I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Students learn the definition of slope and calculate the slope of lines.
Students also learn to consider the slopes of parallel lines and perpendicular lines.
Students learn the definition of slope and calculate the slope of lines.
Students also learn to consider the slopes of parallel lines and perpendicular lines.
Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
Students also learn to graph linear equations using x and y intercepts.
What Makes Candle Making The Ultimate Bachelorette CelebrationWick & Pour
The above-discussed factors are the reason behind an increasing number of millennials opting for candle making events to celebrate their bachelorette. If you are in search of any theme for your bachelorette then do opt for a candle making session to make your celebration memorable for everyone involved.
From Stress to Success How Oakland's Corporate Wellness Programs are Cultivat...Kitchen on Fire
Discover how Oakland's innovative corporate wellness initiatives are transforming workplace culture, nurturing the well-being of employees, and fostering a thriving environment. From comprehensive mental health support to flexible work arrangements and holistic wellness workshops, these programs are empowering individuals to navigate stress effectively, leading to increased productivity, satisfaction, and overall success.
Johnny Depp Long Hair: A Signature Look Through the Yearsgreendigital
Johnny Depp, synonymous with eclectic roles and unparalleled acting prowess. has also been a significant figure in fashion and style. Johnny Depp long hair is a distinctive trademark among the various elements that define his unique persona. This article delves into the evolution, impact. and cultural significance of Johnny Depp long hair. exploring how it has contributed to his iconic status.
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Introduction
Johnny Depp is an actor known for his chameleon-like ability to transform into a wide range of characters. from the eccentric Captain Jack Sparrow in "Pirates of the Caribbean" to the introspective Edward Scissorhands. His long hair is one constant throughout his evolving roles and public appearances. Johnny Depp long hair is not a style choice but a significant aspect of his identity. contributing to his allure and mystique. This article explores the journey and significance of Johnny Depp long hair. highlighting how it has become integral to his brand.
The Early Years: A Budding Star with Signature Locks
1980s: The Rise of a Young Heartthrob
Johnny Depp's journey in Hollywood began in the 1980s. with his breakout role in the television series "21 Jump Street." During this time, his hair was short, but it was already clear that Depp had a penchant for unique and edgy styles. By the decade's end, Depp started experimenting with longer hair. setting the stage for a lifelong signature.
1990s: From Heartthrob to Icon
The 1990s were transformative for Johnny Depp his career and personal style. Films like "Edward Scissorhands" (1990) and "Benny & Joon" (1993) saw Depp sporting various hair lengths and styles. But, his long, unkempt hair in "What's Eating Gilbert Grape" (1993) began to draw significant attention. This period marked the beginning of Johnny Depp long hair. which became a defining feature of his image.
The Iconic Roles: Hair as a Character Element
Edward Scissorhands (1990)
In "Edward Scissorhands," Johnny Depp's character had a wild and mane that complemented his ethereal and misunderstood persona. This role showcased how long hair Johnny Depp could enhance a character's depth and mystery.
Captain Jack Sparrow: The Pirate with Flowing Locks
One of Johnny Depp's iconic roles is Captain Jack Sparrow from the "Pirates of the Caribbean" series. Sparrow's long, dreadlocked hair symbolised his rebellious and unpredictable nature. The character's look, complete with beads and trinkets woven into his hair. was a collaboration between Depp and the film's costume designers. This style became iconic and influenced fashion trends and Halloween costumes worldwide.
Other Memorable Characters
Depp's long hair has also been featured in other roles, such as Ichabod Crane in "Sleepy Hollow" (1999). and Roux in "Chocolat" (2000). In these films, his hair added a layer of authenticity and depth to his characters. proving that Johnny Depp with long hair is more than a style—it's a storytelling tool.
Off-Screen Influenc
Is your favorite ring slipping and sliding on your finger? You're not alone. Must Read this Guide on What To Do If Your Ring Is Too Big as shared by the experts of Andrews Jewelers.
La transidentité, un sujet qui fractionne les FrançaisIpsos France
Ipsos, l’une des principales sociétés mondiales d’études de marché dévoile les résultats de son étude Ipsos Global Advisor “Pride 2024”. De ses débuts aux Etats-Unis et désormais dans de très nombreux pays, le mois de juin est traditionnellement consacré aux « Marches des Fiertés » et à des événements festifs autour du concept de Pride. A cette occasion, Ipsos a réalisé une enquête dans vingt-six pays dressant plusieurs constats. Les clivages des opinions entre générations s’accentuent tandis que le soutien à des mesures sociétales et d’inclusion en faveur des LGBT+ notamment transgenres continue de s’effriter.
