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Title: 2-1%20Functions


1
2-1 Functions
2
What is a Function?
  • Definition __________________________
  • ___________________________________
  • The x values of a function are called the
    ____________________ and all the y values are
    called the _________________
  • ___________________________________
  • X is called the ______________________ while y
    is the ______________________

3
Well, what would a non function look like?
  • Equations that would not be functions
  • __________________________________________________
    ___
  • __________________________________________________
    ___

4
Domain? What was that?
  • -the x values.
  • The easiest way to define the domain ___________
  • _________________________________________
  • _________________________________________
  • ________________________________________
  • _____________________________________
  • b) ____________________________________
  • Either is acceptable.

5
Examples Find the Domain of each
6
Function Notation
  • The algebraic expression
  • is a function. There are LOTS of functions
    out there (any equation you can dream up where an
    x will produce only one y value is a function)
    but I am going to use this one for now. To show
    that something IS a function, it is written like
    this
  • Dont worry! ____________________________
  • _______________________________________

7
OK how do we use it?
  • Lets use the sample from before.
  • 1. Given find
    f(1), f(-2) and f(0).
  • The function is simply an instruction of what to
    do to x. ___________________________________
  • Plug 1 in for all xs and solve for y and put as
    a ordered pair (x,y)
  • f(1) f(-2) f(0)

8
Examples
  • Find the domain of
  • 3.
  • 4.
  • 5.

9
2.2 Graphing Lines
  • Going from an equation to a picture

10
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11
What methods can I use to graph line?
  • ___________________________
  • ___________________________
  • ___________________________
  • ___________________________
  • ___________________________
  • Please graph 2x 3y 6

12
What method can I use to graph line?
  • 2. _____________________________
  • _____________________________
  • Lets review slope for a minute

13
SLOPE
  • Slope
  • Please graph y -3x 4

14
Special Things
  • Parallel Lines
  • Perpendicular Lines
  • Horizontal Lines
  • Vertical Lines

15
Other Review Items
____________ ____________ ____________
16
2.3 Equations of Lines
  • Going the other direction from a picture to the
    equation

17
There are 3 standard forms of equations
  • Slope intercept form
  • ______________
  • Standard form
  • ______________
  • ____________________________
  • 3. Point slope form

18
So, what do you need to have to find the equation
of the line?
Lets try one Slope2 and the y-int 5
19
  • Convert y 1.5 x 6 to standard form.
  • ________________________________
  • __________________________________

2. Convert 10x 2y 3 to slope/intercept
20
Find the equation of the line that has Slope 3,
y intercept 10 Slope 3, x intercept
10 Slope 3, passes through (10, 10)
21
  1. Parallel to -4x 2y 10 and passes through (-1,
    -1)

7. Parallel to x 2y 1 and passes through the
point of intersection of the lines y 3x 2 and
y 2x 1.
22
  • Triangle ABC has vertices A(-4,-2) L(2,8)
    G(6,2)
  • Write the equation of AL

23
  • Triangle ABC has vertices A(-4,-2) L(2,8)
    G(6,2)
  • Find the equation of the perpendicular bisector
  • of LG.
  • Steps
  • ___________________
  • ___________________
  • ____________________
  • 3. ________________

L
G
A
24
  • Triangle ABC has vertices A(-4,-2) L(2,8)
    G(6,2)
  • Find the equation of the altitude to AG
  • Steps
  • 2. ________________

L
G
A
25
2-4 A Variety of Graphs
  • Piecewise Functions

26
What are Piecewise Functions?
  • Piecewise functions are defined
  • ___________________________________
  • ___________________________________
  • ___________________________________
  • ___________________________________

27
Graphing absolute Values
28
How will we graph?
  • ______________________________
  • ______________________________
  • ______________________________
  • ______________________________
  • ______________________________

29
Graphing Absolute Value
  • _______________________________________
  • __________________________________________
  • __________________________________________
  • __________________________________________
  • ______________________________________
  • _________________________________________
  • _________________________________________
  • _________________________________________

30
Examples
  • 1.
  • 2.

31
The next kind of piecewise function
  • The form of this function is similar to this
  • This looks worse than it is. Essentially the
    function is split into multiple functions based
    on particular domains. __________________________
    ____
  • ________________________________________

32
  • _____________________________________
  • _____________________________________
  • x y x
    y x y

33
2-4 Graphing Day 2
  • Greatest Integer Function

34
The Greatest Integer Function y x
  • Rounding down to the nearest integer

35
So, what will each point look like?
  • ______________________________

4 3 2 1
1 2 3 4 5
36
Shifts of Greatest integer Graphs
  • Add/Subtract inside
  • ______________________
    _
  • Add/Subtract outside
  • ______________________
    _
  • Multiply/Divide inside
  • _______________________
  • Multiply/Divide outside?
  • ______________________
    _

37
But how do we graph?
  • You could use the trends to graph. Or, use a
    mathematical method and then look at what you
    predict should happen to double check.
  • The mathematical method will work as long as you
    follow directions!!
  • Well do a problem to learn the method.

