Honors Algebra 2
Spring 2012
Ms. Katz
Day 1: January 30th
Objective: Form and meet study teams. Then work together
to share mathematical ideas and to justify strategies as you
represent geometric objects and order a series of
connected functions to create a desired output.
• Seats
• Problems 1-1 to 1-2
• Introduction: Ms. Katz, Books, Syllabus, Index Card
Homework Record, Expectations
• Conclusion
Homework: Have Parent/Guardian fill out last page of syllabus
and sign; Problems 1-4 to 1-9 AND 1-13 to 1-19; Extra
credit tissues or hand sanitizer (1)
Five Point Star
Function Notation
The f is the name of the function machine,
and the expression to the right of the
equal sign shows what the machine does
to any input.
Function Notation
The f is the name of the function machine and the
value inside the parentheses is the input. The
expression to the right of the equal sign shows
what the machine does to the input.
25
Which do you prefer to write?
f  25
Evaluate f when
OR
x = 25?
5
Function Machines (a)
f x   x
6
gx   x  2
2
x
hx  2  7 k x    1
2

x
-16

k x   
x
1
2

7 

gx   x  2

hx  2x  7
121

f x   x
11
2
Function Machines (b)
f x   x
64
f x   x
x
hx  2  7 k x    1
2

x
8

gx   x  2
2

-36 

k x   

gx   x  2

x
1
2
17
hx  2x  7
131065
2
Support
• www.cpm.org
–
–
–
–
Resources (including worksheets from class)
Extra support/practice
Parent Guide
Homework Help
• www.hotmath.com
– All the problems from the book
– Homework help and answers
• My Webpage on the HHS website
– Classwork and Homework Assignments
– Worksheets
– Extra Resources
Respond on Index Card:
1.
2.
3.
4.
When did you take Algebra 1? Geometry?
Who was your Algebra 1 teacher? Geometry teacher?
What grade do you think you earned in Geometry?
What is one concept/topic from Algebra 1 that Ms. Katz
could help you learn better?
5. What grade would you like to earn in Algebra 2?
(Be realistic)
6. What sports/clubs are you involved in this Spring?
7. My e-mail address (for teacher purposes only) is:
Day 2: January 31st
Objective: Review expectations for class and homework.
Work together to share mathematical ideas and to justify
strategies as you order a series of connected functions to
create a desired output. THEN Draw complete graphs of
functions and identify possible inputs, outputs, and key
points for describing those graphs. You will use a graphing
calculator and develop presentation skills.
•
•
•
•
HW Check and Correct (in red)
Problems 1-10 and 1-12
Problem 1-27
Conclusion
Homework:
Problems 1-20 to 1-26; GET SUPPLIES; Extra
credit tissues or hand sanitizer (1)
Complete Graph
When a problem says graph an equation or
draw a graph:
y
On graph paper:
Plot key points
accurately
(-2,0)
(3,0)
x
Scale your axes
appropriately
(0,-6)
(.5,-6.25)
Label the axes
(with units if
appropriate)
Day 3: February 1st
Objective: Identify the domain and range of functions while
improving your graphing-calculator skills. THEN Find points
of intersection using multiple representations and learn how
to use the [CALC], [TABLE], and [TBLSET] functions on a
graphing calculator.
•
•
•
•
•
HW Check and Correct (in red)
Problems 1-28 to 1-34
Notes
Problems 1-42 and 1-46
Conclusion
Homework:
Problems 1-35 to 1-41 AND 1-47 to 1-53; GET
SUPPLIES; Extra credit tissues or hand sanitizer
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.
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.
Complete Graph
When a problem says graph an equation or
draw a graph:
y
On graph paper:
Plot key points
accurately
(-2,0)
(3,0)
x
Scale your axes
appropriately
(0,-6)
(.5,-6.25)
Label the axes
(with units if
appropriate)
Definitions
Domain
All possible input values (usually x), which
allow the function to work.
Range
All possible output values (usually y),
which result from using the function.
The domain and range help determine the window of a graph.
1-34: Learning Log
Title: Domain and Range
• Describe everything you know about
domain and range.
• Why are the domain and range important
when graphing?
• What calculator buttons allow us to see
the appropriate domain and range of a
graph?
Symbols for Number Set
Natural Numbers: Counting numbers
(maybe 0, 1, 2, 3, 4, and so on)
Integers: Positive and negative counting
numbers (-2, -1, 0, 1, 2, and so on)
Rational Numbers: a number that can be
expressed as an integer fraction
(-3/2, -1/3, 0, 1, 55/7, 22, and so on)
Irrational Numbers: a number that can
NOT be expressed as an integer
fraction (π, √2, and so on)
NONE
Symbols for Number Set
Real Numbers: The set of all rational
and irrational numbers
Real Number Venn
Diagram:
Rational Numbers
Integers
Irrational
Numbers
Natural Numbers
Inequality Notation
Open Dot
and
Parentheses ( )
Closed Dot
and
Brackets [ ]
<
>
≤
≥
Less than (not included)
Greater than (not included)
less than or equal (included)
greater than or equal (included)
Example: Inequalities
Graphically and algebraically represent the following:
All real numbers greater than 11
Graph:
10
Symbolic:
x  11
11
12
OR
11, 
Example: Inequalities
Describe and algebraically represent the following:
-6
-5
-4
Description: All real numbers less than or equal
to -5
Symbolic:
x  5
OR
 , 5
Example: Inequalities
Describe and graphically represent the following:
1 x  5
1,5
OR
Description: All real numbers greater than or
equal to 1 and less than 5
Graph:
1
3
5
Example: Inequalities
Graphically and algebraically represent the following:
All real numbers less than -2 or greater than 4
Graph:
-2
Symbolic:
1
4
x  2 or x  4
OR
 , 2 or  4, 
Day 4: February 2nd
Objective: Find points of intersection using multiple
representations and learn how to use the [CALC], [TABLE],
and [TBLSET] functions on a graphing calculator. THEN
Investigate a function defined by a geometric relationship
and generate multiple algebraic representations for the
function.
•
•
•
•
•
HW Check and Correct (in red)
Wrap-Up Notes
Problems 1-42 and 1-46
Problems 1-54 to 1-58
Conclusion
Homework:
Problems 1-60 to 1-71; Get Supplies!
Team Test Tuesday (?)
Inequality Notation
Open Dot
and
Parentheses ( )
Closed Dot
and
Brackets [ ]
<
>
≤
≥
Less than (not included)
Greater than (not included)
less than or equal (included)
greater than or equal (included)
Multiple Representations
NonAlgebraic
Table
Rule or
Equation
Graph
Context
Algebraic
Solving a System Algebraically
Use the equations to
solve the following
system:
f  x   2 x  5x  6
2
g  x   2 x  x  30
2 x 2  5 x  6  2 x 2  x  30
4 x 2  4 x  24  0
x2  x  6  0
 x  3 x  2  0
x  3 x  2
2
f  3  2  3  5  3  6
f  3  9
2
f  2   2  2   5  2   6
f  2   24
2
3,9 and  2, 24
Using a Table to Solve a System
Use tables to solve the
following system:
f  x   2 x  5x  6
2
g  x   2 x  x  30
2
X
Y
X
Y
-3
39
-3
15
-2
24
-2
24
-1
13
-1
29
0
6
0
30
1
3
1
27
2
4
2
20
3
9
3
9
4
18
4
-6
5
31
5
-25
Day 5: February 3rd
Objective: Investigate a function defined by a geometric
relationship and generate multiple algebraic representations for
the function. THEN Develop an understanding of what it
means to investigate a function as the family of hyperbolas is
investigated.
•
•
•
•
•
HW Check and Correct (in red)
Finish Problems 1-57 to 1-58
Problems 1-78 to 1-83
Start Problems 1-99 to 1-104
Conclusion – [Project will be assigned next week]
Homework:
Problems 1-72 to 1-77 AND 1-84 to 1-90; Supplies!
Team Test Monday? Tuesday?
Domain and Range
y   x  1 x  9
Domain: All ℝ
Range: y  25
Domain: 8  x  9
Range: 7  y  8
Day 6: February 6th
Objective: Develop an understanding of what it means to
investigate a function as the family of hyperbolas is
investigated. THEN Identify what all linear functions have in
common and determine whether relationships in tables and
situations are linear.
•
•
•
•
•
•
HW Check and Correct (in red)
Finish Problems 1-78 to 1-83
Problems 1-99 to 1-104
Assign Project and Review Rubric
Start notes on Exponents if time
Conclusion
Homework:
Problems 1-91 to 1-98 AND 1-105 to 1-111
Ch. 1 Team Test Tomorrow
Ch. 1 Individual Test Friday
Function Investigation Questions
• What is the domain of the function? What
is the range?
• Does the function have symmetry?
• What are the important/key points of this
function? Why are they important?
• What is the shape of the graph?
• Does the function have any “problem
points” or asymptotes? Why do they
happen?
Hyperbola
1
y
x2
What to address:
– Domain and Range
– Key Points (max/min,
intercepts, etc)
– Asymptotes (a line that
the graph of a curve
approaches)
– Symmetry
x -6
-1
0
1
1.5
1.75
1.9
1.99
2
2.01
2.1
2.25
2.5
3
4
5
10
y -.125
-.33
-.5
-1
-2
-4
-10
-100
Ǿ
100
10
4
2
1
.5
.33
.125
Parameter vs. Variable
Variable
(Multiple Values/Vary)
ym
mxx  b
Parameter
(Specific/Constant)
Day 7: February 7th
Objective: Assess Chapter 1 in a team setting. THEN Identify
what all linear functions have in common and determine
whether relationships in tables and situations are linear.
•
•
•
•
•
•
HW Check and Correct (in red)
Chapter 1 Team Test (≤ 50 minutes)
Finish Problems 1-99 to 1-104
Review Project Rubric
Start notes on Exponents if time
Conclusion
Homework:
Problems 1-113 to 1-119 AND CL1-120 to CL1-124
Ch. 1 Individual Test Friday
Day 8: February 8th
Objective: Identify what all linear functions have in common and
determine whether relationships in tables and situations are
linear. THEN Explore, state, and practice the rules for
simplifying exponential expressions.
•
•
•
•
•
HW Check and Correct (in red)
Finish Problems 1-99 to 1-104
Notes on Exponents
Practice – “Exponent Mania”
Conclusion
Homework:
Problems CL1-125 to 1-129 AND Exponent Mania
Ch. 1 Individual Test Friday
Problem 1-112(b) Due Monday, February 13th
1-104: Learning Log
Title: Recognizing Linear Relationships
• How do you recognize a linear relationship
without a graph?
• How can you recognize a linear equation?
• How do you recognize a linear table?
• How do you recognize linear situation?
• What must the rate of change be for every
relationship?
Exponential Notation
Exponent
Base
Base raised to an exponent
Goal
To write simplified statements that contain
distinct bases, one whole number in the
numerator and one in the denominator,
and no negative exponents.
Ex:
9 a b
1