La transidentité, un sujet qui fractionne les Français
Relations and Functions (Algebra 2)
1.
2. Relations and Functions
Analyze and graph relations.
Find functional values.
1) ordered pair 8) function
2) Cartesian Coordinate 9) mapping
3) plane 10) one-to-one function
4) quadrant 11) vertical line test
5) relation 12) independent variable
6) domain 13) dependent variable
7) range 14) functional notation
3. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
Cat 12 28
Cow 15 30
Deer 8 20
Dog 12 20
Horse 20 50
4. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
Cow 15 30
Deer 8 20
Dog 12 20
Horse 20 50
5. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
The ordered pairs for the data are:
Cow 15 30
Deer 8 20
Dog 12 20
Horse 20 50
6. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
The ordered pairs for the data are:
Cow 15 30
(12, 28), (15, 30), (8, 20),
(12, 20), and (20, 50) Deer 8 20
Dog 12 20
Horse 20 50
7. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
The ordered pairs for the data are:
Cow 15 30
(12, 28), (15, 30), (8, 20),
(12, 20), and (20, 50) Deer 8 20
The first number in each ordered pair Dog 12 20
is the average lifetime, and the second
number is the maximum lifetime. Horse 20 50
8. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
The ordered pairs for the data are:
Cow 15 30
(12, 28), (15, 30), (8, 20),
(12, 20), and (20, 50) Deer 8 20
The first number in each ordered pair Dog 12 20
is the average lifetime, and the second
number is the maximum lifetime. Horse 20 50
(20, 50)
9. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
The ordered pairs for the data are:
Cow 15 30
(12, 28), (15, 30), (8, 20),
(12, 20), and (20, 50) Deer 8 20
The first number in each ordered pair Dog 12 20
is the average lifetime, and the second
number is the maximum lifetime. Horse 20 50
(20, 50)
average
lifetime
10. Relations and Functions
This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28
The ordered pairs for the data are:
Cow 15 30
(12, 28), (15, 30), (8, 20),
(12, 20), and (20, 50) Deer 8 20
The first number in each ordered pair Dog 12 20
is the average lifetime, and the second
number is the maximum lifetime. Horse 20 50
(20, 50)
average maximum
lifetime lifetime
11. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
60
50
Maximum Lifetime
40
30
20
10
0 x
0 5 10 15 20 25 30
Average Lifetime
12. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), 60
50
Maximum Lifetime
40
30
20
10
0 x
0 5 10 15 20 25 30
Average Lifetime
13. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), (15, 30), 60
50
Maximum Lifetime
40
30
20
10
0 x
0 5 10 15 20 25 30
Average Lifetime
14. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), (15, 30), (8, 20), 60
50
Maximum Lifetime
40
30
20
10
0 x
0 5 10 15 20 25 30
Average Lifetime
15. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), (15, 30), (8, 20), 60
(12, 20), 50
Maximum Lifetime
40
30
20
10
0 x
0 5 10 15 20 25 30
Average Lifetime
16. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), (15, 30), (8, 20), 60
(12, 20), and (20, 50) 50
Maximum Lifetime
40
30
20
10
0 x
0 5 10 15 20 25 30
Average Lifetime
17. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), (15, 30), (8, 20), 60
(12, 20), and (20, 50) 50
Maximum Lifetime
40
Remember, each point in the coordinate
plane can be named by exactly one 30
ordered pair and that every ordered pair
names exactly one point in the coordinate 20
plane.
10
0 x
0 5 10 15 20 25 30
Average Lifetime
18. Relations and Functions
You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
(12, 28), (15, 30), (8, 20), 60
(12, 20), and (20, 50) 50
Maximum Lifetime
40
Remember, each point in the coordinate
plane can be named by exactly one 30
ordered pair and that every ordered pair
names exactly one point in the coordinate 20
plane.
10
The graph of this data (animal lifetimes) 0 x
lies in only one part of the Cartesian 0 5 10 15 20 25 30
coordinate plane – the part with all Average Lifetime
positive numbers.
20. Relations and Functions
The Cartesian coordinate system is composed of the x-axis (horizontal),
and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane into
four quadrants.
5
Origin
(0, 0)
0
-5 0 5
-5
21. Relations and Functions
The Cartesian coordinate system is composed of the x-axis (horizontal),
and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane into
four quadrants.
You can tell which quadrant a point is in by looking at the sign of each coordinate of
the point.
5
Quadrant II Origin
Quadrant I
( --, + ) ( +, (0, )0)
+
0
-5 0 5
Quadrant III Quadrant IV
( --, -- ) ( +, -- )
-5
22. Relations and Functions
The Cartesian coordinate system is composed of the x-axis (horizontal),
and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane into
four quadrants.
