Download Representing Proportional Relationships

IDENTIFYING & REPRESENTING
FUNCTIONS
Essential Question?
How can you identify & represent
functions?
8.F.1
COMMON CORE STANDARD:
8.F.1 ─ Define, evaluate, and compare functions.
Understand that a function is a rule that assigns to each input
exactly one output. The graph of a function is the set of ordered
pairs consisting of an input and the corresponding output
OBJECTIVES:
• Understand that a function is a rule that assigns to each
input exactly one output.
• Identify whether a relationship is a function from a
diagram, table of values, graph, or equation.
Curriculum Vocabulary
Function (función):
A relationship between an independent variable,
x, and a dependent variable, y, where each value
of x (input) has one and only one value of y
(output).
Relation (relación):
Any set of ordered pairs.
Input (entrada):
A number or value that is entered.
Output (salida):
The number or value that comes out from a
process.
Curriculum Vocabulary
Domain (dominio):
The set of all possible input (x) values.
Range (rango):
The set of all output (y) values.
Continuous graph (gráfica continua):
A graph of points that are connected by a line or
smooth curve on the graph. There are no breaks.
Discrete graph (gráfica discreta):
A graph of isolated points.
Curriculum Vocabulary
Linear Graph (gráfico lineal):
A graph that is a line or a series of collinear
points.
Collinear Points (puntos colineales):
Points that lie in the same straight line.
Non-linear graph (gráfico no lineal):
A graph that is not a line and therefore not a series
of collinear points.
Set (grupo):
A collection of numbers, geometric figures, letters, or
other objects that have some characteristic in
common.
REPRESENTING FUNCTIONS
There are 4 (FOUR) ways to represent a function that we will explore:
1. TABLE
2. MAPPING DIAGRAM
3. EQUATION
4. GRAPH
FUNCTIONS
The diagram below shows the function “add 2.”
Input =
3
Function: Add 2
Output =
5
There is only one possible output for each input.
The function “add 2” is expressed in words.
It can also be:
•
•
•
•
written as the equation y=x+2
represented by a table of values
represented as a mapping diagram
shown as a graph.
IDENTIFYING FUNCTIONS
Look at the following table:
INPUT
OUTPUT
5
11
10
21
15
31
20
41
25
51
For EACH INPUT THERE IS
EXACTLY ONE OUTPUT.
You can notice that there is
NO REPETITION in the
INPUT column.
This table represents a function.
IDENTIFYING FUNCTIONS
Look at the following table:
INPUT
OUTPUT
3
9
3
10
5
25
5
26
7
49
For EACH INPUT THERE IS
MORE THAN ONE OUTPUT.
You can notice that there is
REPETITION in the INPUT
column.
This table DOES NOT represent a function.
IDENTIFYING FUNCTIONS
Let’s examine the following situation:
Carlos wants to buy some apps for his smartphone. Zynga
is offering a game app special. 2 apps will cost him $2.58.
5 apps will cost him $6.45. Help Carlos complete the table.
Number of Apps
Rule
Total Cost
2
5
1
x
There is a ONE to ONE relationship!
This represents a FUNCTION!
IDENTIFYING FUNCTIONS
Let’s use the data we found to create a MAPPING DIAGRAM.
Number of Apps (x)
Total Cost in $ (y)
2
2.58
5
6.45
1
1.29
x
1.29x
Input: Number of Apps
Output: Total Cost in $
1
1.29
2
2.58
5
6.45
There is a ONE to ONE relationship!
This represents a FUNCTION!
IDENTIFYING FUNCTIONS
Does the following mapping diagram represent a function?
5
7
-3
2
10
9
-15
4
15
11
-21
6
20
45
-121
8
25
IDENTIFYING FUNCTIONS
Does the following mapping diagram represent a function?
1
10
2
20
3
11
5
30
4
40
5
50
IDENTIFYING FUNCTIONS
The third way we can represent a function is by writing an
EQUATION.
In the eighth grade,
recognizing if an EQUATION is a FUNCTION is super easy.
If you can get y all alone on one side of the equal sign,
it is a function!
Examples:
𝒚 = 𝟏𝟐𝒙 + 𝟓
𝒚 = 𝟑𝒙𝟐 − 𝒙 + 𝟗
11𝒚 − 𝟒𝒙 =7
IDENTIFYING FUNCTIONS
So far we have seen a function represented as:
• A TABLE
• A MAPPING DIAGRAM
• An EQUATION
Input
1
2
3
Output
3
6
9
4
12
5
15
1
3
2
6
3
9
4
12
5
15
𝑦 = 3𝑥
IDENTIFYING FUNCTIONS
The fourth way to represent a FUNCTION is as A GRAPH:
For A GRAPH to
represent a
FUNCTION, it must
pass the VERTICAL
LINE TEST.
Pass a vertical line
over the entire graph.
If at any time it
touches more than
one point at the same,
it is
IDENTIFYING FUNCTIONS
Is this a function? Continuous or discreet?
IDENTIFYING FUNCTIONS
Is this a function? Continuous or discreet?
IDENTIFYING FUNCTIONS
Is this a function? Continuous or discreet?
IDENTIFYING FUNCTIONS
Is this a function? Continuous or discreet?
IDENTIFYING FUNCTIONS
Is this a function? Continuous or discreet?
IDENTIFYING FUNCTIONS
We have now seen a FUNCTIONS represented as:
• A TABLE
•
No repetition in the input (x-values)
• A MAPPING DIAGRAM
•
Shows a ONE to ONE relationship
• An EQUATION
•
You can get y all alone on one side of the equal sign
• A GRAPH
•
Passes the vertical line test
Input
Output
1
3
1
3
2
6
2
6
3
9
3
9
4
12
4
12
5
15
5
15
𝑦 = 3𝑥
FUNCTIONS
For EACH INPUT
THERE IS EXACTLY
(ONE AND ONLY)
ONE OUTPUT.
IDENTIFYING DOMAIN & RANGE
From a TABLE:
INPUT
OUTPUT
5
11
10
21
15
31
20
41
25
51
DOMAIN: list the x values
{5, 10, 15, 20, 25}
RANGE: list the y values
{11, 21, 31, 41, 51}
IDENTIFYING DOMAIN & RANGE
Identify the domain and range:
INPUT
OUTPUT
2
7
3
7
4
7
5
7
6
7
DOMAIN: list the x values
{2, 3, 4, 5, 6}
RANGE: list the y values
{7}
IDENTIFYING DOMAIN & RANGE
From a GRAPH:
First IDENTIFY all the ORDERED PAIRS
(-10, 4), (-5, 4), (-4, -6), (-3, 8),
(3, 2), (3, -3), (6, 9), (8, 3), (8, -5)
DOMAIN: list the x values
{-10, -5, -4, -3, 3, 6, 8}
RANGE: list the y values
{4, -6, 8, 2, -3, 9, 3, -5}
{-6, -5, -3, 2, 3, 4, 8, 9}
Is the graph continuous or discreet?
IDENTIFYING DOMAIN & RANGE
Identify the domain and range:
Is the graph continuous or discreet?
FOR ACCELERATE CLASSES ONLY

