day 40: march 30 th
DESCRIPTION
Day 40: March 30 th. 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) - PowerPoint PPT PresentationTRANSCRIPT
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 ClassworkProject Due: Wednesday, April 6th (Rubric)
Simplifying Rational Expressions
Simplify the following expressions:
33
xx 16
16xx
3x xx
2
2xx
5 21 2
x xx x
11 1
31 x 3x
51 1x
x 5
1xx
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.
4 4 1 5xx 4 4 1 4x
x
MUST BE MUITLIPLICATION!
Simplifying Rational Expressions
2
3
2 3 20
4 64
x x
x x
2
2 5 4
4 16
x x
x x
2 5 4
4 4 4
x x
x x x
2
2 5or
4 16
x
x x
2 5
4 4
x
x x
Simplify:Can NOT cancel since
everything does not have a common factor and its not in
factored form
CAN cancel since the top and bottom have a
common factor
Factor Completely
Multiplying and Dividing Fractions
a cb d
ywx z
a cb d
w zx y w z
x y
Multiply Numerators
Multiply Denominators
Multiply by the reciprocal (flip)
Remember to Simplify!
Multiply:
Divide:
Multiplying/Dividing Rational Expressions
22 7 3 8
3 2
x x x
x x
22 7 3 8
3 2
x x x
x x
2 2 7 3 8
3 2
x x x x
x x
2 7 3 8
3
x x x
x
Simplify: Half the work is done!
Combine
Rewrite
Cancel
Multiplying/Dividing Rational Expressions
2
2 2
3 15 3 15 18
25 3 10
x x x
x x x
2
2 2
3 15 3 10
25 3 15 18
x x x
x x x
2
3 5 5 2
5 5 3 5 6
x x x
x x x x
1
3x
3 5 5 2
5 5 3 2 3
x x x
x x x x
Simplify:
Flip to turn it into a multiplication
Factor
Factor Completely
Cancel
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)
Adding and Subtracting Fractions
2 13 5 7 3
4 105 32 1
3 5 5 3 7 5 3 24 5 10 2
35 620 20
Add the Numerators
Least Common
Denominator (if you can
find it)Common
Denominator
Remember to Simplify if Possible!
Addition: Subtraction:
10 315 15
1315
2920
Subtract the Numerators
Add/Subtract Rational Expressions
2 2
2 1 4
2 15 2 15
x x
x x x x
2
2 1 4
2 15
x x
x x
2
2 1 4
2 15
x x
x x
2
5
2 15
x
x x
Simplify: Same denominator! Half the work is done!
CAREFUL with subtraction!
Combine Like Terms
5
5 3
x
x x
1
3x
Make sure it can’t be simplified more
Add/Subtract Rational Expressions
2
7 11
2x x
2
7 11 2
22x x
x
x
2 2
7 22
2 2
x
x x
2
7 22
2
x
x
Simplify: Find a Common Denominator
Combine Like Terms
Add/Subtract Rational Expressions
3 5
3 2x x
23 5
3 2
3
2 3
x x
x xx x
3 6 5 15
3 2 2 3
x x
x x x x
3 6 5 15
3 2
x x
x x
Simplify: Find a common denominator
Distribute numerators but leave the
denominators factored
CAREFUL with subtraction
3 6 5 15
3 2
x x
x x
2 21
3 2
x
x x
Combine like Terms
Add/Subtract Rational Expressions
2
8 5
6 2 3 1
x
x x x
8 5
2 3 1 3 1
x
x x x
2 4 5
2 3 1 3 1
x
x x x
4 5
3 1 3 1x x
Simplify: Factor to find a Smaller Common Denominator
4 5
3 1x
1
3 1x
Make sure it can’t be simplified beforehand
Add/Subtract Rational Expressions
2
2
16 3 12
x x
x x
2
4 4 3 4
x x
x x x
2
4 4
43
3 43 4
x x
x x
x
xx
23 6 4
3 4 4 3 4 4
x x x
x x x x
Simplify: Factor to find a Smaller Common Denominator
23 6 4
3 4 4
x x x
x x
2 6
3 4 4
x x
x x
3 2
3 4 4
x x
x x
Make sure it can’t be
simplified more
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)
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) • ClosureHomework: 6-7 (pg 265) AND 6-26 to 6-32 (pgs 270) Project Due: Wednesday, April 6th (Rubric)
Guess my NumberI’m thinking of a number that…
When I… I get… My number is…
• Add four to my numberAND
• 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…
Eleven
3 and -3
Inverse Notation
f x
1f x
Original function
Inverse function
“Undo” Rule
p x 2 x 3 3
x
2
1st Step 2nd Step 3rd Step
p(x)
p -1 (x)
Add 3 Cube Multiply 2
Divide 2 Cube Root Subtract 3
x
2
3
p 1 x x
2
3 3
Start
Only works when there
is one x!
Tables and Graphs of Inverses
y = xLine of Symmetry:
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(0,25)
(2,16)
(6,4)
(10,0)
(14,4)
(18,16)
(20,25)
(4,14)
(4,6)
(0,10)
(16,2)
(16,18)
Function Non-Function
Switch x and y
Switch x and y
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?
