Day 40: March 30th

1
Day 40 March 30th

• Objective Learn how to simplify algebraic
  fractions. THEN Understand how to multiply and
  divide rational expressions and continue to learn
  how to simplify rational expressions.
• Homework Check
• Notes Simplify Rational Expressions
• Rational Expressions 1 W2 (odds)
• Notes Multiplying/Dividing Rational Expressions
• Rational Expressions 2 W2 (odds)
• Closure
• Homework Finish EVENS from Classwork
• Project Due Wednesday, April 6th (Rubric)

2
Simplifying Rational Expressions

• Simplify the following expressions

3
Simplifying Rational Expressions

• A fellow student simplifies the following
  expressions
• Which simplification is correct? Substitute two
  values of x into each to justify your answer.

MUST BE MUITLIPLICATION!
4
Simplifying Rational Expressions
Can NOT cancel since everything does not have a
common factor and its not in factored form
Simplify
Factor Completely
CAN cancel since the top and bottom have a common
factor
5
Multiplying and Dividing Fractions
Multiply Numerators
Multiply
Multiply Denominators
Multiply by the reciprocal (flip)
Divide
Remember to Simplify!
6
Multiplying/Dividing Rational Expressions
Simplify
Half the work is done!
Combine
Rewrite
Cancel
7
Multiplying/Dividing Rational Expressions
Simplify
Flip to turn it into a multiplication
Factor
Factor Completely
Cancel
8
Day 41 March 31st

• Objective Understand how to add and subtract
  rational expressions and continue to learn how to
  simplify rational expressions.
• Homework Check
• Continue to Work on the first 2 worksheets
• Notes Adding and Subtracting Rationals
• Rational Expressions 3 W2
• Closure
• Homework Finish all worksheets
• Project Due Wednesday, April 6th (Rubric)

9
Adding and Subtracting Fractions
Subtraction
Addition
Least Common Denominator (if you can find it)
Common Denominator
Subtract the Numerators
Add the Numerators
Remember to Simplify if Possible!
10
Add/Subtract Rational Expressions
Same denominator! Half the work is done!
Simplify
CAREFUL with subtraction!
Combine Like Terms
Make sure it cant be simplified more
11
Add/Subtract Rational Expressions
Simplify
Find a Common Denominator
Combine Like Terms
12
Add/Subtract Rational Expressions
Simplify
Find a common denominator
Distribute numerators but leave the denominators
factored
CAREFUL with subtraction
Combine like Terms
13
Add/Subtract Rational Expressions
Simplify
Factor to find a Smaller Common Denominator
Make sure it cant be simplified beforehand
14
Add/Subtract Rational Expressions
Simplify
Factor to find a Smaller Common Denominator
Make sure it cant be simplified more
15
Day 42 April 1st

• Objective Consider two functions and identify
  the relationships between the functions and the
  system from which they come.
• Homework Check
• Rational Expressions 4 W2
• Wells Time
• 5-96 (pg 249, RsrcPg)
• Closure Final Challenge
• Homework Finish Worksheet AND 6-8 to 6-15 (pg
  265-266)
• Project Due Wednesday, April 6th (Rubric)

16
Day 43 April 4th

• Objective Learn to find rules that undo
  functions, and develop strategies to justify that
  each rule undoes the other. Also, graph
  functions along with their inverses and make
  observations about the relationships between the
  graphs. THEN Introduction to the term inverse to
  describe undo rules. Also graphing the inverse
  of a function by reflecting it across the line of
  symmetry and write equations for inverses.
• Homework Check
• 6-1 to 6-6 (pgs 263-265)
• Wells Time
• START 6-16 to 6-25 (pgs 267-269, RsrcPg)
• Closure
• Homework 6-7 (pg 265) AND 6-26 to 6-32 (pgs 270)
  
• Project Due Wednesday, April 6th (Rubric)

17
Guess my Number

• Im thinking of a number that

When I I get My number is
Add four to my number AND Multiply by ten -70
Double my number Add four AND Divide by two Five
Square my number Add three Divide by two AND Add one Seven
Double my number Subtract six Take the square root AND Add four Eight
-11
Three
Three and
3 and -3
Eleven
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Inverse Notation
Original function
Inverse function
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Undo Rule
1st Step 2nd Step 3rd Step
p(x)
p -1 (x)
Start
Add 3
Cube
Multiply 2
Divide 2
Cube Root
Subtract 3
Only works when there is one x!
20
Tables and Graphs of Inverses
Switch x and y
Orginal
Inverse
X Y
0 25
2 16
6 4
10 0
14 4
18 16
20 25
X Y
25 0
16 2
4 6
0 10
4 14
16 18
25 20
X Y







Switch x and y
(0,25)
(20,25)
(16,18)
(4,14)
(2,16)
(18,16)
(0,10)
(6,4)
(16,2)
(4,6)
(14,4)
(10,0)
Non-Function
Function
y x
Line of Symmetry
21
6-6 Learning Log

• Title Finding and Checking Undo Rules
• What strategies did your team use to find undo
  rules?
• How can you be sure that the undo rules you found
  are correct?
• What is another name for undo?
• How do the tables of a rule and an undo-rule
  compare? Graph?