You can tell which quadrant a point is in by looking at the sign of each coordinate of
the point.
5
Quadrant II Origin
Quadrant I
( --, + ) ( +, (0, )0)
+
0
-5 0 5
Quadrant III Quadrant IV
( --, -- ) ( +, -- )
-5
The points on the two axes do not lie in any quadrant.
23. Relations and Functions
In general, any ordered pair in the coordinate plane can be written in the form (x, y)
24. Relations and Functions
In general, any ordered pair in the coordinate plane can be written in the form (x, y)
A relation is a set of ordered pairs, such as the one for the longevity of animals.
25. Relations and Functions
In general, any ordered pair in the coordinate plane can be written in the form (x, y)
A relation is a set of ordered pairs, such as the one for the longevity of animals.
The domain of a relation is the set of all first coordinates (x-coordinates) from the
ordered pairs.
26. Relations and Functions
In general, any ordered pair in the coordinate plane can be written in the form (x, y)
A relation is a set of ordered pairs, such as the one for the longevity of animals.
The domain of a relation is the set of all first coordinates (x-coordinates) from the
ordered pairs.
The range of a relation is the set of all second coordinates (y-coordinates) from the
ordered pairs.
27. Relations and Functions
In general, any ordered pair in the coordinate plane can be written in the form (x, y)
A relation is a set of ordered pairs, such as the one for the longevity of animals.
The domain of a relation is the set of all first coordinates (x-coordinates) from the
ordered pairs.
The range of a relation is the set of all second coordinates (y-coordinates) from the
ordered pairs.
The graph of a relation is the set of points in the coordinate plane corresponding to the
ordered pairs in the relation.
28. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
29. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
30. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
31. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
32. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
Domain
-3
0
2
33. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
Domain Range
-3 1
0 2
2 4
34. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
Domain Range
-3 1
0 2
2 4
35. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
Domain Range
-3 1
0 2
2 4
36. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
Domain Range
-3 1
0 2
2 4
37. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
3,1 , 0,2 , 2,4
Domain Range
-3 1
0 2
2 4
one-to-one function
38. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
1,5 , 1,3 , 4,5
39. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
1,5 , 1,3 , 4,5
Domain Range
-1
5
1
3
4
40. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
1,5 , 1,3 , 4,5
Domain Range
-1
5
1
3
4
41. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
1,5 , 1,3 , 4,5
Domain Range
-1
5
1
3
4
42. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
1,5 , 1,3 , 4,5
Domain Range
-1
5
1
3
4
43. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
1,5 , 1,3 , 4,5
Domain Range
-1
5
1
3
4
function,
not one-to-one
44. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6 , 3,0 , 1,1 , 3,6
45. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6 , 3,0 , 1,1 , 3,6
Domain Range
5 6
-3 0
1 1
46. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6 , 3,0 , 1,1 , 3,6
Domain Range
5 6
-3 0
1 1
47. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6 , 3,0 , 1,1 , 3,6
Domain Range
5 6
-3 0
1 1
48. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6 , 3,0 , 1,1 , 3,6
Domain Range
5 6
-3 0
1 1
not a function
49. Relations and Functions
A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.
exactly one
A mapping shows how each member of the domain is paired with each member in
the range.
Functions
5,6 , 3,0 , 1,1 , 3,6
Domain Range
5 6
-3 0
1 1
not a function
50. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
x
(-1,-2) (3,-3)
(0,-4)
51. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
x
(-1,-2) (3,-3)
(0,-4)
52. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x
(-1,-2) (3,-3)
(0,-4)
53. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x
{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
(0,-4)
54. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x
{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
The range is:
(0,-4)
55. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x
{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
The range is:
(0,-4)
{ -4, -3, -2, 3 }
56. Relations and Functions
State the domain and range of the relation shown y
in the graph. Is the relation a function?
(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x
{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
The range is:
(0,-4)
{ -4, -3, -2, 3 }
Each member of the domain is paired with exactly one member of the range,
so this relation is a function.
58. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
59. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
60. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
61. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
62. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
63. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
64. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
65. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.
y
x
66. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a If some vertical line intercepts a
graph in more than one point, graph in two or more points, the
the graph represents a function. graph does not represent a function.
y y
x x
67. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a If some vertical line intercepts a
graph in more than one point, graph in two or more points, the
the graph represents a function. graph does not represent a function.
y y
x x
68. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a If some vertical line intercepts a
graph in more than one point, graph in two or more points, the
the graph represents a function. graph does not represent a function.
y y
x x
69. Relations and Functions
You can use the vertical line test to determine whether a relation is a function.