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)• ClosureHomework: 6-33 to 6-37 (pgs 271) AND 6-44 to 6-53 (pgs 274-
277) Project Due: Wednesday, April 6th (Rubric)
The Rule for an Inverse
23 2 6p x x
6x
1st Step 2nd Step 3rd Step 4th Step
p(x)
p -1 (x)
Add 2 Square Multiply 3
Add 6 Divide 3Square
Root
6
3
x 6
3
x
StartSubtract 6
± Subtract 2
1 62
3
xp x
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.
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.
Restricted Domain
Find the inverse relation of f below:
2f x x 1f x 1f x x 1f x x 0x InverseInverse Function
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
Algebraically Finding an Inverse
6 11y x
6 11x y 11 6x y 11
6xy
116
x y
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.
Algebraically Finding an Inverse
22 7 3x y
23 2 7x y
232 7x y
32 7x y
32 7x y
Find the inverse of the following:
32 7xy
Make sure to check with a table and graph on the calculator.
22 7 3y x
Switch x and y
Solve for y
Because x2=9 has two solutions: 3 & -3
Do not write y-1
Algebraically Finding an Inverse
310
4
yx
34 10x y
3 4 10x y
3 4 10x y
Find the inverse of the following:
1 3 4 10e x x
310
4
xe x Switch x and y
Really y =
Solve for y
Make sure to check with a table and graph on the calculator.
Algebraically Finding an Inverse
Find the inverse of the following: 4 3d x x
4 3x y
3 4x y
23
4
xy
3
4
xy
2
1 3
4
xd x
when 3x
Make sure to check with a table and graph on the calculator.
Switch x and y
Really y =
Solve for y
Restrict the Domain!
Full Parabola (too much)
Only Half Parabola
x=3
Composition of Functions
g xff
g
First
(inside parentheses always first)
Second
f g xOR
Substituting a function or it’s value into another function.
Composition of Functions
Let and . Find:
1 1gf f g 2 5g x x 2 3f x x
21 1 5g
1 4g
1 1 5g
4 2 4 3f
4 11f
4 8 3f
Equivalent StatementsOur text uses the
first one
1 11f g
Plug x=1 into g(x)
first
Plug the result into f(x) last
Composition of Functions
Let and . Find:
f xg 2 5g x x 2 3f x x
2 3f x x
22 3 2 3 5g x x
22 3 4 12 9 5g x x x 2 3 2 3 2 3 5g x x x
24 12 4g f x x x
Plug x into f(x) first
Plug the result into g(x) last
22 3 4 12 4g x x x 22 3 4 12 9 5g x x x
Inverse and Compositions
In order for two functions to be inverses:
AND
g xf x
f xg x
Day 46: April 7th
Objective: Apply strategies for finding inverses to parent graph equations. Begin to think of the inverse function for y=3x. THEN Define the term logarithm as the inverse exponential function or, when y=bx, “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)• ClosureHomework: 6-59 to 6-66 (pgs 279-280) AND 6-72 to
6-80 (pgs 283-284) Project Due: Monday, April 4th (Rubric)
Silent Board Game
x g x
x g x
83
32
5
1
2
110
16
442
31.6
64
6
21
0
0.25
21
2
1
2
0.2
~ 2.3
1
8
3 g x 2log x
Silent Board Game
x g x
x g x
83
32
5
1
2
110
16
442
31.6
64
621
0
0.25
21
2
1
2
0.2
~ 2.3
1
8
3
g x 2log x
Logarithm and Exponential Forms
5 = log2(32)
25 = 32
Logarithm Form
Exponential Form
Base Stays the
BaseLogs Give you
Exponents
Input Becomes
Output
Examples
Write each equation in exponential form
1.log125(25) = 2/3
2.Log8(x) = 1/3
Write each equation in logarithmic form
1.If 64 = 43
2.If 1/27 = 3x
1252/3 = 25
81/3 = x
log4(64) = 3
Log3(1/27) = x
Original Inverse
Inverse of an Exponential Equation
Log’s give you exponents!
2xy 2yx
2logy xOR
The logarithm base a of b is the exponent you put on a to get b:
i.e. Logs give you exponents!
Definition of Logarithm
if and only
lo
if
ga
x
b x
a b
a > 0
and
b > 0
6-71: Closure
7
3
75
1.210
32
log 49
log 81
log 5
log 10
log 2w
2
4
7
1.2
w + 3
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)
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 y=log(x).
• Homework Check• Logarithms and Graphs Packet (Extra)• Wells Time• 6-93 and 6-95 (pgs 288-289)• ClosureHomework: 6-96 to 6-105 (pgs 290-291)
6-83: Learning LogTitle: 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 x=0 (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 0<b<1?• What does the graph look like when b=1?• What does the graph look like when b>1?
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 doesn’t have the subscript 10 because it stands for the common logarithm:
log10100 = log100
Logarithmic Function
logby xParent Equation
Graphing Form
logby a x h k
Example: Exponential
log 3 2y x y = 2
x = 3
Transformation: Shift the parent graph three units to the right and two units up.
New Equation:
Transformation: