2-1%20Functions

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2-1 Functions
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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 ______________________

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Well, what would a non function look like?

• Equations that would not be functions
• __________________________________________________
  ___
• __________________________________________________
  ___

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Domain? What was that?

• -the x values.
• The easiest way to define the domain ___________
• _________________________________________
• _________________________________________
• ________________________________________
• _____________________________________
• b) ____________________________________
• Either is acceptable.

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Examples Find the Domain of each
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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! ____________________________
• _______________________________________

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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)

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Examples

• Find the domain of
• 3.
• 4.
• 5.

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2.2 Graphing Lines

• Going from an equation to a picture

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What methods can I use to graph line?

• ___________________________
• ___________________________
• ___________________________
• ___________________________
• ___________________________
• Please graph 2x 3y 6

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What method can I use to graph line?

• 2. _____________________________
• _____________________________
• Lets review slope for a minute

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SLOPE

• Slope
• Please graph y -3x 4

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Special Things

• Parallel Lines
• Perpendicular Lines
• Horizontal Lines
• Vertical Lines

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Other Review Items
____________ ____________ ____________
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2.3 Equations of Lines

• Going the other direction from a picture to the
  equation

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There are 3 standard forms of equations

• Slope intercept form
• ______________
• Standard form
• ______________
• ____________________________
• 3. Point slope form

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

2. Convert 10x 2y 3 to slope/intercept
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Find the equation of the line that has Slope 3,
y intercept 10 Slope 3, x intercept
10 Slope 3, passes through (10, 10)
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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.
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• Triangle ABC has vertices A(-4,-2) L(2,8)
  G(6,2)
• Write the equation of AL

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• 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
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• 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
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2-4 A Variety of Graphs

• Piecewise Functions

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What are Piecewise Functions?

• Piecewise functions are defined
• ___________________________________
• ___________________________________
• ___________________________________
• ___________________________________

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Graphing absolute Values
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How will we graph?

• ______________________________
• ______________________________
• ______________________________
• ______________________________
• ______________________________

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Graphing Absolute Value

• _______________________________________
• __________________________________________
• __________________________________________
• __________________________________________
• ______________________________________
• _________________________________________
• _________________________________________
• _________________________________________

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Examples

• 1.
• 2.

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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. __________________________
  ____
• ________________________________________

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• _____________________________________
• _____________________________________
• x y x
  y x y

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2-4 Graphing Day 2

• Greatest Integer Function

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The Greatest Integer Function y x

• Rounding down to the nearest integer

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So, what will each point look like?

• ______________________________

4 3 2 1
1 2 3 4 5
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Shifts of Greatest integer Graphs

• Add/Subtract inside
• ______________________
  _
• Add/Subtract outside
• ______________________
  _
• Multiply/Divide inside
• _______________________
• Multiply/Divide outside?
• ______________________
  _

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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.

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Method

• ______________________________
• _________________________________

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

• Graph

4. Graph
3 2 1
-1 1 2 3
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2-5 Systems of Equations

• Finding a solution that works for multiple
  equations

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Warm Up

• Please graph on one set of axes the following

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Solutions for multiple equations?

• That is, where 2 lines intersect.
• How can 2 lines intersect?

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What methods have you already learned for finding
where 2 line intersect?

• _______________________________________
• _______________________________________
• __________________________________________
• __________________________________________
• _______________________________________
• __________________________________________
• __________________________________________

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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

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Steps to solve 3 Equations 3 Variables
1. __________________________________ 2.
__________________________________ 3.
___________________________________ 4.
___________________________________ 5.
___________________________________
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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?

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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.
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• 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??

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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?
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6. The sum of two numbers is 20. The larger is
5 less than twice the smaller. What are the
numbers??
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2-6 Graphing Quadratic Functions
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No more linear functions

• What happens graphically when an equations high
  power is 2?
• _____________________________
• _____________________________

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The Parabola (The Picture)
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Looking at Trends
5 4 3 2 1
-4 -3 -2 -1 1 2 3
4
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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.

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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?
• ________________________

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The Parabola (The Equation)

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

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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).

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Examples
4
1
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4
2
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• Function Increasing and Decreasing
• _____________________________
• _____________________________

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

• Day 1
• Completing the Square

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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

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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.
• ______________________________
• ______________________________
• ______________________________

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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

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Perfect Square Trinomials

• Is there a relationship between the red term and
  the blue term?

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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.

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Example

• Graph

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Example

• 2. Solve

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• Graph

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• Using Completing the square
• _________________________________
• _________________________________

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Graph

• 2.

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

• The Quadratic Formula

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Quadratic Equation
The numbers for the variables come from
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The Discriminant

• ______________________________
• ______________________________
• ______________________________

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

• ______________________________
• ______________________________
• ______________________________

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2.8 Quadratic Applications

• Word Problems

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In the Word Problems

• Essentially you will see three things.
• __________________________________________________
  ______
• __________________________________________________
  __________________________________________________
  ____________
• __________________________________________________
  __________________________________

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Maximize? Minimize

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

________
________
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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
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1. The sum of two numbers is 40. Find the two
   numbers if their product is a maximum.

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

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