear Functions (Day 2)
4F
Lesson 6:
Graphing Linear Equations (Day 2)
(
To graph a line (linear equation), we first want to make sure the equation is in slope intercept form
). We will then use the slope and the y-intercept to graph the line.
Graphing Linear Equations:
Ex. Point Slope Form
To Graph: Solve the equation for slope intercept form.
Ex. Standard Form
To Graph: Solve the equation for slope intercept form.
37
Exercise #1: Graph
Exercise #2: Graph
Exercise #3: Graph
38
Exercise #4: Graph
Exercise #5: Graph
Exercise #6: Graph
39
Exercise #7: Graph
Exercise #8: Graph
Exercise #9: Graph
40
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Graphing Linear Functions (Day 2)
4F HW
Lesson 6 HW:
1.) Which point is located on the line represented by the equation
[1] ( 4, 5)
[2] ( 5, 4)
[3] (3, 4)
[4] ( 3,4)
2.) Which equation is equivalent to
[1]
[2]
[3]
[4]
3.) What are the coordinates of the y-intercept of the line:
[1] (0, 5)
[2] (5, 0)
[3] (0, -11)
[4] (-11, 0)
4.) Which of the following is an equation of a line in standard form?
[1]
[2]
[3]
[4]
5.) The line whose equation is
[1] slope = y-intercept = 4
[3] slope =
y-intercept = 4
has:
[2] slope =
y-intercept = 4
[4] slope =
y-intercept = 4
6.) Graph
41
7.) Graph
8.) Graph
9.) Graph
42
10.) Graph
11.) Graph
43
Review Section:
12.) Which statement is not always true?
[1] The product of two irrational numbers is irrational.
[2] The product of two rational numbers is rational.
[3] The sum of two rational numbers is rational.
[4]The sum of a rational number and an irrational number is irrational.
13.) 2 A satellite television company charges a one-time installation fee and a monthly service charge. The
total cost is modeled by the function
. Which statement represents the meaning of each part of
the function?
[1] y is the total cost, x is the number of months of service, $90 is the installation fee, and $40 is the
service charge per month.
[2] y is the total cost, x is the number of months of service, $40 is the installation fee, and $90 is the
service charge per month.
[3] x is the total cost, y is the number of months of service, $40 is the installation fee, and $90 is the
service charge per month.
[4] x is the total cost, y is the number of months of service, $90 is the installation fee, and $40 is the
service charge per month.
14.) Which point is not on the graph represented by
[1]
[2]
[3]
[4]
?
44
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Writing Equations in Slope-Intercept
4G
Lesson 7:
Writing Equations in Slope-Intercept Form
One skill that we need to become fluent at in Algebra 1 is created the equation of a linear function. We will
concentrate on learning how to form equations in the slope-intercept form that we have been working with.
Given a linear function
The Slope-Intercept Form of a Linear Function
, it can be expressed in equation form by:
where the two parameters are
average rate of change = slope =
and
y-intercept of the line.
Exercise #1: Consider the linear function whose graph is shown.
(a) Determine an equation in the form
for this line.
(b) Test your equation for the value
When the y-intercept is an integer, such as in the last exercise, it is fairly easy to get the exact relationship
between x and y. Lets try another graphical problem where the y-intercept is not an integer.
Exercise #2: Consider the linear function whose graph is shown.
(a) Find the equation of the linear function show in slope-intercept form.
(b) Test your equation for the value of
.
45
We need to be able to find the equation for a linear function if we know two points that lie on it. Notice that this
means we have to determine the value of the two parameters with two pieces of information.
Exercise #3: Find the equation of the line that passes through each of the following pairs of points in
slope-intercept form.
(a)
and
(b)
and
(c)
and
46
(d)
(e)
and
and
Exercise #4: A car is traveling along a straight road. After one hour, the car is 72 miles from Chicago. After three
hours, the car is 188 miles from Chicago. Determine an equation for the distance, , the car is from Chicago after
hours if the relationship between and is linear.
47
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Writing Equations in Slope-Intercept
4G HW
Lesson 7 HW:
1.) Each of the following lines has a slope ad y-intercept that can be determined by examining the graph. For each,
state the slope, the y-intercept and then write the equation in slope-intercept form.
(a)
(b)
Slope: _______________________
Slope: _______________________
y-intercept: ________________
y-intercept: ________________
Equation: ____________________________
Equation: ____________________________
2.) Each of the following lines has a slope that can be determined by examining the graph. Use another point on the
line to solve for the exact y-intercept. Then state the equation of the line.
(a)
(b)
Slope: _______________________
Slope: _______________________
Solve for the y-intercept.
Solve for the y-intercept
Equation: ____________________________
Equation: ____________________________
48
3.) Find the equation of the line that passes through each of the following pairs of points in slope-intercept form.
(a)
and
(b)
(c)
and
and
49
(d)
and
4.) A steady snow fall is coming down outside. Amanda decides to measure the depth of the snow on the ground.