Vertical Line Test
If no vertical line intersects a If some vertical line intercepts a
graph in more than one point, graph in two or more points, the
the graph represents a function. graph does not represent a function.
y y
x x
70. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
1960 4.7
1970 5.2
1980 5.5
1990 5.5
2000 6.1
71. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
1980 5.5
1990 5.5
2000 6.1
72. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Population
5
(millions)
2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
73. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
74. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
75. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
76. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
77. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
78. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
79. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
80. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
81. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
2
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
82. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
Notice that no vertical line can be drawn that
2
contains more than one of the data points.
1
0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
83. Relations and Functions
The table shows the population of Indiana over the last several Population
Year
decades. (millions)
1950 3.9
We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2
Population of Indiana
8 1980 5.5
7
6
1990 5.5
Use the vertical
Population
5
(millions)
line test. 2000 6.1
4
3
Notice that no vertical line can be drawn that
2
contains more than one of the data points.
1
0 Therefore, this relation is a function!
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year
87. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values.
x y
-1 -1
0 1
1 3
2 5
88. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
89. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range.
90. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range.
Domain is all real numbers.
91. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range.
Domain is all real numbers.
Range is all real numbers.
92. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers.
Range is all real numbers.
93. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers.
Range is all real numbers.
94. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers.
Range is all real numbers.
95. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers.
Range is all real numbers.
96. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers.
Range is all real numbers.
97. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers. The graph passes the vertical line test.
Range is all real numbers.
98. Relations and Functions
Graph the relation y 2 x 1 2) Graph the ordered pairs.
1) Make a table of values. y
7
6
x y 5
4
-1 -1
3
2
0 1
1
0 x
1 3
-1
2 5 -2
-3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. 4) Determine whether the relation is a function.
Domain is all real numbers. The graph passes the vertical line test.
Range is all real numbers.
For every x value there is exactly one y value,
so the equation y = 2x + 1 represents a function.
102. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
1) Make a table of values.
x y
2 -2
-1 -1
-2 0
-1 1
2 2
103. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2
104. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2
3) Find the domain and range.
105. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
106. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
107. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2 4) Determine whether the relation is a function.
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
108. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2 4) Determine whether the relation is a function.
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
109. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2 4) Determine whether the relation is a function.
3) Find the domain and range.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
110. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2 4) Determine whether the relation is a function.
3) Find the domain and range. The graph does not pass the vertical line test.
Domain is all real numbers,
greater than or equal to -2.
Range is all real numbers.
111. Relations and Functions
Graph the relation x y 2 2 2) Graph the ordered pairs.
y
1) Make a table of values. 7
6
5
x y
4
2 -2 3
2
-1 -1 1
0 x
-2 0 -1
-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2 4) Determine whether the relation is a function.
3) Find the domain and range. The graph does not pass the vertical line test.
Domain is all real numbers, For every x value (except x = -2),
greater than or equal to -2. there are TWO y values,
Range is all real numbers. so the equation x = y2 – 2
DOES NOT represent a function.
112. Relations and Functions
When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.
113. Relations and Functions
When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.
The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.
114. Relations and Functions
When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.
The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.
Equations that represent functions are often written in function notation.
115. Relations and Functions
When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.
The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.
Equations that represent functions are often written in function notation.
The equation y = 2x + 1 can be written as f(x) = 2x + 1.
116. Relations and Functions
When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.
The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.
Equations that represent functions are often written in function notation.
The equation y = 2x + 1 can be written as f(x) = 2x + 1.
y
The symbol f(x) replaces the __ , and is read “f of x”
117. Relations and Functions
When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.
The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.
Equations that represent functions are often written in function notation.
The equation y = 2x + 1 can be written as f(x) = 2x + 1.
y
The symbol f(x) replaces the __ , and is read “f of x”
The f is just the name of the function. It is NOT a variable that is multiplied by x.
118. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
119. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
120. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
121. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
Therefore, if f(x) = 2x + 1
122. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
Therefore, if f(x) = 2x + 1
Then f(4) = 2(4) + 1
123. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
Therefore, if f(x) = 2x + 1
Then f(4) = 2(4) + 1
f(4) = 8 + 1
124. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
Therefore, if f(x) = 2x + 1
Then f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9
125. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
Therefore, if f(x) = 2x + 1
Then f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9
NOTE: Letters other than f can be used to represent a function.
126. Relations and Functions
Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1
This is written as f(4) and is read “f of 4.”
The value f(4) is found by substituting 4 for each x in the equation.
Therefore, if f(x) = 2x + 1
Then f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9
NOTE: Letters other than f can be used to represent a function.
EXAMPLE: g(x) = 2x + 1