38
Method
  • ______________________________
  • _________________________________

39
3. _____________________________________ ________
_______________________________
4. Graph
3 2 1
-1 1 2 3
40
Lets try another one
  • Graph

4. Graph
3 2 1
-1 1 2 3
41
2-5 Systems of Equations
  • Finding a solution that works for multiple
    equations

42
Warm Up
  • Please graph on one set of axes the following

43
Solutions for multiple equations?
  • That is, where 2 lines intersect.
  • How can 2 lines intersect?

44
What methods have you already learned for finding
where 2 line intersect?
  • _______________________________________
  • _______________________________________
  • __________________________________________
  • __________________________________________
  • _______________________________________
  • __________________________________________
  • __________________________________________

45
What method do you have to use?
  • Unless specified (i.e. follow directions) you may
    use ANY method you want. ? I want you to be
    happy.
  • Examples

46
Steps to solve 3 Equations 3 Variables
1. __________________________________ 2.
__________________________________ 3.
___________________________________ 4.
___________________________________ 5.
___________________________________
47
  1. A golfer scored only 4s and 5s in a round of 18
    holes. His score was 80. How many of each score
    did he have?

48
2. Tuition plus Room/Board at a local college is
24,000. Room/Board is 400 more than one-third
the tuition. Find the tuition.
49
  • 3. Mr. Tem bought 7 different shirts for the
    coaches of his baseball team. The blue long
    sleeved shirts cost 30 each and the white short
    sleeved shorts cost 20 each. If he paid a total
    of 160, how many of each shirt did he buy??

50
4.. Rob invests money, some at 10 and some at
20 earning 20 in interest per year. Had the
amounts invested been reversed, he would have
received 25 in interest. How much has he
invested all together?
51
6. The sum of two numbers is 20. The larger is
5 less than twice the smaller. What are the
numbers??
52
2-6 Graphing Quadratic Functions
53
No more linear functions
  • What happens graphically when an equations high
    power is 2?
  • _____________________________
  • _____________________________

54
The Parabola (The Picture)
55
Looking at Trends
5 4 3 2 1
-4 -3 -2 -1 1 2 3
4
56
So, we see some trends
  • We probably wont use trends much like absolute
    values, one easy way to graph parabolic functions
    is to plot the vertex and then plot 2 points on
    either side of the x coordinate of the vertex.

57
The Parabola (The Equation)
  • From what we saw, these are the trends
  • Add/Subtract inside the squared quantity?
  • ______________________
    __
  • Add/Subtract outside the squared quantity?
  • ______________________
    __
  • Multiply/Divide inside or outside?
  • ________________________

58
The Parabola (The Equation)
  • a ____________________________
  • (h, k) _________________________
  • If a lt 0, what will happen to the graph?

59
So what will we do with this information?
  1. Determine the vertex (h, k).
  2. Find 1 x values on either side of h and plug them
    in to find 2 points to graph.
  3. If asked to, determine domain (hint what CANT
    you put in?)
  4. If asked to, determine range (hint decide
    up/down orientation then think about where you
    will move from the vertex).

60
Examples
4
1
61
4
2
62
  • Function Increasing and Decreasing
  • _____________________________
  • _____________________________

As we go left to right until we hit x-2, what
are the y values doing?
-2
63
2-7 The Quadratic Formula and Completing the
Square
  • Day 1
  • Completing the Square

64
When the directions are graph
  • In the last section the graphs were already in
    parabolic form, which makes graphing easy. The
    vertex is right there to see.
  • What if instead you are asked to graph
  • __________________________________
  • How would we go about graphing this one?
  • By just plotting points, will we be able to find
    the vertex easily? Not necessarily

65
Completing the Square
  • Completing the square is the way to convert a
    parabola in quadratic form to parabolic form
    so that you can find the vertex easily.
  • ______________________________
  • ______________________________
  • ______________________________

66
Completing the Square
  • In this method you complete the square by
    adding the same thing to both sides of an
    equation so as to create a perfect square
    trinomial. Then by factoring and isolating f(x),
    you will have parabolic form.
  • Easier than it sounds with a little review

67
Perfect Square Trinomials
  • Is there a relationship between the red term and
    the blue term?

68
This is what you will add to both sides of a
quadratic equation.
  • ______________________________
  • This will create a factorable perfect square
    trinomial. Then, depending on whether you want
    to solve or graph you go from there. Well do an
    example of each to see both paths.

69
Example
  • Graph

70
Example
  • 2. Solve

71
  • Graph

72
  • Using Completing the square
  • _________________________________
  • _________________________________

73
Graph
  • 2.

74
Solve by completing the square
75
2-7 The Quadratic Formula and Completing the
Square
  • The Quadratic Formula

76
Quadratic Equation
The numbers for the variables come from
77
The Discriminant
  • ______________________________
  • ______________________________
  • ______________________________

___________________
___________________
___________________
78
Example
79
Graph then solve
Convert to Parabolic form
Now Solve using QF
80
Inequalities
Solve (ie find the x-ints)
1 3 5
__________________
-4
__________________________________________________
__________________________________
81
Easy Way to solve Inequalities
  • ______________________________
  • ______________________________
  • ______________________________

82
2.8 Quadratic Applications
  • Word Problems

83
In the Word Problems
  • Essentially you will see three things.
  • __________________________________________________
    ______
  • __________________________________________________
    __________________________________________________
    ____________
  • __________________________________________________
    __________________________________

84
Maximize? Minimize
  • Why would we be talking about maximizing or
    minimizing with quadratic word problems?
  • ______________________________

________
________
85
Example 1. I have 80 feet of fence to make a
garden which will have one wall of my house as a
border. Find the dimensions so that the area is
a maximum.
House
Garden
86
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87
  1. The sum of two numbers is 40. Find the two
    numbers if their product is a maximum.

88
  1. Find two consecutive positive integers such that
    the sum of their squares is 113. (notice! No
    maximum/minimum)

89
The sum of a number and its square is 72. Find
the number
90
The sum of 2 numbers is 12. Find the numbers if
the product of one and twice the other is a
maximum.
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