After 4 hours, the snow is at a depth of 9 inches, and after 8 hours it is at a depth of 14 inches.
(a) Express the information given in this problem
(b) Find the slope of the line that passes through
as two coordinate pairs,
, where h is the
these two points. What are its units?
number of hours and d is the depth of the snow.
(c) Find the equation of the line that passes through (d) What was the depth when the snowfall began
these two points in
form.
(h = 0)? What would the depth be after 12 hours?
50
5.) What are the coordinates of the one point shared in common between the two linear functions given below?
Do you remember what the name of this one shared point is?
Do you remember what this type of problem is called?
Review Section:
6.) The function
represents the height,
after it is dropped. A realistic domain from this function is:
[1]
[2]
[3]
[4]
, in feet, of an object from the ground after seconds
7.) The table shows the average diameter of a pupil in a person’s eye as he or she grows older.
What is the average rate of change, in millimeters per year, of a person’s pupil diameter from age 20 to age 80?
[1] 2.4
[2] 0.04
[3] 2.4
[4] 0.04
8.) Sam and Jeremy have ages that are consecutive odd integers. The product of their ages is 783. Which equation
could be used to find Jeremy’s age, , if he is the younger man?
[1]
[2]
[3]
[4]
51
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Modeling with Linear Functions
4H
Lesson 8:
Modeling with Linear Functions
When we use equations to model real-world phenomena we often look to lienar models first because they are the
easiest to use and understand. We can now use our skills from the last few days to model real-world linear
phenomena. Don’t ever forget these two facts about linear models:
Critical Linear Model Facts
All linear models in the form
have two parameters, the slope, m, and the y-intercept, b:
1.) The slope, m, always tells us how fast the output is changing relative to the input
2.) The y-intercept, b, always tells us “how much” we start with, or the outputs starting value (at x = 0)
Exercise #1: Jane has $450 in her savings account at the beginning of the year. She places money in the account at
the rate of $5 per week. We want to model the amount of money she has in savings, , as a function of the number of
weeks she has been saving, .
(a) Fill out the table for some of the number of weeks.
(b) Use information in the givens or in the table to
Show the calculations that result in your answer.
write an equation for the savings, , as a linear function
of the weeks she has been saving, .
(c) If Jane saves for exactly one year, what is the range
in her savings over the year? Show how you arrived
at your answer.
(d) Why would it not make sense to evaluate
of this linear function?
In other words, what types of numbers belong in the domain
(e) Use two points from the table to verify that the rate of change for the function is 5. How do the units show up
in the calculation?
52
Sometimes the infromation we have about the linear relationship does not include the starting value. Let’s take a look
at that type of situation.
Exercise #2: Kirk is driving along a long-road at a constant speed. He is driving directly towards Denver. He knows
that after 2-hours of driving he is 272 miles from Denver. After 3 and a half hours he is 176 miles from Denver.
(a) Summarize the information given in the problem
(b) Calculate
(Include units)
as two ordered pairs, where the number of hours, ,
is the input and the distance from Denver
is the output.
(c) You should have found that the rate of change was
negative. Why is it? Explain what is physically happening
to result in this negative rate of change.
(d) Assuming the relationship is linear (which it
would be at a constant speed), write an equation for
the distance D as a linear function of the hours h.
(e) How far did Kirk start from Denver? Show the
work that leads to your answer.
(f) After how many hours will Kirk arrive in Denver?
Show the work that leads to your answer.
Exercise #3: Erin is walking away from a light pole at a rate of 4 feet per second. If she starts at a distance of 6 feet
from the light pole, which of the following gives her distance, , from the light pole after walking for seconds?
[1]
[2]
[3]
[4]
53
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Modeling with Linear Functions
4H HW
Lesson 8 HW:
1.) Water is building up in a bathtub. After 2 minutes there are 12 gallons of water and after 4 minutes,
there are 20 gallons of water. What is the average rate at which water is entering the bathtub from
to
minutes? Show how you calculated the rate.
[1] 8 gallons per minute
[2] 10 gallons per minute
[3] 6 gallons per minute
[4] 4 gallons per minute
2.) Francisco is saving money in an account. At the beginning of the year, he has $56 in savings and puts in
another $4 per week. Which of the following equations models the amount of savings, s, as a function of the
number of weeks, w, Francisco has been saving?
[1]
[2]
+ 56
[3]
[4]
3.) Maria charges $15 for every 2 hours that she babysits. Answer the following questions based on this information.
(a) How much would Maria chrge for working 5 hours?
(b) Fill out the table below for the amount that Maria
makes as she babysits and graph the relationship
on the grid provided.
(c) Write an equation for the amount, , tht Maria makes as a function of the number of hours, , that she babysits.
Keep in mind that Maria will make $0 for babysitting for 0 hours.
54
4.) The temperature is falling outside at a steady rate of 4 degrees Fahrenheit every hour. If the temperature
starts at 68 Fahrenheit do the following.
