Transcript Rational expressions, inverses, and logs power point

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)
Simplifying Rational Expressions
Simplify the following expressions:
16 x
16 x
1
x 3
x 3
x
x
x 5
x 1

1
2
x
2
x
 1 
x
3
x 2
x 2

x
3
x
3
x 5
x 1
1 
x 5
x 1
1
Simplifying Rational Expressions
A fellow student simplifies the following
expressions:
4x
x
 4 1  4
4 x
x
 4 1  5
Which simplification is correct? Substitute
two values of x into each to justify your
answer.
MUST BE MUITLIPLICATION!
Simplifying Rational Expressions
Simplify:
2 x  3x  20
4 x3  64 x
2
 2 x  5  x  4 
4 x  x  16 
Can NOT cancel since
everything does not have a
common factor and its not in
factored form
Factor Completely
2
 2 x  5 x  4
4 x  x  4  x  4 
CAN cancel since the
top and bottom have a
common factor
2x  5
2x  5
or
4x  x  4
4 x 2  16 x
Multiplying and Dividing Fractions
Multiply:
a
b
Divide:
w
x
 
c
d
Multiply by the
reciprocal (flip)
 
y
z
w
x
ac
bd
Multiply
Numerators
Multiply
Denominators
 
z
y
Remember to Simplify!
w z
x y
Multiplying/Dividing Rational
Expressions
 x  2   x  7  3x  8
2
Simplify:
x3
x2
Half the work is done!
 x  2   x  7  3x  8 
 x  3 x  2 
2
 x  2 x  2 x  7 3x  8
 x  3 x  2
 x  2  x  7  3x  8
x3
Combine
Rewrite
Cancel
Multiplying/Dividing Rational
Expressions
3x  15 3x  15 x  18
 2
2
25  x
x  3x  10
2
Simplify:
3x  15 x  3x  10 Flip to turn it into a
multiplication
 2
2
25  x 3x  15 x  18
3  x  5
x  5  x  2 


Factor
2
 5  x  5  x  3  x  5 x  6 
2
3  x  5
x  5 x  2


  x  5 5  x  3  x  2 x  3
1

x3
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
Subtraction:
Addition:
2
3
2
3

1
5
  
5
5
10
15

13
15
1 3
5 3
3
15
Common
Denominator
Add the
Numerators
7
4
7
4

3
10
  

5
5
35
20
Least
Common
Denominator
(if you can
find it)
3 2
10 2
6
20 Subtract the
29
20
Numerators
Remember to Simplify if Possible!
Add/Subtract Rational Expressions
Simplify:
2x 1
x4
 2
2
x  2 x  15 x  2 x  15
 2 x  1   x  4 
x 2  2 x  15
2x 1  x  4
x 2  2 x  15
x 5
x 2  2 x  15
x 5
 x  5 x  3
1
x3
Same denominator!
Half the work is done!
CAREFUL with
subtraction!
Combine Like
Terms
Make sure it can’t be
simplified more
Add/Subtract Rational Expressions
Simplify:
7 11

2
2x
x
Find a Common
Denominator
7 11 2 x
 
2
2x
x 2x
7
22 x
 2
2
2x
2x
7  22 x
2 x2
Combine
Like Terms
Add/Subtract Rational Expressions
Simplify:
3
5

x3 x 2
Find a common
denominator
3  x  2
5  x  3



x  3  x  2 x  2  x  3
Distribute numerators
3x  6
5 x  15

but leave the
 x  3 x  2  x  2 x  3 denominators factored
 3x  6   5x  15
 x  3 x  2
CAREFUL with
subtraction
3x  6  5 x  15
 x  3 x  2
2 x  21
Combine like Terms
 x  3 x  2 
Add/Subtract Rational Expressions
Simplify:
8x
5

2
6 x  2 x 3x  1
8x
5

2 x  3x  1 3x  1
2x  4
5

2 x  3x  1 3x  1
4
5

3x  1 3x  1
45
3x  1
1
3x  1
Factor to find a Smaller
Common Denominator
Make sure it can’t be
simplified beforehand
Add/Subtract Rational Expressions
Simplify:
x2
x

2
x  16 3 x  12
Factor to find a Smaller
Common Denominator
x2
x

 x  4  x  4  3  x  4 
x  4

x2
3
x
 

 x  4 x  4 3 3 x  4  x  4
3x  6
x2  4 x

3  x  4  x  4  3  x  4  x  4 
 3x  6    x 2  4 x 
Make sure it can’t
be simplified more
3  x  4  x  4 
x  3 x  2 

x2  x  6

3  x  4  x  4 
3  x  4  x  4 
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)
• Closure
Homework: 6-7 (pg 265) AND 6-26 to 6-32 (pgs 270)
Project Due: Wednesday, April 6th (Rubric)
Guess my Number
I’m thinking of a number that…
When I…
I get…
My number is…
• Add four to my number
AND
• Multiply by ten
-70
-11
• Double my number
• Add four
AND
• Divide by two
Five
Three
• Square my number
• Add three
• Divide by two
AND
• Add one
Seven
Three
3 and
and…
-3
• Double my number
• Subtract six
• Take the square root
Eleven
Eight
Inverse Notation
f
f  x
Original
function
 x
Inverse
function
1
“Undo” Rule
px  2x  3
3
1st Step
2nd Step
3rd Step
Start
p(x)