4 3 2
 6a b

2 1 2 2
c
8 4
9b c

12
4a
Exploration
Evaluate the following without a calculator:
34 = 81
33 = 27
32 =
9
31 = 3
Describe a pattern and find the answer for:
30 = 1
Zero Power
0
a =1
Anything to
the zero
power is one
Can “a” equal zero?
No.
You can’t divide by 0.
Exploration
Simplify:
=
x
7
Product of a Power
If you multiply
powers having the
same base, add the
exponents.
a
mn
Example
Simplify:
=
11
3x
Exploration
Simplify:
x


5
3
 x
15
Power of a Power
To find a power of a
power, multiply the
exponents.
a
mn
Example
Simplify:
2ss
 t  4t =
6
2
3
3
2
13 11
8s t
Exploration
Simplify:
z x
2
5
z x
10
5
Power of a Product
a b
m
If a base has a
product, raise each
factor to the power
m
Example
Simplify:
3x  2xy  =
2
5
4
7 20
-288x y
Exploration
Complete
the tables
(with
fractions)
by finding
the pattern.
55
3125
54
625
53
125
52
25
51
5
50
1
5-1
1/5
5-2
1/25
1/125
1/625
5-3
5-4
1
25
1
24
1
23
1
22
1
21
1
20
1
2 1
1
2 2
1
2 3
1
2 4
1/32
1/16
1/8
¼
½
1
2
4
8
16
A Negative Exponent
A simplified
expression has
no negative
exponents.
1
m
a
1
m

a
m
a
Example
Simplify:
2
12x y
3
8x
1
5
3x

2y
Exploration
Simplify:
=
Quotient of a Power
a
To find a quotient of
a power, subtract
the bottom exponent
from the top if the
bases are the same.
mn
a0
Example
Simplify:
6
2x y
x

2 3
2
6x y
3y
4
Exploration
Simplify:
6
a
a

 
6
b
b
 
6
Power of a Quotient
m
a
m
b
To find a power of a quotient,
raise the denominator and
numerator to the same power.
Example
Simplify:
2
3 
 
y 
2x y 
  5  
 x 
2
7
3
23
8y
9
9x
Day 9: February 9th
Objective: Represent exponential growth with a diagram, table,
equation, and graph. Write equations based on the patterns in
tables, recognize patterns of exponential growth, and use
equations to make predictions.
• HW Check and Correct (in red)
• Problems 2-1 to 2-5
• Conclusion/Notes
Homework:
Problems 2-6 to 2-12 (finish thru 2-19 if you don’t
want weekend HW aside from the project)
Ch. 1 Individual Test TOMORROW
Problem 1-112(b) Due Monday, February 13th
Multiplying Like Bunnies
Case 1: Start with 2 rabbits; each pair has 2
babies per month
Month TOTAL
0
1
2
3
2
4
8
16
Babies
born this
month
2
4
8
…
Multiplying Like Bunnies
Case 2: Start with 10 rabbits; each pair has 2
babies per month
Month TOTAL
0
1
2
3
10
20
40
80
Babies
born this
month
10
20
40
…
Multiplying Like Bunnies
Case 3: Start with 2 rabbits; each pair has 4
babies per month
Month TOTAL
0
1
2
3
2
6
18
54
Babies
born this
month
4
12
36
…
Multiplying Like Bunnies
Case 4: Start with 2 rabbits; each pair has 6
babies per month
Month TOTAL
0
1
2
3
2
8
32
128
Babies
born this
month
6
24
96
…
Day 10: February 10th
Objective: Assess Chapter 1 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Calculator Portion…put calculator away when finished
• Second: Non-Calculator Portion (ask for it)
• Third: Correct last night’s homework
Homework:
Finish your project!
Problems 2-13 to 2-19
Individual Take Home: Problem 1-112(b) Due Monday!
Make sure the rubric is already attached by 3rd block!
Day 11: February 13th
Objective: Generate data and model the data with tables, rules,
and graphs. Calculate the rebound ratio when a ball bounces.
THEN Introduce an example of exponential decay.
•
•
•
•
•
•
HW Check and Correct (in red)
Hand in Project with Rubric Attached!
Wrap-Up Lesson 2.1.1 (LL Stuff)
Problems 2-21 to 2-23
Problems 2-30 to 2-35
Conclusion
Homework:
Problems 2-24 to 2-29 AND 2-36 to 2-41
Exponential Table
There is a constant
multiplier
between
consecutive
output values.
X
Y
-1
5.67
x3
0
17
x3
1
51
x3
2
3
153
459
x3
Exponential Graph
Notes:
Horizontal
Asymptote
Recursive Formula
A formula that requires the previous terms
in order to find the value of the next term.
Example: 2, 4, 8, 16, …
Recursive Formula:
Doubles
Explicit Formula
A formula that requires the number of the
term in order to find the value of the next
term.
Example: 2, 4, 8, 16, …
Plug in with
Parentheses
in the
calculator!
Explicit Formulas:
Initial
Rate
y  2  2
x
Month
y   2
 x 1
Exponential Equation
A function whose input (x) is located in the
exponent.
Rate
y  a b
Initial
x
Example: Jason has $17 and quadruples his money every
month. Write an equation to represent the situation.
y  17  4 
x
2-5: Learning Log
Title: Exponential Functions
• Describe everything you know about
exponential functions.
• What operation must be in the equation?
• What do their graphs look like?
• What patterns are in the tables?
• Draw examples of all the representations.
Rebound Ratio
Rebound Height
Starting Height
Continuous Graph
The points of the
graph are
connected.
Therefore, there
are no holes or
breaks in it.
Discrete Graph
The graph is
made up of
separate points.
Starting Height vs. Rebound Height
Rebound Height
Rebound Height
y = mx + b
Starting Height
Rebound
Starting
Ratio
Height
Day 12: February 14th
Objective: Introduce an example of exponential decay. THEN
Introduce sequences and sort them into groups based on
patterns in their representations. Also, identify sequences
generated by adding a constant as arithmetic, and those
generated by multiplying a constant as geometric.
•
•
•
•
HW Check and Correct (in red)
Problems 2-30 to 2-35
Problems 2-42, 2-43, and 2-45
Conclusion
Homework:
Problems 2-46 to 2-60
Reminders/Notes
• You are responsible for content in Math
Notes boxes – make sure you review them.
• There is a non-calculator portion to every
test. Work on your pacing.
• All tests are cumulative.
• Sometimes homework problems introduce
topics that won’t be taught in class.
• You get out of this class what you put in –
make sure you are doing your part.
• If you need help, please see me before it’s
too late. As Mellor’s sign states, “TODAY is
the day to worry about your grade.”
Bounce vs. Rebound Height
Rebound Height
Rebound Height
Bounce
Number
y = ab
Bounce Number
Discrete!
Starting
Height
x
Rebound
Ratio
Exponential Function
y  200  r 
n
Where r is the rebound ratio, n is the
bounce number, and y is the height of
the ball after the nth bounce.
Summary of Bounce Labs
Lesson 2.1.2:
The height of a ball’s rebound grows constantly as
the drop height grows, so it makes sense that
this would be a linear model.
Lesson 2.1.3:
The height of each bounce is a constant multiple of
its previous height, so it makes sense that, if left
to bounce repeatedly, the ball’s height would
shrink exponentially.
Day 13: February 15th
Objective: Introduce sequences and sort them into groups
based on patterns in their representations. Also, identify
sequences generated by adding a constant as arithmetic,
and those generated by multiplying a constant as
geometric. THEN Learn the vocabulary and notation for
arithmetic sequences as formulas for the nth term are
developed.
•
•
•
•
•
HW Check and Correct (in red)
Finish Problems 2-43, and 2-45
Problems 2-61 to 2-70
Start Problem 2-78 if time
Conclusion
Homework:
Problems 2-71 to 2-77
Sequences vs. Functions
Sequence: t(n)
Domain (n) = Positive Integers (sometimes 0)
Range (t(n)) = Can be all Real numbers
The Graph is Discrete
Function: f(x)
Domain (x) = Can be all Real numbers
Range (f(x))= Can be all Real numbers
The Graph is Continuous
Arithmetic Sequences
A sequence which has a constant difference
between terms. The rule is linear.
Example: 1, 4, 7, 10, 13,…
n
t(n)
0
1
1
4
2
7
3
10
4
13
+3
+3
+3
+3
(generator is +3)
Discrete
t  n   3n  1
Geometric Sequences
A sequence which has a constant ratio
between terms. The rule is exponential.
Example: 4, 8, 16, 32, 64, …
n
t(n)
0
4
1
8
2
16
3
32
4
64
x2
x2
x2
x2
(generator is x2)
Discrete
t  n   4  2
n
Day 14: February 16th
Objective: Use geometric sequences to solve problems involving
percent increase and decrease. Also, identify multipliers both
to classify the sequences as geometric and to write equations
for those sequences. THEN Recognize that sequences are
functions with domains limited to non-negative integers. Use
Guess and Check or graphical methods to solve exponential
equations.
•
•
•
•
•
•
HW Check and Correct (in red)
Review Problems 2-62 to 2-68
Review Chapter 1 Individual Test (& general comments)
Problems 2-78 to 2-82
Problems 2-92 to 2-97
Conclusion
Homework:
Problems 2-86 to 2-91 AND 2-98 to 2-105
Day 15: February 17th
Objective: Use geometric sequences to solve problems involving
percent increase and decrease. Also, identify multipliers both
to classify the sequences as geometric and to write equations
for those sequences. THEN Recognize that sequences are
functions with domains limited to non-negative integers. Use
Guess and Check or graphical methods to solve exponential
equations.
•
•
•
•
HW Check and Correct (in red)
Problems 2-78 to 2-82
Problems 2-92 to 2-97
Conclusion
Homework:
Problems 2-106 to 2-108 AND 2-130 to 2-134 AND
Revisit other homework problems that you
may have had troubles with – slow down,
redo and regroup for Monday.
Generator for a Percent Increase
What is a 15% increase of 100?
First Step:
.
.
= 0.15
.15.%
2 1
Second Step:
100  100  0.15  100 1  0.15  100 1.15
Multiplier
Generator for a Percent Decrease
Example: What is the multiplier for a 17%
decrease?
First Step:
.
.
= 0.17
.17.%
2 1
Second Step:
1  0.17  0.83
Multiplier
πPod Problem
Week
Sales
0 100 = 100.1.150
1 115 = 100.1.15 = 100.1.151
2 132.25 = 100.1.15.1.15 = 100.1.152
3 152.09 = 100.1.15.1.15.1.15 = 100.1.153
4 174.9 = 100.1.15.1.15.1.15.1.15 = 100.1.154
n
100.1.15n
Day 16: February 21st
Objective: Recognize that sequences are functions with domains
limited to non-negative integers. Use Guess and Check or
graphical methods to solve exponential equations. THEN Write
rules for arithmetic and geometric sequences, identifying the
first term as term number one rather than term zero. THEN
Identify equivalent expressions and develop and share
algebraic strategies for demonstrating equivalence.
•
•
•
•
•
HW Check – Compare answers with teammates, please!
Finish Problems 2-92 to 2-97
Review Problems 2-106 to 2-108 [Do 2-109]
Problems 2-118 to 2-120
Conclusion
Homework:
Problems 2-110 to 2-117 AND 2-122 to 2-129
Chapter 2 Team Test Thursday
Exponential Function
Time
Initial
y  a b
x
Rate
Exponential Growth
A(t) = P ( 1 + r )
t
A(t): Amount as a function in terms of t
P: Principal (starting amount)
t: Time after starting point
r: Decimal increase (% ÷ 100)
Learning Log
Exponential Decay
A(t) = P ( 1 – r )
t
A(t): Amount as a function in terms of t
P: Principal (starting amount)
t: Time after starting point
r: Decimal decrease (% ÷ 100)
Learning Log
2-94: Learning Log
Title: Sequences vs. Functions
• Is a sequence a function?
• What makes a sequence different than
most functions?
• If a sequence and a function have the
same rule, how are they different?
• What are the restrictions on the domain of
a sequence?
Working Backwards for a Rule
Sequences start with n=1 now!
First find the generator and the n=0 term.
Then write the equation:
1
0
Ex 1:
2
3
4
40, 36, 32, 28, 24, …
–4
t  n   4n  40
0
3
 ,
Ex 2:
5
1
2
3
4
3, -15, 75, -375, …
x-5
3
n
t  n     5 
5
Day 17: February 22nd
Objective: Identify equivalent expressions and develop and share
algebraic strategies for demonstrating equivalence. THEN Use
an area model to multiply expressions. Factor expressions and
demonstrate equivalence. THEN Solve equations by first
rewriting them as simpler equivalent equations.
•
•
•
•
•
•
HW Check and Correct (in red)
Finish Problems 2-118 to 2-120
Review Problems 2-130 to 2-134
Problems 2-143 to 2-147
Hand Back and Review Projects
Conclusion
Homework:
Problems 2-135 to 2-142 AND 2-149 to 2-156
Chapter 2 Team Test Tomorrow
Properties
Distributive Property
a(b + c) = ab + ac
(a + b)(c + d) = ac + ad + bc + bd
Associative Property
Addition: a + (b + c) = (a + b) + c
Multiplication: a(bc) = (ab)c
Commutative Property
Addition: a + b = b + a
Multiplication: ab = ba
Two Butt Cheeks
When there is addition or subtraction:
 x  3
 x  3 x  3
2
x  3x  3x  9
2
x  6x  9
2
Day 18: February 23rd
Objective: Solve equations by first rewriting them as simpler
equivalent equations. THEN Assess Chapter 2 in a team
setting.
•
•
•
•
HW Check and Correct (in red)
Start Problems 2-143 to 2-147
Chapter 2 Team Test
Conclusion
Homework:
Problems CL2-157 to CL2-165
Chapter 2 Individual Test Tuesday
Day 19: February 24th
Objective: Solve equations by first rewriting them as simpler
equivalent equations. THEN Investigate the family of functions
y = bx. Make and justify statements about the behaviors of
graphs in this family.
•
•
•
•
HW Check and Correct (in red)
Work a little on Problems 2-144 to 2-147
Problems 3-1 to 3-6
Conclusion
Homework:
Problems 3-7 to 3-21
Chapter 2 Individual Test Tuesday
Solutions to Equations in 2-144
a. x = -5, 4
d. x = 2
b. x = -2, ½
e. x = 2 and y = -5
c. x = 1
f. x = -5 and y = 3
Begin to Investigate Exponentials
If the graph shows
nothing, try:
ZOOM – 4:ZDecimal
3-2 needs SKETCHES OF GRAPHS
and SUMMARY STATEMENTS
3-2: Possible Exponential Graphs
1  b  0
Answer the following questions:
• What is the shape of the graph? (Sketch)
• What are the domain and range?
• What are the intercepts? Or other key
points?
• Are there any points that don’t work? Or
asymptotes?
b0
0  b 1
b 1
b 1
Graphs of y=bx
bX
<0
x
y=(-2)x
-3
-.125
-2
.25
-1
-.5
0
1
1
-2
2
4
3
-8
4
16
Graphs of y=bx
b=0
0<b<1
x-axis where x>0
Decreasing, asymptote y=0
b>0
b=1
Horizontal line y=1
No x-intercepts.
The y-intercept
is (0,1).
There is never
a vertical
asymptote
b>1
Increasing, asymptote y=0
Day 20: February 27th
Objective: Deepen and extend our knowledge of exponential
functions by examining the relationships between different
representations of those functions. Generalize the roles of a
and b in y = a·bx. THEN Apply our knowledge of linear and
exponential functions to investigate the relationship between
simple and compound interest.
***NEW SEATS***
• HW Check and Correct (in red)
• Wrap-Up Problem 3-6
• Review Chapter 2 Team Test
• Problems 3-22, 23, 25
• Start Problems 3-34 to 3-38
• Exponential Graph Online
Homework:
Problems 3-26 to 3-33 AND STUDY!
Chapter 2 Individual Test Tomorrow
3-6: Learning Log
Title: Investigating y = bx
• What values of b are acceptable?
• What values of b are unacceptable? Why?
• How does changing the value of b affect a
graph?
• 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?
• What is the horizontal asymptote?
• Why is there no vertical asymptote?
Graphs of y=bx
0<b<1
Domain: All Reals
Decreasing, asymptote y=0
x-intercepts: None
Range: y  0
y-intercept: (0,1)
Vertical asymptote: None
For any Exponential:
b=1
Horizontal line y=1
b>0
b>1
Increasing, asymptote y=0
Range: y  1
Range: y  0
Exponential Function Web
Table
Rule or
Equation
Graph
Context
Graphs of y=abx
The initial value
f  x  1
b
f  x
x
-3
-2
-1
0
1
2