(a) Fill out the table below for the outside
(b) Write a linear equation that relates the
temperature during the time it is cooling down.
Fahrenheit temperature, F, to the time in hours, t,
that it has been falling.
(c) According to your equation, what is the
temperature when t = 2.75 hours?
(d) If this cooling continues at this constant rate,
how many hours will it take for the temperature to
reach the freezing point of water? Show your work.
5.) The population of deer in a park is growing over the years. The table below gives the population found
in a survey by local wildlife officials.
(a) Find the average rate that the deer population is changing over each time interval below:
From 2000 to 2003
From 2003 to 2006
From 2006 to 2009
(b) Why does this calculation indicate a
linear relationship?
(c) If t stands for the number of years since 2000,
write an equation for the deer population, p, as
a function of t.
(d) What does your model predict the deer
population to be in the year 2014?
(e) How many years will it take for the deer
population to reach 500? Round to the nearest year.
55
Review Section:
6.) If
[1]
[3]
then equals:
[2]
[4]
7.) Which of thses numbers is a solution for
[1]2
[2] 1
[3] 3
[4] 5
8.)A hotel charges $20 for the use of its dining room and $2.50 a plate for each dinner. An association gives a
dinner and charges $3 a plate but invites four nonpaying guests. If each person has one plate, how many
paying persons must attend for the association to collect the exact amount needed to pay the hotel?
[1] 60
[2] 44
[3] 40
[4] 20
56
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
More Linear Modeling
4I
Lesson 9:
More Linear Modeling
Although it can be challenging, it is critically important that students who exit Algebra I have a good ability
to deal with linear relationships. In this lesson we get more practice modeling linear phenomena.
Exercise #1: A warehouse is keeping track of its inventory of cardboard boxes. At the beginning of the
month, they had a supply of 1,275 boxes left. They use boxes at a rate of 75 per day.
(a) How many boxes are left after 10 days? Show
(b) Which of the following linear equations
the calculation that leads to your answer.
correctly models the number of boxes left, n,
after d-days?
[1]
[2]
[3]
[4]
(c) If the warehouse needs to order more boxes
when their supply reaches 150, how many days
can they wait?
(d) If after 5 days, they start using boxes at a rate
of 90 per day, how many days will it be before they
run out of boxes? Show the work that leads to
your answer.
57
We want to feel very comfortable with quickly and accurately determining linear models. Keep in mind that
the two most important aspects of any linear model are its rate of change (slope) and its starting value (yintercept).
Exercise #2: The cost, c, in dollars of running a particular factory that produces w-widgets can be modeled
using the linear function.
(a) How do you interpret the fact that
(b) Give a physical interpretation for the
two parameters in this equation, 1.25 and 2175.
Exercise #3: Biologists estimate that the number of deer in Rhode Island in 2003 was 1,028, and in 2008 it
had grown to 1,488. Biologists would like to model the deer population, p, as a function of the years, t, since
2000.
(a) Represent the information we have been told
(b) Calculate from 2003 to 2008. Include
as two coordinate points. Be careful to know
proper units in your answer.
what your values of time are for each year.
(c) Give a physical interpretation of the value
you found in part (b).
(d) Determine a linear relationship between the
deer population, p, and the years since 2000, t.
(e) How many deer does this model predict were in
Rhode Island in the year 2000? What does this
represent about the linear function?
(f) How many deer does the model predict for
Rhode Island now?
58
Exercise #4: Water is draining out of a bathtub such that the volume still left, g-gallons, is shown as a
function of the number of minutes, m, it has been draining.
(a) Calculate the average rate of change of g over
the interval
. Include proper units.
(b) Calculate the average rate of change of g over
the interval
. Include proper units.
(c) Why can we say that the relationship between m and g is not linear?
59
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
More Linear Modeling
4I HW
Lesson 9 HW:
1.) A water tank is being filled by pumps at a constant rate. The volume of water in the tank V, in gallons, is
given by the equation:
where is the time, in minutes, the pump has been on
(a) At what rate, in gallons per minute, is the
(b) How many gallons of water were in the tank
water being pumped into the tank?
when the pumps were turned on?
(c) What is the volume in the tank after two hours
of the pumps running?
(d) The pumps will turn off when the volume in the
tank hits 10,000 gallons. To the nearest minute,
after how long does this happen?
2.) A solar lease customer built up an excess of 6,500 kilowatt hours (kwh) during the summer using his
solar panels. When he turned his electric heat on, the excess began to be used up at a rate of 50 kilowatt
hours per day.
(a) If E represents the excess left and d represents
(b) How much of the excess will be left after one
the number of days since the heat has been turned
month (use a month length of 30 days)?
on, write an equation for E in terms of d.
(c) If the heat will need to be turned on for 5 months, will the excess be enough to last through this time
period? Justify your answer.
60
3.) As Evin is driving her car, she notices that after 1 hour her gas tank has 7.25 gallons left and after 4 hours
of driving, it has 3.5 gallons of gas left in it.
(a) Represent this information as two coordinate
(b) Find the slope between these two points.
pairs in the form
, where h is the number
Using proper units, interpret this slope.
of hours driven and g is the gallons of gas left.
(c) Assuming the relationship between h and g
is linear, find an equation for g in terms of h.
(d) According to this equation, after how many
hours of driving would Evin run out of gas?
4.) The population of Champaign, Illinois is given for three years in the table below:
(a) Using 1970 as t = 0, create a linear model
from the first data points in this table to predict
the population, p, as a function of the number of
years since 1970, .
(b) If this model is used to predict the population
of Champaign in the year 2012, will the model
overestimate or underestimate the actual
population? Explain.
61
Review Section:
5.) Juan has a cellular phone that costs $12.95 per month plus 25¢ per minute for each call. Tiffany has a
cellular phone that costs $14.95 per month plus 15¢ per minute for each call. For what number of minutes
do the two plans cost the same?
6.) A cake recipe calls for 1.5 cups of milk and 3 cups of flour. Seth made a mistake and used 5 cups of flour.
How many cups of milk should he use to keep the proportions correct?
[1]1.75
[2] 2
[3] 2.25
[4] 2.5
7.) Which point lies on the line whose equation is 2x  3y  9?
[1] (1,3)
[2] (1, 3)
[3] (0, 3)
[4] (0,3)
8.) Which graph represents the solution to the inequality:
[1]
[2]
[3]
[4]
9.) In a recent town election, 1,860 people voted for either candidate A or candidate B for the position of
supervisor. If candidate A received 55% of the votes, how many votes did candidate B receive?
[1] 186
[2] 837
[3] 1,023
[4] 1,805
62
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Strange Lines
4J
Lesson 10:
Strange Lines – Horizontal & Vertical
Although they don’t fit the classic linear model, it is important to understand how we write equations for
horizontal and vertical lines. The first exercise will illustrate the idea. Never forget, though, that when we
create an equation for a curve, it simply describes what all points on the curve share in common.
Exercise #1: Shown below are a horizontal line and a vertical line.
Horizontal Line:
Name two coordinate points
Vertical Line:
Name two coordinate points
What do they share?
What do they share?
What is the lines equation?
What is the lines equation?
Horizontal and Vertical Lines
Horizontal Line
Vertical Line
(Constants can be determined by using any point that the line passes through)
H–O–Y
H- horizontal line
O – zero slope
Y – y = # (equation)
V–U–X
V- vertical line
U – undefined slope
X – x = # (equation)
Ex:
H- horizontal line
O – zero slope
Y – y = # (equation)
Ex:
V- vertical line
U – undefined slope
X – x = # (equation)
63
Exercise #2: Which of the following equations represents a vertical line that passes through
[1]
[3]
[2]
[4]
Exercise #3: For each of the following, give the equation of the line shown or described.
(g) The equation of a vertical line passing through
the point
(h) The equation of a horizontal line passing
through the point
Exercise #4: Sketch the region bounded by the three lines whose
equations are given below. Label each with its equation. Find the
area of the triangular region enclosed by the lines. You may want
to use your calculator to create a table of values of the first line or
simply use facts about the slope and y-intercept.
64
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Strange Lines
4J HW
Lesson 10 HW:
1.) For each of the following, give the equation of the line shown.
2.) Write the equations of lines that fit the following descriptions. Sketch a picture if needed.
(a) A vertical line that passes through the
(b) A horizontal line that passes through
point
the point
(c) A line parallel to the x-axis that passes
through the point
(d) A line perpendicular to the x-axis that passes
through the point
3.) A rectangle is surrounded by the lines whose equations are shown below.
Graph these lines and find the area of the rectangle enclosed by them.
65
4.) Each of the following lines are either horizontal, vertical, or slanted. Label each with its type and then
graph on the grid. Label each with its equation.
TYPE
(a)
______________________
(b)
______________________
(c)
______________________
(d)
______________________
(e)
______________________
5.) The triangular region shown below is bordered by one vertical line, one horizontal line, and one slanted
line. State the equation of each line and determine the triangle’s area.
Vertical Line: ________________________________
Horizontal Line: ________________________________
Slanted Line: ________________________________
Area: ________________________________
66
Review Section:
6.) The Edison Lightbulb Company tests 5% of their daily production of lightbulbs. If 500 bulbs were tested
on Tuesday, what was the total number of bulbs produced that day?
[1] 25
[2] 1,000
[3] 10,000
[4] 100,000
7.) Which number is in the solution set of the inequality 5x  3  38?
[1] 5
[2] 6
[3] 7
[4] 8
8.) What is the slope of the line represented by the equation 4x + 3y = 12?
[1]
[2]
[3]
[4]
9.) In a baseball game, the ball traveled 350.7 feet in 4.2 seconds. What was the average speed of the ball,
in feet per second?
[1] 83.5
[2] 177.5
[3] 354.9
[4] 1,472.9
10.) The formula for the volume of a pyramid is V = Bh. What is h expressed in terms of B and V?
[1] h = VB
[2] h 
[3] h 
[4] h  3VB
67
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Abs. Value & Step Functions
4K
Lesson 11:
Absolute Value and Step Functions
There are two very interesting functions that can be considered related to linear, the absolute value
function and the step function. Let’s start with the simpler of the two, the absolute value.
Exercise #1: The absolute value gives us the “size” or magnitude of a number. Find each of the following.
(a)
=
(b)
(c)
(d)
Exercise #2: Consider the absolute value function
(a) Evaluate
and
.
. Do the following:
(b) Fill out the table below and graph the function
over this interval.
(c) What is the minimum value of the function on
this interval?
(d) Over what domain interval is
increasing?
Exercise #3: Consider the absolute value function
(a) Evaluate
and
.
. Do the following:
(b) Fill out the table below and graph the function
over this interval.
(c) What did the negative symbol do to the function?
(d) Over what domain interval is
decreasing?
68
Exercise #4: For the function
calculations that lead to your answer.
[1] 10
[2] 2
[3] 12
[4] 4
which of the following is the value of
Show the
Step functions are another type of function that is related to the linear family. It’s graph will reflect its well
chosen name.
Exercise #5: Consider the step function given by
(a) Evaluate each of the following. After you do your
evaluation, write down what coordinate point must
lie on the graph as a result of the calculation.
(b) Graph the step function on the grid to the right.
Step Functions can arise in the real world whenever the output to a particular function is constant over
particular ranges. Here’s an example.
Exercise #6: At a local amusement park, the park charges an admission based on age. Graph the amount of
money a person would have to pay for admission based on their age. Remember that someone who is one
day short of 4 years old can consider themselves three and under.
69
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Abs. Value & Step Functions
4K HW
Lesson 11 HW:
Evaluate each of the following using the provided functions:
1.)
2.)
3.)
4.)
5.)
6.)
7.) Consider the absolute value function
(a) Evaluate
and
without a calculator.
(b) Graph this function over the interval
Create your table below.
only on the interval
.
.
(c) Over which of the following intervals is
always increasing? Circle the correct choice.
[1]
[2]
[3]
[4]
(d) State the range of
on this domain interval.
70
8.) Are the two expressions
and
equivalent? Give evidence to support your yes or no
answer. Remember, for expressions to be equivalent, they must have the same value for all values of the
input variable, .
9.) For each of the following step functions, produce a graph on the grid given.
(a)
(b)
10.) Postage rates on envelopes are a great example of step functions. There is a fixed price for a certain
range of weights and then another fixed price for another range of weights, etcetera. Below is the graph of
one such price structure.
(a) According to this graph, what would be the
postage rate on a letter weighing 1.5 ounces?
(b) What would be the postage rate on a letter
weighing exactly 3.0 ounces?
(c) Write a piecewise defined function for the
postage rates:
(d) Why would it be incorrect to state that the
range of this function is
71
Review Section:
11.) When solving the equation
step. Which property justifies Emily’s first step?
[1] addition property of equality
[2] commutative property of addition
[3] multiplication property of equality
[4] distributive property of multiplication over addition
, Emily wrote
as her first
12.) Officials in a town use a function, , to analyze traffic patterns.
represents the rate of traffic
through an intersection where is the number of observed vehicles in a specified time interval. What
would be the most appropriate domain for the function?
[1]
[2]
[3]
[4]
13.) If
[1]
[2]
[3]
[4]
and
then
equals:
72
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
The Truth About Graphs
4L
Lesson 12:
The Truth About Graphs
At this point we’ve looked at graphs of linear functions and more general functions as simply being plots of
input/output pairs. And, for functions, this makes a lot of sense. But, more generally, we want to be able to
define points that lie on the graph of an equation or on an inequality with a simple test/definition.
Graphing Equations and Inequalities
The connection between graphs and equations/inequalities is a simple one:
1.) Any coordinate pair
that makes the equation of inequality true lies on the graph
2.) The entire graph is a collection of all of the
pairs that make the equation of inequality true.
Exercise #1: Consider the linear equation
(a) Does the point
lie on the graph of this
equation? Justify your answer.
Exercise #2: The equation
Justify how you found your answer.
(b) Does the point
lie on the graph of this
equation? Justify your answer.
describes a parabola. Does the point
lie on its graph?
Inequalities can also be graphed and we will concentrate on that tomorrow. But, in this lesson we can
certainly determine if particular points will lie on the graph of an inequality.
Exercise #3: Determine for each of the following inequalities whether the point given lies on the graph.
(a)
for
(b)
for
(c)
for
(d)
for
73
We can even determine, with some additional calculations, whether a point is a solution to a system of
equations or a system of inequalities. You’ve studied systems before and we will devote the next unit to
them. But, with a simple definition you can “easily” tell whether points are solutions.
