Relations and Functions (Algebra 2)

Slide 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

Slide 3: Relations and Functions Average This table shows the average lifetime 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

Slide 4: Relations and Functions Average This table shows the average lifetime 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

Slide 5: Relations and Functions Average This table shows the average lifetime 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

Slide 6: Relations and Functions Average This table shows the average lifetime 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), Deer 8 20 (12, 20), and (20, 50) Dog 12 20 Horse 20 50

Slide 7: Relations and Functions Average This table shows the average lifetime 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), Deer 8 20 (12, 20), and (20, 50) Dog 12 20 The first number in each ordered pair is the average lifetime, and the second number is the maximum lifetime. Horse 20 50

Slide 8: Relations and Functions Average This table shows the average lifetime 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), Deer 8 20 (12, 20), and (20, 50) Dog 12 20 The first number in each ordered pair is the average lifetime, and the second number is the maximum lifetime. Horse 20 50 (20, 50)

Slide 9: Relations and Functions Average This table shows the average lifetime 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), Deer 8 20 (12, 20), and (20, 50) Dog 12 20 The first number in each ordered pair is the average lifetime, and the second number is the maximum lifetime. Horse 20 50 (20, 50) average lifetime

Slide 10: Relations and Functions Average This table shows the average lifetime 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), Deer 8 20 (12, 20), and (20, 50) Dog 12 20 The first number in each ordered pair is the average lifetime, and the second number is the maximum lifetime. Horse 20 50 (20, 50) average maximum lifetime lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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 x 0 0 5 10 15 20 25 30 Average Lifetime

Slide 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) x 0 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.

Slide 19: Relations and Functions The Cartesian coordinate system is composed of the x-axis (horizontal), -5 0 5

Slide 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

Slide 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 Origin Quadrant II Quadrant I ( --, + ) ( +, (0, )0) + 0 -5 0 5 Quadrant III Quadrant IV ( --, -- ) ( +, -- ) -5

Slide 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 Origin Quadrant II 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.

Slide 23: Relations and Functions In general, any ordered pair in the coordinate plane can be written in the form (x, y)

Slide 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.

Slide 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.

Slide 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.

Slide 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.

Slide 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.

Slide 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

Slide 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.

Slide 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

Slide 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

Slide 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 Range Domain -3 1 0 2 2 4

Slide 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 Range Domain -3 1 0 2 2 4 one-to-one function

Slide 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

Slide 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 Range Domain -1 5 1 3 4

Slide 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 Range Domain -1 5 1 3 4 function, not one-to-one

Slide 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

Slide 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 Range Domain 5 6 -3 0 1 1

Slide 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 Range Domain 5 6 -3 0 1 1 not a function

Slide 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 Range Domain 5 6 -3 0 1 1 not a function

Slide 50: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) x (-1,-2) (3,-3) (0,-4)

Slide 51: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) } x (-1,-2) (3,-3) (0,-4)

Slide 52: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) } The domain is: x (-1,-2) (3,-3) (0,-4)

Slide 53: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) 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)

Slide 54: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) 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)

Slide 55: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) 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 }

Slide 56: Relations and Functions State the domain and range of the relation shown y in the graph. Is the relation a function? (2,3) (-4,3) 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.

Slide 57: Relations and Functions You can use the vertical line test to determine whether a relation is a function.

Slide 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

Slide 66: Relations and Functions You can use the vertical line test to determine whether a relation is a function. Vertical Line Test If some vertical line intercepts a If no vertical line intersects a graph in two or more points, the graph in more than one point, graph does not represent a function. the graph represents a function. y y x x

Slide 70: Relations and Functions Population The table shows the population of Indiana over the last several Year (millions) decades. 1950 3.9 1960 4.7 1970 5.2 1980 5.5 1990 5.5 2000 6.1

Slide 71: Relations and Functions Population The table shows the population of Indiana over the last several Year (millions) decades. 1950 3.9 1960 4.7 We can graph this data to determine if it represents a function. 1970 5.2 1980 5.5 1990 5.5 2000 6.1

Slide 72: Relations and Functions Population The table shows the population of Indiana over the last several Year (millions) decades. 1950 3.9 1960 4.7 We can graph this data to determine if it represents a function. 1970 5.2 Population of Indiana 1980 5.5 8 7 1990 5.5 6 Population 5 (millions) 2000 6.1 4 3 2 1 0 7 ‘70 ‘80 ‘50 ‘60 ‘90 ‘00 0 Year

Slide 73: Relations and Functions Population The table shows the population of Indiana over the last several Year (millions) decades. 1950 3.9 1960 4.7 We can graph this data to determine if it represents a function. 1970 5.2 Population of Indiana 1980 5.5 8 7 1990 5.5 6 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 7 ‘70 ‘80 ‘50 ‘60 ‘90 ‘00 0 Year

Slide 82: Relations and Functions Population The table shows the population of Indiana over the last several Year (millions) decades. 1950 3.9 1960 4.7 We can graph this data to determine if it represents a function. 1970 5.2 Population of Indiana 1980 5.5 8 7

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