22
Day 44 April 5th

• Objective Introduction to the term inverse to
  describe undo rules. Also graphing the inverse
  of a function by reflecting it across the line of
  symmetry and write equations for inverses. THEN
  Use the idea of switching x and y-values to learn
  how to find an inverse algebraically. Also learn
  about compositions of functions and use
  compositions f(g(x)) and g(f(x)) to test
  algebraically whether two functions are inverses
  of each other.
• Homework Check
• Finish 6-16 to 6-25 (pgs 267-269 , RsrcPg)
• Wells Time
• 6-38 to 6-42 (pgs 272-274)
• Closure
• Homework 6-33 to 6-37 (pgs 271) AND 6-44 to 6-53
  (pgs 274-277)
• Project Due Wednesday, April 6th (Rubric)

23
The Rule for an Inverse
1st Step 2nd Step 3rd Step 4th Step
p(x)
p -1 (x)
Start
Add 2
Square
Multiply 3
Subtract 6
Square Root

Add 6
Divide 3
Subtract 2
24
Vertical Line Test
If a vertical line intersects a curve more than
once, it is not a function. Use the vertical
line test to decide which graphs are functions.
25
Horizontal Line Test
If a horizontal line intersects a curve more than
once, the inverse is not a function. Use the
horizontal line test to decide which graphs have
an inverse that is a function.
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Restricted Domain

• Find the inverse relation of f below

Inverse
Inverse Function
27
Day 45 April 6th

• Objective Use the idea of switching x and
  y-values to learn how to find an inverse
  algebraically. Also learn about compositions of
  functions and use compositions f(g(x)) and
  g(f(x)) to test algebraically whether two
  functions are inverses of each other.
• Homework Check
• 6-38 to 6-42 (pgs 272-274)
• Closure
• Homework 6-44 to 6-53 (pgs 274-277)
• Project Due Today

28
Algebraically Finding an Inverse
Find the inverse of the following
Switch x and y
Solve for y
Do not write y-1
Make sure to check with a table and graph on the
calculator.
29
Algebraically Finding an Inverse
Find the inverse of the following
Switch x and y
Solve for y
Do not write y-1
Because x29 has two solutions 3 -3
Make sure to check with a table and graph on the
calculator.
30
Algebraically Finding an Inverse
Find the inverse of the following
Switch x and y
Really y
Solve for y
Make sure to check with a table and graph on the
calculator.
31
Algebraically Finding an Inverse
Only Half Parabola
Find the inverse of the following
Switch x and y
Really y
Solve for y
Restrict the Domain!
Full Parabola (too much)
x3
Make sure to check with a table and graph on the
calculator.
32
Composition of Functions
Substituting a function or its value into
another function.
Second
g
f
First (inside parentheses always first)
OR
33
Composition of Functions

• Let and
  . Find

Equivalent Statements
Our text uses the first one
Plug x1 into g(x) first
Plug the result into f(x) last
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Composition of Functions

• Let and
  . Find

Plug the result into g(x) last
Plug x into f(x) first
35
Inverse and Compositions

• In order for two functions to be inverses
• AND

36
Day 46 April 7th

• Objective Apply strategies for finding inverses
  to parent graph equations. Begin to think of the
  inverse function for y3x. THEN Define the term
  logarithm as the inverse exponential function or,
  when ybx, y is the exponent to use with base b
  to get x.
• Homework Check
• 6-54 to 6-58 (pgs 277-279)
• Wells Time
• 6-67 to 6-71 (pgs 281-282)
• Closure
• Homework 6-59 to 6-66 (pgs 279-280) AND 6-72 to
  6-80 (pgs 283-284)
• Project Due Monday, April 4th (Rubric)

37
Silent Board Game




38
Silent Board Game




39
Logarithm and Exponential Forms
Logarithm Form
5 log2(32)
Base Stays the Base
Input Becomes Output
Logs Give you Exponents
25 32
Exponential Form
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Examples

• Write each equation in exponential form
• log125(25) 2/3
• Log8(x) 1/3
• Write each equation in logarithmic form
• If 64 43
• If 1/27 3x

1252/3 25
81/3 x
log4(64) 3
Log3(1/27) x
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Inverse of an Exponential Equation
Original Inverse

OR

• Logs give you exponents!

42
Definition of Logarithm
The logarithm base a of b is the exponent you put
on a to get b i.e. Logs give you exponents!
a gt 0 and b gt 0
43
6-71 Closure
2
4
7
1.2
w 3
44
Day 47 April 8th

• Objective Assess Chapters 1-5 in an individual
  setting.
• Homework Check
• Midterm Exam
• Closure
• Homework 6-84 to 6-92 (pgs 286-287)

45
Day 48 April 11th

• Objective Develop methods to graph logarithmic
  functions with different bases. Rewrite
  logarithmic equations as exponential equations
  and find inverses of logarithmic functions. THEN
  Look into the base of the log key on the
  calculator. Also extend our knowledge of general
  equations for parent functions to transform the
  graph of ylog(x).
• Homework Check
• Logarithms and Graphs Packet (Extra)
• Wells Time
• 6-93 and 6-95 (pgs 288-289)
• Closure
• Homework 6-96 to 6-105 (pgs 290-291)

46
6-83 Learning Log

• Title The Family of Logarithmic Functions
• What is the general shape of the graph?
• What happens to the value of y as x increases?
• How is the graph related to the exponential
  graph?
• What is the Domain? Range?
• Why is the x-intercept always (1,0)?
• Why is the line x0 (y-axis) always an asymptote?
• Why is there no horizontal asymptote?
• How does the graph change if b changes?
• What does the graph look like when 0ltblt1?
• What does the graph look like when b1?
• What does the graph look like when bgt1?

47
Common Logarithm
Ten is the common base for logarithms, so log(x)
is called a common logarithm and is shorthand for
writing log10(x). You read this as the
logarithm base 10 of x.
Our calculator has the button log . It doesnt
have the subscript 10 because it stands for the
common logarithm log10100 log100
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Logarithmic Function
Parent Equation
Graphing Form
49
Example Exponential
Transformation Shift the parent graph three
units to the right and two units up.
Transformation
New Equation
y 2
x 3