y  ab
x
The multiplier
y=a(b)x
xb
(0,a)
a
a is the starting point
Context:
b is the multiplier
Day 21: February 28th
Objective: Assess Chapter 2 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Calculator Portion…put calculator away when finished
• Second: Non-Calculator Portion (ask for it)
• Third: Check your work & hand the test in to Ms. Katz
• Fourth: Correct last night’s homework & start tonight’s
Homework:
Problems 3-39 to 3-47
Day 22: February 29th
Objective: Apply our knowledge of linear and exponential
functions to investigate the relationship between simple and
compound interest. THEN Represent exponential decay in
multiple ways and further investigate the effect when the
exponent is 0 or negative.
• HW Check and Correct (in red)
– Discuss 3-29 to 3-31 and 3-40
• Problems 3-34 to 3-38
• Exponential Graph Online
• Problems 3-48 to 3-52
Homework:
Problems 3-53 to 3-61
3-40: Tickets for a Concert
w
P(w)
-3
119.07
-2
128.60
-1
138.89
0
150
1
162
2
174.96
3
188.96
4
204.07
5
220.40
6
238.03
7
257.07
8
277.04
÷1.08
÷1.08
÷1.08
x1.08
x1.08
x1.08
x1.08
x1.08
x1.08
x1.08
x1.08
162
 1.08
150
P  w   150 1.08 
w
Don’t change the value of b in the
equation to calculate past outputs. Use
negative inputs:
P  13  150 1.08
 $55.15
13
Interest Example
Find an equation for the following context:
Fred invests $12,000 in an account that
offers 3.2% annual interest compounded
annually.
.
.0.3.2% = 0.032
2
The initial value
1
y  12000 1  0.032 
Decimal Rate
x
Interest Example
Find an equation for the following context:
Fred invests $12,000 in an account that offers
3.2% annual interest compounded semiannually.
.0.3.2% = 0.032
2
1
y  12000 1
The initial value

Decimal Rate
0.032 2 x
2
Number of
Intervals
Day 23: March 1st
Objective: Represent exponential decay in multiple ways and
further investigate the effect when the exponent is 0 or
negative.
• HW Check and Correct (in red)
• Wrap-Up Problems 3-34 to 3-38
• Problems 3-48 to 3-52
Homework:
Problems 3-62 AND 3-64 to 3-71
Interest
Simple:
A  P  Prt
Compound:
 r 
A  P1 
 n 
nt
P = Principal Amount (original)
r = rate ( % ÷ 100 )
t = time in years

