lesson 1-1 relations and functions - mr. long's mathjefflongnuames.weebly.com › uploads › 5 ›...
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Math-3
Lesson 1-1
Relations and Functions
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Vocabulary
Relation: A “mapping” or pairing of input values to output values.
Function: A relation where each input has exactly one output.
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Your turn:
Describe how a relation is
1) Similar to
1) Different from
a function?
Both have inputs matched to outputs.
One input to a relation can be matched with
two or more outputs.
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Is it a function or a relation?
input output
2
3
-4
-5
2
3
4
5
input output
2
3
-4
-5
2
3
4
5
Something to think about:
in math OR doesn’t mean
one or the other. It means
one or the other or both!!
Both
Both
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Is it a function or a relation?
input output
2
3
-4
-5
2
3
4
5
input output
2
3
-4
-5
2
3
4
5
Something to think about:
in math OR doesn’t mean
one or the other. It means
one or the other or both!!
Both
Relation only
The input value “2” has
two outputs (“2” and “5”)
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Is it a function or a relation?
input output
2
3
-4
-5
2
3
4
5
input output
2
3
-4
-5
2
3
4
5
(There aren’t any pairings.)
Neither
Both
Each input has exactly one
output (even though it’s
the same output for each).
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Which is a “mapping”?
http://www.flickr.com/photos/sanchome/525890022/
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Name 6 ways to show a relation between inputand output values.
Ordered Pairs: (2, 4), (3, 2), (-4, 3), etc.
Data table: x 2 3 -4y 4 2 3
Graph:
Equation: y = 2x + 1
Mappinginput output
2
3
-4
2
3
4
Function notation: f(2) = 4
Note: not all of the relations above are the same.
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Your turn:What are 6 ways you can show a relation
between input and output ?
There are actually more ways to show a relation. Next time we’ll talk about sequences.
Does anyone remember the two different ways sequences can be defined?
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Domain: the set made up of the input values.
Range: the set made up of the output values.
This is too simplistic!!!
Domain: the set made up of the input values for
which there is a corresponding output value.
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Identify the Domain
1. (2, 4), (3, 5), (-4, 2)
2. x 6 9 -2y 4 7 3
3.4.
input output
2
3
-4
-5
2
3
4
-3 1 3
2
3
-4
-5
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Determining if the relation is linear.
What are 6 ways you can show a relationbetween input and output ?
Graph:
The slope needs to be constant.
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Data table: Is the data linear?
x f(x)-2 -70 -52 -34 -16 18 3
10 512 715 9
1st “difference”
2 x
in change
2 x
In engineering we often
refer to the change (or
difference) as the “delta”,
(using the Greek letter)
2 x
2 x
2 x
2y
2y
2y
2y
2y
If the 1st difference for
both input and output
(“x” and “y”) is always
the same then the
relation is linear.
Why is that?3 x
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Your turn: Which data set is linear?
x f(x)-4 -7-3 -5-2 -3-1 -10 11 32 53 74 9
x g(x)-4 32-3 18-2 8-1 20 01 22 83 184 32
x f(x)0 01 12 1.43 1.74 2.05 2.26 2.47 2.68 2.89 3
A B C
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Your turn: Is the data linear? If so, what is the equation that “fits” the data?
x f(x)-4 -7-3 -5-2 -3-1 -10 11 32 53 74 9
bmxy
What is this number?
The value of ‘y’ when x = 0.
bmy )0(
by
),0( b
1 mxy
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Your turn: What is the equation that fits the linear data?
x f(x)-4 -7-3 -5-2 -3-1 -10 11 32 53 74 9
1 mxy
What is this number?
Slope:
12 xy
2 y
in change
x''in change
y''in changem
1 x
in change
21
2m
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Vocabulary:
Parent function: The most basic function in
a family of functions.
For lines: y = x is the “parent function”
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Graph the simplest line of all.
y = x
1
1
Your turn:
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Your turn: What is the equation that fits the data?
x f(x)
-4 -9
-2 -6
0 -3
2 0
4 3
6 6
8 9
10 12
12 15
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Your turn: What is the difference between the two representations?
x f(x)
-4 -7
-3 -5
-2 -3
-1 -1
0 1
1 3
2 5
3 7
4 9
1
1
3
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What is the difference between the three representations?
x f(x)
-2 -3
-1 -1
0 1
1 3
1 1
3
1
3
1
Discrete Discrete Continuous
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What is the domain of each?
x f(x)
-2 -3
-1 -1
0 1
1 3
1 1
3
1
3
1
D = {x = -2, -1, 0, 1}Discrete Continuous
D = {-2 ≤ x ≤ 1}
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Linear Relationships
Does the grade a person earns vary linearly with the number of hours he/she studies?
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Linear Relationships
Does the amount of natural gas used by a family vary linearly with the outside temperature?
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Does the height of person relate linearly to his/her weight?
Using this data, how could we predict the weight of a person who is 7 feet tall?
bmxy
)75 ,35(),( yx
2535-60 x
55
75130
y
Slope: 2.225
55
x
y
m
bxy 2.2
b )35(2.275
2)"84(2.2 y
)"84'7(
b )35(2.2752b
lb. 189y
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Linear RelationshipsIs height of a falling object linear with time?
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Linear “Correlation” between the quantities being measured
“Positive” refers to what in the upper left graph?
“Negative” refers to what in the lower left graph?
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Pure Math: Equations, graphs, tables of numbers, ordered pairs, mappings, and proofs that are just math and are not being used to relate to the physical world around us
Applied Math: The use of equations, graphs, tables of numbers, ordered pairs, and mappings that are used to model relationships between quantities in the real world.
Engineering: The use of applied math and science to design machines and tools for use in the real world.
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Graphing Points on your calculator
Push the “stat” button
There are 3 “pull
down” menu’s.
You are in the
“edit” menu.
Push “enter”
You will enter the
x-values into “L1”
(list 1) and y-values
into “L2”.
You have to
clear L1
Move cursor until L1
heading is highlighted.
Push “clear”
Notice that only the
bottom box was cleared.
Push “enter”
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Your are now ready to enter values into the two lists.
With the cursor
in L1, type “-4”
Move the cursor to L2
x f(x)
-4 -7
-3 -5
-2 -3
-1 -1
0 1
1 3
2 5
3 7
4 9
Push “enter”
Enter the rest of the
x-values into L1.
Enter the y-value into L2
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Turn on the ability to graph points
Push “y = “Move the cursor to
“Plot1” then hit
enter.
Clear the equation.
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The minimum x-value that
will be graphed is -2.5.Check to see if the window
is “big” enough.
x f(x)
-4 -7
-3 -5
-2 -3
-1 -1
0 1
1 3
2 5
3 7
4 9
Push the “zoom” button
Push “6”
Will all the
points show in
the window?
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Push “graph”
x f(x)
-4 -7
-3 -5
-2 -3
-1 -1
0 1
1 3
2 5
3 7
4 9
22xy
12 xy
xy
Which equation is it?
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Your turn: Enter the data into L1 and L2
x f(x)
-4 -9
-2 -6
0 -3
2 0
4 3
6 6
8 9
10 12
12 15
Your turn: Plot the data on your calculator.
Your turn: Is the data linear?