SYSTEMS OF EQUATIONS
A system of equations is a collection of two or more equations joined by the AND truth condition.
Because the AND condition is only true when all of its components are true, the solution set of a system is:
The collection of all points that result in all equations or inequalities being true.
Exercise #4: Determine if the point
is a solution to the system of equations shown below.
Justify your answer.
and
Exercise #5: Does the point
and
lie in the solution set of the system of inequalities show below?
You can even mix equations and inequalities because the answer always depends on whether all
conditions are true or not.
Exercise #6: Is the point
a solution to the system shown below? Justify your answer carefully.
and
74
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
The Truth About Graphs
4L HW
Lesson 12 HW:
1.) Which of the following points lie on the graph of
[1]
[2]
[3]
[4]
2.) Which one of the following points does not lie on the graph
[1]
[2]
[3]
[4]
3.) Which of the following points would not lie on the line
[1]
[2]
[3]
[4]
4.) For the inequality
determine if each of the following points does or doesn’t lie in its solution.
Show the work that leads to your answer.
(a)
(b)
(c)
5.) Determine if the point
and
(d)
is a solution to the system of equations shown below. Justify your answer.
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6.) One of the following two points lies in the solution set of the system of inequalities below. Determine
which point it is and explain why your choice lies in the solution set and the other does not.
and
7.) James quickly sketched the graph of
and
His graph is shown. Explain how you know that his graph is inaccurate.
8.) The point
lies on the line
, for some value of .
(a) If
will the point
lie on the line?
(b) Find the value of
How can you tell?
will lie on the line.
for which the point
76
Review Section:
9.) Which value of x satisfies the equation
[1]8.25
[3] 8.89
[2]19.25
[4] 44.92
10.) A company that manufactures radios first pays a start-up cost, and then spends a certain amount of
money to manufacture each radio. If the cost of manufacturing radios is given by the function
, then the value 5.25 best represents:
[1] the start-up cost
[2] the profit earned from the sale of one radio
[3] the amount spent to manufacture each radio
[4] the average number of radios manufactured
11.) A ball is thrown into the air from the edge of a 48-foot-high
cliff so that it eventually lands on the ground. The graph below
shows the height, y, of the ball from the ground after x seconds.
For which interval is the ball’s height always decreasing?
[1]
[2]
[3]
[4]
12.) Given:
Which expression results in a rational number?
[1] L + M
[2] M + N
[3] N + P
[4] P + L
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Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Graphs of Linear Inequalities
4M
Lesson 13:
The Graphs of Linear Inequalities
So, we have graphed linear functions and in the last lesson learned that the points that lie on a graph are
simply the
pairs that make the equation true. Graphing an inequality in the xy-plane is the same.
Graphing Inequalities
To graph an inequality simply means to plot (or shade) all
pairs that make the inequality true.
Exercise #1: Consider the inequality
(a) Determine whether each of the following points lies in the solution set (and thus on the graph of)
the given inequality.
(b) Graph the line
on the grid below in dashed form. Why are the points that lie on this line not
part of the solution set of the inequality?
(c) Plot the three points from part (a) and use them
to help you share the proper region of the plane
that represents the solution set of the inequality.
(d) Choose a fourth point that lies in the region you
shaded and show that it is in the solution set of
the given inequality.
(e) The point
region. Is
cannot be drawn on the graph grid above, so it is difficult to tell if it falls in the shaded
part of the solution set of this inequality? Show how you arrive at your answer.
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There are some challenges to graphing linear inequalities, especially if the out, , has not been solved for. Let’s look at
the worst case scenario.
Exercise #2: Consider the inequality
(a) Rearrange the left-hand side of this inequality
using the commutative property of addition.
(b) Solve this inequality for
properties of inequality.
by applying the
(c) Shade the solution set of this inequality on the
graph paper below.
(d) Pick a point in the shaded region and show that
it is a solution to the original inequality.
The final type of inequality that we should be able to graph quickly and effectively is one that involves either
a horizontal line or a vertical line.
Exercise #3: Shade the solution set for each of the following inequalities in the
planes provided First, state in
your own words the
pairs that the inequality is describing.
(a)
(b)
Explain in your own words:
Explain in your own words:
79
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Graphs of Linear Inequalities
4M HW
Lesson 13 HW:
1.) Determine which of the following points lie in the solution set of the inequality
do not. Justify each choice.
(a)
(b)
(c)
and which
(d)
2.) Which of the following points lies in the solution set of the inequality
[1]
[2]
[3]
[4]
3.) Which of the following points does not lie in the solution set to the inequality
[1]
[2]
[3]
[4]
?
4.) Which of the following linear inequalities is shown graphed?
[1]
[2]
[3]
[4]
5.) Graph the solution set to the inequality shown. State one point that lies in the solution set and one point
that does not lie in the solution set.
