Title: 2-1%20Functions
12-1 Functions
2What 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 ______________________
3Well, what would a non function look like?
- Equations that would not be functions
- __________________________________________________
___ - __________________________________________________
___
4Domain? What was that?
- -the x values.
- The easiest way to define the domain ___________
- _________________________________________
- _________________________________________
- ________________________________________
- _____________________________________
- b) ____________________________________
- Either is acceptable.
5Examples Find the Domain of each
6Function 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! ____________________________
- _______________________________________
7OK 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)
8Examples
- Find the domain of
- 3.
- 4.
- 5.
92.2 Graphing Lines
- Going from an equation to a picture
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11What methods can I use to graph line?
- ___________________________
- ___________________________
- ___________________________
- ___________________________
- ___________________________
- Please graph 2x 3y 6
12What method can I use to graph line?
- 2. _____________________________
- _____________________________
-
-
- Lets review slope for a minute
13SLOPE
- Slope
- Please graph y -3x 4
-
14Special Things
- Parallel Lines
- Perpendicular Lines
- Horizontal Lines
- Vertical Lines
15Other Review Items
____________ ____________ ____________
162.3 Equations of Lines
- Going the other direction from a picture to the
equation
17There are 3 standard forms of equations
- Slope intercept form
- ______________
-
- Standard form
- ______________
- ____________________________
- 3. Point slope form
18So, 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
20Find the equation of the line that has Slope 3,
y intercept 10 Slope 3, x intercept
10 Slope 3, passes through (10, 10)
21- 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
252-4 A Variety of Graphs
26What are Piecewise Functions?
- Piecewise functions are defined
- ___________________________________
- ___________________________________
- ___________________________________
- ___________________________________
27Graphing absolute Values
28How will we graph?
- ______________________________
- ______________________________
- ______________________________
- ______________________________
- ______________________________
29Graphing Absolute Value
- _______________________________________
- __________________________________________
- __________________________________________
- __________________________________________
- ______________________________________
- _________________________________________
- _________________________________________
- _________________________________________
30Examples
31The 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
332-4 Graphing Day 2
- Greatest Integer Function
34The Greatest Integer Function y x
- Rounding down to the nearest integer
35So, what will each point look like?
- ______________________________
4 3 2 1
1 2 3 4 5
36Shifts of Greatest integer Graphs
- Add/Subtract inside
- ______________________
_ - Add/Subtract outside
- ______________________
_ - Multiply/Divide inside
- _______________________
- Multiply/Divide outside?
- ______________________
_
37But 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.
38Method
- ______________________________
- _________________________________
-
393. _____________________________________ ________
_______________________________
4. Graph
3 2 1
-1 1 2 3
40Lets try another one
4. Graph
3 2 1
-1 1 2 3
412-5 Systems of Equations
- Finding a solution that works for multiple
equations
42Warm Up
- Please graph on one set of axes the following
43Solutions for multiple equations?
- That is, where 2 lines intersect.
- How can 2 lines intersect?
44What methods have you already learned for finding
where 2 line intersect?
- _______________________________________
- _______________________________________
- __________________________________________
- __________________________________________
- _______________________________________
- __________________________________________
- __________________________________________
45What 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
46Steps to solve 3 Equations 3 Variables
1. __________________________________ 2.
__________________________________ 3.
___________________________________ 4.
___________________________________ 5.
___________________________________
47- 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?
482. 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??
504.. 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?
516. The sum of two numbers is 20. The larger is
5 less than twice the smaller. What are the
numbers??
522-6 Graphing Quadratic Functions
53No more linear functions
- What happens graphically when an equations high
power is 2? - _____________________________
- _____________________________
54The Parabola (The Picture)
55Looking at Trends
5 4 3 2 1
-4 -3 -2 -1 1 2 3
4
56So, 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.
57The 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?
- ________________________
58The Parabola (The Equation)
- a ____________________________
- (h, k) _________________________
- If a lt 0, what will happen to the graph?
59So what will we do with this information?
- Determine the vertex (h, k).
- Find 1 x values on either side of h and plug them
in to find 2 points to graph. - If asked to, determine domain (hint what CANT
you put in?) - If asked to, determine range (hint decide
up/down orientation then think about where you
will move from the vertex).
60Examples
4
1
614
2
62- Function Increasing and Decreasing
- _____________________________
- _____________________________
As we go left to right until we hit x-2, what
are the y values doing?
-2
632-7 The Quadratic Formula and Completing the
Square
- Day 1
- Completing the Square
64When 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
65Completing 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. - ______________________________
- ______________________________
- ______________________________
66Completing 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
67Perfect Square Trinomials
- Is there a relationship between the red term and
the blue term?
68This 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.
69Example
70Example
71 72- Using Completing the square
- _________________________________
- _________________________________
73Graph
74Solve by completing the square
752-7 The Quadratic Formula and Completing the
Square
76Quadratic Equation
The numbers for the variables come from
77The Discriminant
- ______________________________
- ______________________________
- ______________________________
___________________
___________________
___________________
78Example
79Graph then solve
Convert to Parabolic form
Now Solve using QF
80Inequalities
Solve (ie find the x-ints)
1 3 5
__________________
-4
__________________________________________________
__________________________________
81Easy Way to solve Inequalities
- ______________________________
- ______________________________
- ______________________________
822.8 Quadratic Applications
83In the Word Problems
- Essentially you will see three things.
- __________________________________________________
______ - __________________________________________________
__________________________________________________
____________ - __________________________________________________
__________________________________
84Maximize? Minimize
- Why would we be talking about maximizing or
minimizing with quadratic word problems? - ______________________________
________
________
85Example 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
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87- The sum of two numbers is 40. Find the two
numbers if their product is a maximum.
88- Find two consecutive positive integers such that
the sum of their squares is 113. (notice! No
maximum/minimum)
89The sum of a number and its square is 72. Find
the number
90The sum of 2 numbers is 12. Find the numbers if
the product of one and twice the other is a
maximum.