n = number of intervals
Day 24: March 5th
Objective: Represent exponential decay in multiple ways and
further investigate the effect when the exponent is 0 or
negative. THEN Use what is known about exponential growth
to write equations for exponential functions presented as
graphs. THEN Complete the exponential multiplerepresentations web, solidifying connections between the table,
equation, graph, and context representations of an exponential
function.
•
•
•
•
•
HW Check and Correct (in red)
Review Chapter 2 Individual Test
Problems 3-48 to 3-52
Review Problems 3-62 to 3-63
Problems 3-72 to 3-77
Homework:
Problems 3-78 to 3-86 AND 3-87 to 3-88
Chapter 3 Team Test Thursday
Penny Lab/Half-Life
200
y  100  0.5 
x
x
y
-3
800
-2
-1
0
100
1
2
3
4
5
3
6
÷0.5
÷0.5
200
÷0.5
100
x0.5
50
x0.5
25
x0.5
12.5
x0.5
6.25
x0.5
3.125
x0.5
1.5625
400
3-63: Learning Log
Title: Graph → Rule for Exponential
Functions
Methods for creating an exponential rule
given a graph:
• The y-intercept for “a,” and if you have
consecutive terms divide the higher term
by the lower term to find “b”
• Making a table and then use guess and
check
Day 25: March 6th
Objective: Complete the exponential multiple-representations web,
solidifying connections between the table, equation, graph, and
context representations of an exponential function. THEN Find
equations of linear and exponential functions by using known
quantities to solve for a missing parameter. Also, interpret
fractional exponents. THEN Find linear and exponential equations
given two points. Also, evaluate roots with the calculator by
converting to fractional exponent notation.
•
•
•
•
•
HW Check and Correct (in red)
Continue Working on Problems 3-72 to 3-77
Review Problems 3-87 to 3-88
Problems 3-89 to 3-94
Problems 3-105 to 3-108
Homework:
Problems 3-95 to 3-104 AND 3-109 to 3-116
Chapter 3 Team Test Thursday
Exponential Function Web
Table
Rule or
Equation
Graph
Context
Example: 3-89
Find the equation of an exponential function with
an asymptote at y = 0 that passes through the
points (0,5) and (3,320).
x
a 5
y  5 b
320  5  b    5
3
64   b 
3
3
64  b
4b
y  5  4
x
Day 26: March 7th
Objective: Find linear and exponential equations given two points.
Also, evaluate roots with the calculator by converting to fractional
exponent notation. THEN Write and solve a system of exponential
functions in the context of investigating used-car prices.
•
•
•
•
HW Check and Correct (in red)
Wrap-Up Fractional Exponents (3 slides – LL)
Problems 3-105 to 3-108
Problems 3-117 to 3-120
Homework:
Problems 3-121 to 3-130 (and start closure?)
Chapter 3 Team Test Tomorrow
Radical Property
a b
ab
ONLY
when a≥0 and b≥0
Exponents into Radical Notation
b
 b
q
p/q
p
or
=
q
b 
p
Generally b≥0
Example
Evaluate the following without a
calculator:
64
5 6

6
   2
5
64
5
 32
System of Exponential Equations
Find an exponential function that passes
through (2,16) and (6,256). Substitute into either
Substitute into
y=abx twice
256  ab
6
÷ (16  ab
2
Divide #s
16  b
4
16  b
2b
4
Larger
exponent
first
)
Subtract
Exponents
Find the
Root
equation to find a
16  a  2 
16  a  4
4a
y  4  2
x
2
Day 27: March 8th
Objective: Assess Chapter 3 in a team setting. THEN
Collect non-linear data, fit an equation to the data, and
use the equation to make predictions.
•
•
•
•
HW Check and Correct (in red) Quickly!
Wrap-Up Problem 3-117
Chapter 3 Team Test
Start Problems 4-1 to 4-4
Homework: Problems CL3-131 to 3-137 AND 4-5 to 4-12
Chapter 3 Individual Test Thursday
By the End of the Chapter…
You will be able to easily sketch graphs
similar to the following by just looking at
the equations:
1
2
y  x  3 10
2
1
y
6
x 2
x 10  y  3  25
2
2
Shrinking Targets Lab
Radius (mm)
Weight (grams)
78
3.5
71
3.0
61
2.2
55
1.7
46.5
1.3
34
0.7
27
0.4
22
0.3
Day 28: March 9th
Objective: Collect non-linear data, fit an equation to the
data, and use the equation to make predictions. THEN
Connect transformations of parabolas with their
equations in graphing form.
• HW Check and Correct (in red) Quickly!
• Problems 4-1 to 4-4
• Problems 4-13 to 4-17
Homework: Problems 4-18 to 4-33
Chapter 3 Individual Test Thursday
Finish Closure (CL3-138 to 3-142) while
you study for Thursday’s test
[won’t be checked for points]
Modeling our Data
Which measurement of a circle directly
affects its weight?
Area
What is the equation for the area of a circle?
A  r
2
What is the name of the graph of the area of
a circle?
Quadratic/Parabola
What we know about Transforming y=x2
y=ax
2
The further the number you multiply by is from
zero, the steeper the parabola.
The closer the number you multiply by is to
zero, the wider the parabola.
Vertical Dilations
Transform!
Vertical
Vertical
Compression
Stretch
Day 29: March 12th
Objective: Connect transformations of parabolas with
their equations in graphing form.
• HW Check and Correct (in red) Quickly!
• Problems 4-13 to 4-17
• Conclusion
Homework: Problems CL3-138 to 3-142
 I changed my mind…get points tomorrow
Chapter 3 Individual Test Thursday
Day 30: March 13th
Objective: Graph quadratic equations without making
tables. Also, rewrite quadratic equations from standard
form into graphing form.
•
•
•
•
•
HW Check
Quick Look @ Team Tests
Wrap-Up Problems 4-16 to 4-17
Start Problems 4-34 to 4-38
Conclusion
Homework: Problems 4-39 to 4-45
Chapter 3 Individual Test Thursday
Graphing Form for a Parabola
y = a(x – h) + k
2
same
( h, k ): The Vertex
opposite
The value of a
Positive: Opens Up
If it Increases: Vertical Stretch
Negative: Opens Down If it Decreases: Vertical Compression
Example
Plot :
y = 2(x+3)
2 –3 – 5
Use
Find
the
Drawthe
thestretch
stretch
factor
to
vertex
Parabola
factor the
translate
“first points”
2
Day 31: March 14th
Objective: Graph quadratic equations without making
tables. Also, rewrite quadratic equations from standard
form into graphing form.
• HW Check and Correct (in red) Quickly!
• Problems 4-35 to 4-38
• Conclusion
Homework: Study like it’s your job!
Chapter 3 Individual Test Tomorrow
Graphing Form to Standard Form
y  4  x  1  7
y  4  x  1 x  1  7
2
y