One point in the solution:
One point not in the solution:
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6.) rearrange the inequality below so that it is easier to graph and then sketch its solution set on the grid
given. Be careful when dividing by a negative and remember to switch the inequality sign.
One point in the solution:
One point not in the solution:
7.) Graph the solution set to each of the following inequalities.
(a)
(b)
81
Review Section:
8.) Sam and Odel have been selling frozen pizzas for a class fundraiser. Sam has sold half as many pizzas as
Odel. Together they have sold a total of 126 pizzas. How many pizzas did Sam sell?
[1] 21
[2] 42
[3] 63
[4] 84
9.) The Jamison family kept a log of the distance they traveled
during a trip, as represented by the graph below. During which
interval was their average speed the greatest?
[1] the first hour to the second hour
[2] the second hour to the fourth hour
[3] the sixth hour to the eighth hour
[4] the eighth hour to the tenth hour
10.) The graph of
is shown.
Which point could be used to find
[1] A
[2] B
[3] C
[4] D
11.) The formula for the volume of a cone is
[1]
[2]
[3]
[4]
. The radius
of the cone may be expressed as:
82
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Introduction to Sequences
4N
Lesson 14:
Introduction to Sequences
A sequence is a very special type of function. When students first encounter sequences, they often think of
them as just a list of numbers in some particular order (and then they have to find the pattern). We will start
with the technical definition of a sequence in terms of a function.
Sequence Definition
A sequence is a function whose set of inputs, the domain, is a subset of the natural numbers, i.e.
A sequence is often shown as an ordered list of numbers, called the terms or elements of the sequence.
Sequence function notation can be tricky.
Exercise #1:Consider the sequence below. If we represent this sequence with the letter , then:
(a) Find
(b) Find
(c) Find
(d) Find
(e) Find
(f) Solve for :
Sequences are functions. The key here is that the input is simply the number’s place in line so to speak and
the output is the actual number in the list.
Exercise #2: Consider the sequence defined by the formula
(a) Write out the first five elements of this sequence.
(b) Graph the sequence on the grid shown below for
.
(c) Why shouldn’t we connect the points plotted with a
continuous straight line?
(d) What is the 21st term of this sequence? Show how you
arrived at your answer.
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Sequences can be defined by a classic function formula, like what we saw in Exercise #2, and they also can
be defined recursively. A recursive formula is one where each term in the sequence depends on a term
or terms that came before it.
Exercise #3: Consider a sequence of numbers given by the following definition:
(a) Give a common sense interpretation for this
recursive function rule.
(b) Write out the rule for the first 4 terms and
evaluate each one of them (except ).
One of the most famous of all recursively defined sequences is known as the Fibonacci Sequence. Let’s
play around with it in the next exercise.
Exercise #4: The Fibonacci Sequence is defined recursively as follows:
(a) How do you interpret this recursive rule?
Write it down in your own words.
(b) Write down the rule for
and determine their values.
and
Sequences often show up in the real world, where they are defined in terms of a recursive process.
Exercise #5: Kirk is trying to train for the marathon. His first month, he runs 5 miles per workout. He adds
an additional 3 miles to his workout for each month that he trains.
(a) Fill out the table below for the amount of miles
(c) Graph this sequence for
.
he runs as a function of how many months he has
been running.
(b) Give a recursive definition for the sequence
Don’t forget to give an initial value.
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Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Introduction to Sequences
4N HW
Lesson 14 HW:
1.) Consider the sequence below. If we represent this sequence with the letter , then do the following:
(a) Find
(b) Find
(c) Find
(d) Find
(e) Find
(f) Find a recursive definition for
the sequence
2.) Consider the sequence defined in the table below.
(a) Find
(b) Find
(c) Find a recursive definition for
the sequence
.
3.) Consider a sequence of numbers given by the definition
and
(a) Write out the first 4 terms of this sequence.
(b) Find the value of
Show calculations.
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4.) Erin is traveling abroad this summer and would like to have a bit of spending cash while she’s overseas.
She has 100 dollars already saved and she plans on saving 40 dollars a month.
(a) Fill out the table below for the amount of money
(c) Graph this sequence for
she saves as a function of how many months she
has been saving.
(b) Give a recursive definition for the sequence
. Don’t forget to give an initial value.
5.) A sequence is defined recursively as follows:
(a) How do you interpret this recursive rule?
Write it down in your own words.
and
(b) Write down the rule for
and determine their values.
and
6.) The diagrams below represent the first three terms of a sequence.
Assuming the pattern continues, which formula determines
the number of shaded squares in the th term?
[1]
[2]
[3]
[4]
7.)A sunflower is 3 inches tall at week 0 and grows 2 inches each week. Which function(s) shown below can
be used to determine the height,
of the sunflower in n weeks?
I.
II.
III.
where
[1] I and II
[2] III, only
[3] II, only
[4] I and III y
86
Review Section:
8.) Which statement is not always true?
[1] The product of two irrational numbers is irrational.
[2] The product of two rational numbers is rational.
[3] The sum of two rational numbers is rational.
[4] The sum of a rational number and an irrational number is irrational.
9.) What is one point that lies in the solution set of the system of inequalities graphed below?
[1] (7,0)
[2] (0,7)
[3] (3,0)
[4] (-3,5)
10.) The owner of a small computer repair business has one employee, computations. who is paid an hourly
rate of $22. The owner estimates his weekly profit using the function P(x) = 8600 - 22x. In this function, x
represents the number of
[1] computers repaired per week
[2] hours worked per week
[3] customers served per week
[4] days worked per week
87
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Arithmetic Sequences
4O
Lesson 15:
Arithmetic Sequences
There are many types of sequences, but there is one that is related to linear functions and in fact is a type of
discrete linear function. These are known as arithmetic sequences. Let’s illustrate one first.
Exercise #1: Evin is saving money for to buy a new toy. She already has $12 in her account. She gets an
allowance of $4 per week and plans to save $3 in her account.
(a) Fill out the table below for the amount of
(c) What’s wrong with the graph of the
money Evin has after n weeks of saving?
sequence shown below?
(b) Write a recursive definition for this sequence.
(d) Evin proposes the following explicit formula for
the amount of savings, a, as a function of the number
of weeks saved, n. Is the formula correct? Test it!
Arithmetic sequences are ones where the terms in the list increase or decrease by the same amount
given a unit increase in the index (where the number is in line).
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Exercise #2: An arithmetic sequence is given using the recursive definition:
Which of the following is the value of
Show the work that leads to your answer.
[1] 24
[2] 12
[3] 21
[4] 15
and
.
Arithmetic sequences are relatively easy to spot and are easy to fill in, so to speak.
Exercise #3: For each of the following sequences, determine if it is arithmetic based on the information
given. If it is arithmetic, fill in the missing blank. If it is not, show why.
(a) 5, 9, 13, ___________, 21, 25
(b) 5, 10, 20, 40, ___________, 160
(c) 7, 4, 1, ___________, 5,
8
(d) 64, 16, 4, ___________, ,
Finding a specific term in an arithmetic sequence without listing them sometimes can be a challenge, but not
if you take your time and really think about it.
Exercise #4: Consider an arithmetic sequence whose first three terms are given by: 4, 14, 24
(a) What is the 4th term? How many times was
(b) Use what you learned in part (a)
th
10 added to 4 to get to the 4 term? Show a
to find the value of
, 10th the term.
diagram to illustrate this.
(c) Write a recursive formula for the
on the term number n.
based
(d) Write an explicit formula for
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Exercise #5: Seats in a small amphitheater follow a pattern where each row has a set number of seats more
than the last row. If the first row has 6 seats and the fourth row has 18, how many seats does the last row,
which is the 20th, have in it? Show your work to justify your response.
90
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Arithmetic Sequences
4O HW
Lesson 15 HW:
and
Show the work that leads to your answer.
1.) An arithmetic sequence is given using the recursive definition:
following is the value of
[1] 14
[3] 6
[2] 2
[4] 4
. Which of the
2.) For each of the following sequences, determine if it is arithmetic based on the information given. If it is arithmetic,
fill in the missing blank. If it is not, show why.
(a) 12, 24, 36, _____ , 60, 72
(b) 10000, 1000, ________, 10, 1
(c) _____ , 24, 20, 16, 12, 8
(d)
3.) Given the sequence defined by the explicit formula
recursive definition and graph the sequence on the interval
______,
write out the first four terms. Then, create a
.
4.) Which of the following is an arithmetic sequence?
[1] 2, 4, 8, 16, 32, 64
[2] 50, 45, 40, 35, 30
[3] 1, 1, 2, 3, 5, 8, 13
[4] 1,
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5.) Mike is building a tower out of paper cups. In each row (counting from the floor up), there are two less cups than
the row below it. The first row has 26 cups in it.
(a) State the number of cups in the second, third,
(b) Give a recursive definition for this arithmetic
and fourth rows.
sequence.
(c) How many cups will be in the 11th row? Show the calculation that leads to your answer.
6.) Eric considers the sequence of numbers given by the following definition:
decides the first 4 numbers are:
4, 11, 18, 25
and
and
(a) Interpret in your own words, what the sequence is saying and what he actually did.
(b) What should the first four numbers be?
92
Review Section:
7.) Which domain would be the most appropriate set to use for a function that predicts the number of
household online-devices in terms of the number of people in the household?
[1] integers
[2] irrational numbers
[3] whole numbers [4] rational numbers
8.) Which function has the same
[1]
[2]
[3]
[4]
intercept as the graph below?
9.) Fred is given a rectangular piece of paper. If the length of Fred’s piece of paper is represented by
and the width is represented by
, then the paper has a total area represented by:
[1]
[2]
[3]
{4}
10.) The graph of a linear equation contains the points
on the graph?
[1]
[3]
and
. Which point also lies
[2]
[4]
93