Add 3
p -1 (x)
Divide 2
Cube
Multiply 2
Cube Root Subtract 3
x 
1
p x   3   3
2 
Only works
when there
is one x!
Tables and Graphs of Inverses
Switch
x and y
Orginal
(0,25)
(20,25)
(2,16)
(18,16)
(6,4)
(14,4)
(10,0)
Function
X
0
2
6
10
14
18
20
Y
25
16
4
0
4
16
25
X
25
16
4
0
4
16
25
Y
0
2
6
10
14
18
20
Inverse
Switch
x and y
(16,18)
(4,14)
(0,10)
(4,6)
(16,2)
Non-Function
Line of Symmetry: y = x
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)
• Closure
Homework: 6-33 to 6-37 (pgs 271) AND 6-44 to 6-53 (pgs 274277)
Project Due: Wednesday, April 6th (Rubric)
The Rule for an Inverse
p  x   3 x  2  6
2
1st Step
2nd Step
3rd Step
4th Step
Start
p(x)
p -1
(x)
Add 2
Add 6
Square
Divide 3
Multiply 3 Subtract 6
Square
Root
±
x6
p  x  
2
3
1
Subtract 2
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:
f  x  x
x0
2
ff fxxx xx
11 1
Inverse
Inverse
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
Find the inverse of the following:
Switch x and y
y  6 x 11
x  6 y 11
x  11  6 y
x 11
y 6
x 11

y
6
Solve for y
Do not write y-1
Make sure to check with a table and graph on the calculator.
Algebraically Finding an Inverse
Find the inverse of the following: y  2  x  7   3
2
Switch x and y
x  2  y  7  3
2
Solve for y
x  3  2  y  7
x 3
2


x 3
2
x 3
2
  y  7
 y7
y
2
2
2
Because x =9 has
two solutions: 3 & -3
x 3
2
7
Do not write y-1
7  y
Make sure to check with a table and graph on the calculator.
Algebraically Finding an Inverse
Find the inverse of the following: e  x  
 x 10 3
4
Switch x and y
y  10 

x
Really y =
3
4
Solve for y
3
4 x   y  10 
3
3
e
1
 x 
3
4 x  10
4x  y 10
4x  10  y
Make sure to check with a table and graph on the calculator.
Algebraically Finding an Inverse
Only Half Parabola
Find the inverse of the following: d  x   4 x  3
Switch x and y
x  4 y 3
Really y =
Solve for y
x 3  4 y
Full Parabola
x 3
 y (too much)
4
2
 x 3

 y
 4 
x=3
Restrict the
Domain!
 x 3
d  x  

 4 
2
1
when x  3
Make sure to check with a table and graph on the calculator.
Composition of Functions
Substituting a function or it’s value into another
function.
Second
f
 g  x 
g
f
First
(inside parentheses
always first)
f g  x
OR
Composition of Functions
Let f  x   2x  3 and g  x   x2  5 . Find:
Our text uses the
first one
f
 g 1  f
g 1   1  5
2
Plug x=1
into g(x)
first
g 1
Equivalent
Statements
f  4  2  4  3
g 1   1  5
f  4  8  3
g 1  4
f  4  11
f  g 1  11
Plug the
result into
f(x) last
Composition of Functions
Let f  x   2x  3 and g  x   x2  5 . Find:
g  f  x 
Plug the result into g(x) last
g  2 x  3    2 x  3  5
g  2x  3    2x  3 2x  3  5
g  2 x  3    4 x 2  12 x  9   5
g  2x  3  4x2 12x  9  5
2
g  2x  3  4x 12x  4
2
Plug x into
f(x) first
f  x   2x  3
g  f  x   4x2 12x  4
Inverse and Compositions
In order for two functions to be inverses:
f
 g  x   x
AND
g  f  x   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)
• 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)
Silent Board Game
x8
g  x
32
1
2
1
4 3 64
16
3 5 1 0 4 2
x2 0
g  x
1
0.25
1
 2 
1.6
2 0.2
1
2
~ 2.3
g  x   log2  x 
1
8
6
3
Silent Board Game
x 1 0
g  x
1
8
0.2
  3
~ 2.3
1
0.25
2
1
 2 1 0
x2 3 4 8
g  x
1
1.6
2
1
2
16 32 64
2 3 4 5 6
g  x   log2  x 
Logarithm and Exponential Forms
Logarithm Form
5 = log2(32)
Logs Give
you
Exponents
Input
Becomes
Output
Base
Stays the
Base
5
2 = 32
Exponential Form
Examples
Write each equation in exponential form
1.log125(25) = 2/3
1252/3 = 25
2.Log8(x) = 1/3
81/3 = x
Write each equation in logarithmic form
3
1.If 64 = 4
log4(64) = 3
2.If 1/27 = 3x
Log3(1/27) = x
Inverse of an Exponential Equation
Original
Inverse
y2
x2
x
OR
y
y  log2  x 
Log’s give you exponents!
Definition of Logarithm
The logarithm base a of b is the
exponent you put on a to get b:
log a b  x
a>0
if and only if
and
a b
b>0
x
i.e. Logs give you exponents!
6-71: Closure
log 7  49   2
log 3  81  4
  7
10   1.2
2   w + 3
7
log 5 5
log10
log 2
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)
• Closure
Homework: 6-96 to 6-105 (pgs 290-291)
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 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
Parent Equation
y  logb  x 
Graphing Form
y  a  logb  x  h  k
Example: Exponential
Transformation: Shift the parent graph three
units to the right and two units up.
New Equation:
y=2
y  log  x  3  2
x=3