Slide 1 / 68 - NJCTLcontent.njctl.org/.../functions-4-2015-11-30-1-slide-per-page.pdf · Slide 9 /...
68
Slide 1 / 68
Transcript of Slide 1 / 68 - NJCTLcontent.njctl.org/.../functions-4-2015-11-30-1-slide-per-page.pdf · Slide 9 /...
Slide 3 / 68
click on the topic to go to that sectionTable of Contents
• Relationships and Functions
• Domain and Range
• Vertical Line Test
• Linear Vs. Non-Linear Functions
• Glossary
Slide 5 / 68
State CapitalNew Jersey TrentonNew York Albany
Colorado DenverRhode Island Providence
x y5 72 9
0 8-4 6
Person GrandparentEric ArthurEric Ellie
Maria CamineMaria Susan
x y4 32 8
0 84 6
A function is a relation where every input (x) has exactly one output (y).
Let's look at a few examples of relations. Some are functions and some are not.
A relation is a set of ordered pairs.
Relationships and Functions
Slide 6 / 68
State CapitalNew Jersey TrentonNew York Albany
Colorado DenverRhode Island Providence
Person GrandparentEric ArthurEric Ellie
Maria CamineMaria Susan
Relationships and FunctionsLet's look at these a little closer.
This is a function. Each state (input) has only one capital (output).
This is not a function. Each person (input) has more than one grandparent. (output).
Slide 7 / 68
x y5 72 9
0 8-4 6
x y4 32 8
0 84 6
Relationships and Functions
Each number in the x (input) column has only one y (output) as an answer so it is a function.
Notice that 4 has an output of both 3 and 6. Therefore it cannot be a function.
Slide 8 / 68
x y8 35 8
3 8-1 6
x y4 32 8
0 84 6
A B
Let's compare the last problem with another one. How do they differ? Are either of them functions?
Relationships and Functions
In table A, there is one x (4) that has two different y's (3 & 6). This is NOT a function.
In table B, there are two different x's (3 & 5) that have the same y (8). This is a function.
click to reveal
Slide 9 / 68
SarahJohnDwayneAlanJada
June 8Oct. 28March 18Oct. 28Dec. 29
SarahJohnDwayneAlanJada
April 9Jan. 23Sept. 14Feb. 23May 5July 19
Consider this: Two people in a class (input) may have the same birthday (output). However, any one person (input) does not have more than one birthday (output).
Relationships and Functions
John and Alan have thesame birthday. That can happen. It is a function.
Dwayne's birthday is February 23 and September 14. This is not possible. It is not a function.
Slide 10 / 68
Input Output
ABCD
1234
Relationships and Functions
Notice that there are different ways to express functions.
They may be in words, a table, ordered pairs, a graph, or like the last slide, a function map.
In a function map one side is the input and the other is output. An arrow extends from each input element to its matching output element.Example:
Slide 11 / 68
x y2 63 74 85 9
Relationships and Functions
Functions can also be written as sets of ordered pairs. All of the x coordinates are the input and all of the y coordinates are the output.
To figure out if sets of ordered pairs are functions, make a function map or table first.
{(2, 6), (3, 7), (4, 8), (5, 9)}
This is a function because every input has exactly one output.
Slide 12 / 68
1 Is the following a function?
YesNo
1234
5432
Slide 13 / 68
2 Is the following a function?
YesNo
x y1 25 79 213 617 5
Ans
wer
Slide 14 / 68
3 Is the following a function?
YesNo
A person's social security number
Slide 15 / 68
4 Is the following a function?
YesNo
x y3 24 54 79 69 12
Ans
wer
Slide 16 / 68
5 Is the following a function?
YesNo
Any number times 3.
Slide 17 / 68
6 Is the following a function?
YesNo 1
234
567891011
Slide 18 / 68
7 Is the following a function?
YesNo
{(1, 2), (2, 3), (3, 4), (4, 5)}
Slide 19 / 68
8 Is the following a function?
YesNo
{(1, 2), (1, 3), (2, 4), (3, 4), (3, 5)}
Slide 20 / 68
9 Is the following a function?
YesNo
City State
Englewood New Jersey
Englewood Colorado
Springfield Maryland
Springfield Illinois
Springfield Montana
Ans
wer
Slide 21 / 68
10 A relationship between x and y is defined by the
equation , where x is the input and y is
the output. Which statements about the relationship
are true? Select each correct statement.
A y is a function of x.
B The graph of the relationship is a line.
C When the input is -3, the output is 4.
D When the input is -2, the output is 3.
E The y-intercept of the relationship is (0,1).From PARCC EOY sample test non-calculator #4
Slide 22 / 68
11 When the input to a function is -2, the output is 4. Which statement about this function must be true?
A An input of -2 has infinitely many possible outputs
B An input of -2 has exactly one possible output
C An output of 4 has infinitely number of inputs
D A output of 4 has exactly one input
From PARCC EOY sample test non-calculator #10
Slide 24 / 68
Days 1 2 3 4
$ Earned 125 42 98 106
Domain and Range
Any set of numbers that is the input of a relation or function is called the domain.
Any set of numbers that is the output of a relation or function is called the range.
Domain: {1, 2, 3, 4} Range: {42, 98, 106, 125}
Slide 25 / 68
x 1 2 3 4y 6 7 8 9 1
2
34
6
78
9Domain Range
Domain and Range
Which numbers are part of the domain and which are part of the range?
Rule: x + 5 = y
Slide 26 / 68
Domain Range1
83
10
5 127 14
9
16
Domain and Range
How about when we have ordered pairs?
Which numbers are in the domain? Which are in the range?
{(1,8), (3, 10), (5, 12), (7, 14), (9, 16)}
Slide 27 / 68
12 Which numbers are in the domain?
246810
1020304050
A 2B 10C 12D 8E 20
Slide 28 / 68
13 Which numbers are in the range?
A 2B 10C 12D 8E 20
246810
1020304050
Slide 29 / 68
14 Which numbers are part of the domain?
A 12B 20C 24D 11E 9
{(24, 12), (22, 11), (20, 10), (18, 9)}
Slide 30 / 68
15 Which numbers are part of the range?
A 12B 20C 24D 11E 9
{(24, 12), (22, 11), (20, 10), (18, 9)}
Slide 32 / 68
Vertical Line Test
A function can also be expressed as a graph.
Slide 33 / 68
Function Function Not a Function Not a Function
Vertical Line Test
In order to verify that a graph is a function you can use the Vertical Line Test.In order to be a function, there may be only one point that goes through any given vertical line on the graph.
Slide 34 / 68
Slope = 0 Undefined slope
Vertical Line Test
A linear equation with a slope of zero is a function.
Prove that a linear equation with an undefined slope cannot be a function.
Slide 35 / 68
16 Is this a function?
YesNo
Slide 36 / 68
17 Is this a function?
YesNo
Slide 37 / 68
18 Is this a function?
YesNo
Slide 38 / 68
19 Is this a function?
YesNo
Slide 39 / 68
20 Is this a function?
YesNo
Slide 40 / 68
21 Is this a function?
YesNo
Slide 41 / 68
22 Is this a function?
YesNo
Slide 43 / 68
A Linear Function is a graph that is represented by a straight line. As an equation it is written in slope intercept form: y=mx+b (see Graphing Equations Unit).
Linear functions will never have an exponent larger than 1 in the equation.
y = 2x - 3 y = -1/4x +4
Linear Functions
Slide 44 / 68
y = 2x +3
+1+1+1+1
+3+3+3+3
x y-2 -1-1 10 31 52 7
Linear Functions
Linear Functions are incremental.
Slide 45 / 68
y = x 2 y = x 3 + 2x 2 - x
Non-Linear Function
A Non-linear Function is any function that is not represented as a straight line.
Non-linear functions often have exponents in their equation.
Slide 46 / 68
y = x 2 + 2
+1+1+1+1
-3-1+1+3
x y-2 6-1 30 21 32 6
Non-Linear Function
Non-linear functions are not incremental.
Slide 47 / 68
23 The following is a linear function.
TrueFalse
Slide 48 / 68
24 The following is a linear function.
TrueFalse
Slide 49 / 68
25 The following is a linear function.
TrueFalse
Slide 50 / 68
26 The following is a linear function.y = 5x + 7
TrueFalse
Slide 51 / 68
27 The following is a linear function.
TrueFalse
3 94 165 256 36
Ans
wer
Slide 52 / 68
28 The following is a linear function
TrueFalse
Slide 53 / 68
29 The following is a linear function.
TrueFalse
Slide 54 / 68
30 The following is a function.
TrueFalse
y = 2x 2 + 3
Slide 55 / 68
31 The following is a function.
TrueFalse
x 3 5 7 9y 13 17 21 25 A
nsw
er
Slide 56 / 68
32 The graph of a nonlinear function is shown on the coordinate plane.
In the graph y is the function of x. When the input of the function is -4, what is the output of the function?
From PARCC EOY sample test non-calculator #14
A -5
B -1
C 1
D 5
Slide 57 / 68
33 The graph shows y as a function of x.
Students type their answers here
From PARCC EOY sample test non-calculator #6
For each interval in the table, indicate whether the function is increasing, decreasing, or neither increasing nor decreasing over the interval.
Interval Increasing Descreasing Neither
-7<x<-3-3<x<-1-1<x<11<x<33<x<55<x<7
Ans
wer
Slide 58 / 68
34 Classify each equation as defining y as a linear or non-linear function of x. Select one cell per column. Students type their answers here
From PARCC EOY sample test non-calculator #17
Slide 60 / 68
Back to
Instruction
Domain
Any set of numbers that is the input of a relation or function.
Domain: {1,2,3,4}
Domain: {10,25,30,45}
Domain: {1,2,3,4}
# of people 10 25 30 45
$ Earned 100 420 980 1060
{(1,2.3),(2,5),(3,10),(4,11)}
Slide 61 / 68
Back to
Instruction
Function
A relation where every input (x) has exactly one output (y).
Slide 62 / 68
Back to
Instruction
Function MapA map where one side is the input and the other is output and an arrow extends from each input element to its matching output
element.
Slide 64 / 68
Back to
Instruction
Linear Function
A graph that is represented by a straight line.
Slide 65 / 68
Back to
Instruction
Non-Linear Function
Any function that is not represented as a straight line.
Slide 66 / 68
Back to
Instruction
Range
Any set of numbers that is the output of a relation or function.
Range: {100,420,980,1060}
# of people 10 25 30 45
$ Earned 100 420 980 1060 {(1,2.3),(2,5),(3,10),(4,11)}
Range: {5,4,3,2}
Range: {2.3,5,10,11}
Slide 68 / 68
Back to
Instruction
Vertical Line TestA test that shows if an equation is a function by
examining the graph of the function and seeing if there is only one point that goes through any
given vertical line on the graph.
![[TRAVEL TRAILERS AND FIFTH WHEELS]media.rvusa.com/library/Fleetwood2007_te_b.pdf · Single Slide-Out Sleeps 6* 2652BS Single Slide-Out Sleeps 9* PANTRY SOFA BED 40" X 68" FOLD DOWN](https://static.fdocuments.net/doc/165x107/5f957db323226162cd1759c7/travel-trailers-and-fifth-wheelsmediarvusacomlibraryfleetwood2007tebpdf.jpg)