Objective:
 To
identify dependent & independent quantities
 To identify the domain and range of a function
 To recognize, evaluate, and express functions using
function notation.
Curriculum Vocabulary
Dependent quantity (cantidad de dependientes):
When one quantity depends no another in a
problem situation, it is said to be the dependent
quantity.
Independent quantity (cantidad independiente):
The quantity that the dependent quantity
depends upon is called the independent
quantity.
Circle the independent quantity and underline the dependent quantity in each
statement:
• the number of hours worked and the money earned.
• your grade on a test and the number of hours you studied.
• the number of people working on a particular job and the time it takes to
complete a job.
• the number of games played and the number of points scored.
• the speed of a car and how far the driver pushes down on the gas pedal.
IDENTIFYING DOMAIN & RANGE
From a GRAPH:
Is the graph continuous or discreet?
Continuous Functions
Most CONTINUOUS
FUNCTIONS are shown on the
graph by using arrows.
It means that the x-values of the
function continue off to infinity in
both directions.
DOMAIN:
The domain of a continuous function
with arrows in both directions will always
be ALL REAL NUMBERS.
RANGE:
The range of a continuous function will
vary. In this case the y-values also go
off to infinity, so the domain is also ALL
REAL NUMBERS.
IDENTIFYING DOMAIN & RANGE
From a GRAPH:
Continuous Functions
DOMAIN:
ALL REAL NUMBERS.
RANGE:
In this function what is the y-value?
You can see that the y-value will
ALWAYS be 6.
{6}
IDENTIFYING DOMAIN & RANGE
From a GRAPH:
Non-functions
DOMAIN:
In this graph the x-value will always
be -2.
{-2}
RANGE:
Here the y-values go off to infinity in
both directions.
ALL REAL NUMBERS
IDENTIFYING DOMAIN & RANGE
From a GRAPH:
Non-functions
DOMAIN:
For geometric shapes and line
segments, you need to determine
where the x-values are trapped.
In this example, the triangle is trapped
between -5 and 9.
-5≤x≤9
RANGE:
For geometric shapes and line
segments, you need to determine
where the y-values are trapped.
In this example, the triangle is trapped
between 1 and 8.
1≤y≤8
IDENTIFYING DOMAIN & RANGE
From a GRAPH:
What about this one?
DOMAIN:
Since it is a continuous function the
domain is:
ALL REAL NUMBERS.
RANGE:
This functions lowest value is -6.
Then it goes up from there to positive
infinity.
y≥-6
FUNCTION FORM

FUNCTION FORM means to get y all
alone on one side of the equal sign.
y  stuff
FUNCTION NOTATION
Function Notation is another name for
the letter y.
 The same way a person whose name is
José, might be known as Pepe, function
notation is another name for the same
thing.
 Function notation looks like this:

y  f (x)

We say y equals f of x. This does NOT
mean multiply!
FUNCTION NOTATION
y  f (x)
In function form, the variable that is all
alone, y, is the DEPENDENT VARIABLE.
The variable in the parenthesis, x, is the
INDEPENDENT VARIABLE.
The value of y depends on what you plug
in for x.
Function Notation
y  2x  3
f (x)  2x  3
when x  1, y  5
f (1)  5
when x  2, y  7
f (2)  7
when x  3, y  9
f (3)  9
when x  4, y  11
f (4)  11
f ( 4)  5
g(x)  x
2
h(x)  3x  2
Evaluate the following.
1) g(4)  16
5) h(4)  g(1) 
2) h( 2)  8
6) h( 5)  g( 2) 
3) g( 3)  9
4) h(5)  13
10  1  11
17  4  68
7) g  h(3)  
g(7)  49
8) h  g(2)  
h(4)  10
Evaluate the function over the domain,
x = -1, x = 0, x = 2.
1) f (x)  4x
{4, 0, 8 }
2) g(x)  3x  9
{12,  9,  3 }
3) h(x)  x  1
2
{ 0,  1, 3 }