4
x

x

x

1

7


Same
2
a! y  4  x  2 x  1  7
2
y  4 x  8x  4  7
2
y  4 x  8x  3
2
Day 32: March 15th
Objective: Assess Chapter 3 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Calculator Portion…put calculator away when finished
• Second: Non-Calculator Portion (ask for it)
• Third: Check your work & hand the test in to Ms. Katz
• Fourth: Correct last night’s homework
• Fifth: Work on Problems 4-37 and 4-38 and then re-do 4-40
Homework:
Problems 4-37 and 4-38 AND 4-52 to 4-58
Day 33: March 16th
Objective: Learn how to write quadratic equations for
situations using the graphing form of the parabola y = a(x
– h)2 + k. Specifically, develop an algebraic strategy for
finding the value of the stretch factor, a. THEN Transform
the graphs of y = bx, y = 1/x, y = √x, and y = x3.
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Review Problems 4-37 to 4-38
Problems 4-46 to 4-50
Start Problems 4-59 to 4-63
Conclusion
Homework: Problems 4-51 AND 4-64 to 4-70
Standard Form to Graphing Form
Use an algebraic method to write
2
y  2 x  4 x  30 in graphing form.
1. Find the value of a: 2
2. Find the x-intercepts
0  2 x  4 x  30
2
0  x  2 x  15
0   x  5 x  3
2
x  5  0 x 3  0
x  5 x  3
3. Average the x-intercepts for h
x
53
2

2
2
 1
4. Substitute h into the rule for k
y  2  1  4  1  30  32
2
5. Substitute a, h, k into the
graphing form
y  2  x  1  32
2
WARNING: This method does not work if there are no x-intercepts
Standard Form to Graphing Form
Use an algebraic method to write
2
y  2 x  4 x  30 in graphing form.
1. Find the value of a: 2
2. Find the x-intercepts
0  2 x2  4 x  30
0  x  2 x  15
2
2 
 2 2  41 15
21
x
2 64
x 2
x  5 x  3
3. Average the x-intercepts for h
x
53
2

2
2
 1
4. Substitute h into the rule for k
y  2  1  4  1  30  32
2
5. Substitute a, h, k into the
graphing form
y  2  x  1  32
2
WARNING: This method does not work if there are no x-intercepts
Finding the Equation of a Line
Find the equation of a line that passes
through the point (3,5) and has a slope of
2.
y  mx  b
y  2x  b
5  2  3  b
5 6b
1  b
y  2x 1
Finding a Quadratic Equation with the
Vertex and Another Point
A rabbit jumped over a 3ft-high fence. The highest point the rabbit
reached was 3 feet and it landed 8 feet from where it jumped.
Assume the rabbit follows a parabolic path. Sketch a graph and find
the equation for the height of the rabbit verses the horizontal
Since we know the vertex
distance it has traveled.
Sketch: One Possibility:
(4,3)
Substitute into y=a(x-h)2+k:
Plug in the vertex y
 a  x  4  3
2
0  a 0  4  3
Solve for a
2
0  a  4   3
0  16a  3
Plug in a,h,&k
3  16a
2
3
3

Equation: y   16  x  4   3
16  a
Plug in another point
Label all
the known
points
(0,0)
(8,0)
2
Different Forms For a Quadratic
Parent Graph: y = x2
Factored Form: y = __( __ )( __ )
Standard form: y = ax2 + bx + c
Graphing Form: y = a(x – h)2 + k
Vertex (locator point): ( h, k)
Vertical Compression: a  1
Open up: a  0
Vertical Stretch: a  1
Open down: a  0
The “a” is the same in standard and graphing form!
Day 34: March 19th
Objective: Transform the graphs of y = bx, y = 1/x, y = √x, and y =
x3. THEN Identify the point (h,k) for parabolas, hyperbolas,
cubics, and square root graphs, and relate the Point-Slope form
of a line to (h,k). Consolidate all of the understanding of parent
graphs and general equations in a toolkit. THEN Use our
knowledge of transformations to write a general equation for a
family of functions based on an absolute value parent graph.
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problems 4-59 to 4-63
PPT Examples on Transformations
Graphing Form Packet (Up through Linear)
Problems 4-99 to 4-101
Conclusion
Homework:
Problems 4-71 to 4-85
Function Transformations
Family
Parent
Graphing
Cubic
yx
y  ax  h   k
3
Calculator
3
 1 
 1 
1
y  a
 k y  a
Hyperbola y 
 k
x  h 
x  h 

x



Square
Root
y  x y a x h k y  a


x
Exponential y  b



xh 
y  ab

k
x  h  k
y  a b
 xh
k
Graphing Form
( h, k ): The Key Point
The value of a
Positive: Same Orientation
If it Increases: Vertical Stretch
Negative: Flipped
If it Decreases: Vertical Compression
Parent Graph: When a=1, h=0, and k=0
Quadratic
2
y  a  x  h  k
Hyperbola
y  a  x1 h   k
Exponential
 xh
y  ab
k
Cubic
3
y  a  x  h  k
Square Root
y  a xh k
Linear
y  a  x  h  k
Example: Quadratic
Transformation: Shift the parent graph three
units to the right and four units up.
y=4
New Equation:
(3,4)
y   x  3  4
2
x=3
Example: Cubic
Transformation: Flip the parent graph and shift
it five units up.
y=5
(0,5)
New Equation:
y    x  5
3
x=0
Example: Hyperbola
Transformation: Shift the parent graph four
units to the left and three units down.
New Equation:
(-4,3)
y = -3
x = -4
y
1
x4
3
Example: Square Root
Transformation: Shift the parent graph six units
to the left.
x = -6
New Equation:
y=0
(-6,0)
y  x6
Example: Exponential
Transformation: Shift the parent graph five units
to the right and two units up. Then stretch the
graph by a factor of 3.
a=3
y=2
(5,2)
x=5
New Equation:
y  3 2
x 5
2
Linear Function
Parent Equation
yx
Graphing Form
y  a  x  h  k
Point:
(h,k)
Slope:
a
Unless specified, you do not need to have the
answer in y=mx+b form!
Example: Linear
Transformation: A line with slope ½ that passes
through the point (-6,4).
Slope = ½
New Equation:
(-6,4)
y=4
y
1
2
 x  6  4
Point-Slope Form
y  ax  h  k
x = -6
Slope
Point
Day 35: March 20th
Objective: Use our knowledge of transformations to write a general
equation for a family of functions based on an absolute value
parent graph. THEN Use what we know about transforming
parabolas to make conjectures about transforming relations,
specifically sleeping parabolas and circles. Also, define the
meaning of a non-function (relation).
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problems 4-99 to 4-101 & Add to Graphing Forms Packet
Problems 4-112 to 4-117
Start Problems 4-128 to 4-134
Conclusion
Homework:
Problems 4-91 to 4-98 AND 4-103 to 4-111
Ch. 4 Team Test Thursday
Ch. 4 Individual Test next week
Absolute Value in a TI
1. Hit MATH
2. Go to the right once to NUM
3. Choose 1:abs(
Absolute Value Function
Parent Equation
y x
Graphing Form
y  a xh k
a) MATH
Absolute value can be found
in the calculator:
b) Right to NUM
c) 1. abs(
Example: Absolute Value
Transformation: Flip the parent graph and shift
it three units to the left and four units up.
(-3,4)
y=4
New Equation:
y   x3 4
x = -3
Sleeping Parabola
y x
2
Parent:
y x
Graphing Form:
Calculator:
y  a x  h  k
y1  a x  h  k
y2  a x  h  k
Circle
x  y  25
2
Parent:
2
y   25  x
Graphing Form:
2
y  a 25   x  h   k
2
y1  a 25   x  h   k
2
Calculator:
y2  a 25   x  h   k
2
Day 36: March 21st
Objective: Learn how to convert a parabola into graphing form by
completing the square. THEN Extend the idea of completing the
square to change circles written in standard form into graphing
form.
•
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Bell Ringer! (Next slide)
Review Ch. 3 Individual Test
Finish Problems 4-116 to 4-117
Problems 4-128 to 4-134
Start Problems 4-144 to 4-146
Conclusion
Homework:
Problems 4-119 to 4-127 AND 4-135 to 4-143
Ch. 4 Team Test Tomorrow [NO CALCULATORS!]
Ch. 4 Individual Test next week
Bell Ringer
Distribute the following:
1.
2.
3.
4.
y = (x – 2)2
y = (x + 3)2
y = (x + 4)2
y = (x – 6)2
1.
2.
3.
4.
Factor the following:
y = x2 + 8x + 16
y = x2 – 16x + 64
y = x2 + 20x + 100
y = x2 – 9x + 20.25
Is there a pattern when comparing a, b, and c when it is in
standard form vs factored form?
Equation for a Circle
Example
x  y  25
2
Center:
(0,0)
2
Radius:
25  5
Graphing Form
 x  h   y  k 
2
Center:
(h,k)
2
r
Radius:
2
r r
2
Example: Circle
Transformation: A circle centered at (4,-1)
whose radius is 4.
x=4
New Equation:
 x  4    y  1
2
y = -1
(4,-1)
Center:
Radius:
Is a circle a function? NO!
2
 16
(4,-1)
16  4
Day 37: March 22nd
Objective: Assess Chapter 4 in a team setting. THEN Learn how to
convert a parabola into graphing form by completing the square.
THEN Extend the idea of completing the square to change
circles written in standard form into graphing form.
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Chapter 4 Team Test – No Calculators
Finish Problems 4-131 to 4-134
Start Problems 4-144 to 4-146
Conclusion
Homework:
Problems 4-148 to 4-155
Ch. 4 Individual Test Next Friday
Day 38: March 23rd
Objective: Extend the idea of completing the square to change
circles written in standard form into graphing form.
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Summarize Problems 4-131 to 4-134
Problems 4-144 to 4-146
Start Lesson 5.1.1
Conclusion
Homework:
Problems CL4-156 to CL4-166
Ch. 4 Individual Test Next Friday – Start Studying!
Perfect Square
A polynomial that can be factored into the
following form:
2
(x + a)
Bell Ringer
Distribute the following:
1.
2.
3.
4.
y = (x – 2)2
y = (x + 3)2
y = (x + 4)2
y = (x – 6)2
1.
2.
3.
4.
Factor the following:
y = x2 + 8x + 16
y = x2 – 16x + 64
y = x2 + 20x + 100
y = x2 – 9x + 20.25
Is there a pattern when comparing a, b, and c when it is in
standard form vs factored form?
Completing the Square
x2 + bx + c is a perfect square if:
1 
c   b
2 
2
The value of c will always be positive.
Always write out all of your work. It will help you soon.
Completing the Square
Find the c that completes the square:
1. x2 + 50x + c
2. x2 – 22x + c
3. x2 + 15x + c
Factoring a Completed Square
2
If x + bx + c is a perfect square,
then it will easily factor to
1 

x

b


2 

2
Perfect Squares: Parabolas & Circles
Find the vertices of the following graphs and state
whether they are maximums or minimums.
• y = (x + 5)2 – 5
• y = -(x + 3)2 + 1
• y = -3(x – 7)2 + 8
• y = 4(x – 52)2 – 74
State the length of the radius and the coordinates of
the center for each circle below:
• ( x – 2 )2 + ( y + 7 )2 = 64
• x2 + y2 = 36
• ( x + 4 )2 + ( y + 11 )2 = 5
• ( x + 3 )2 + y2 = 175
A new Equation?
What will the graph of the following look like:
x  4 x  y  2 y  11
2
2
Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square:
y = x2 + 8x + 25
y = (x2 + 8x + 16 ) + 25 – 16
y = (x + 4)2 + 9
1 
 8
2 
2
 4
16
Vertex:
(-4, 9)
2
Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square:
y = 3x2 – 18x – 10
y = 3(x2 – 6x + 9 ) – 10 – 3 9
y = 3(x – 3)2 – 10 – 27
y = 3(x – 3)2 – 37
Vertex:
1

  6 
2

2
 3 
9
(3, -37)
2
Standard to Graphing: Circle
Find the center and radius of the equation by completing the square:
x2 + y2 + 6x – 12y – 9 = 0
x2 + 6x + y2 – 12y – 9 = 0
+9+9
x2 + 6x + y2 – 12y = 9
(x2 + 6x + 9 ) + (y2 – 12y + 36 ) = 9 + 9 +
36
(x + 3)2 + (y – 6)2 = 54
2
1 
 6 
2 
Center:
 3
2
2
2
1

  12    6   36
2

 9
(-3, 6)
Radius:
54  9 6 3 6
Solving Graphically
How did you use the
graph to solve:
(x+3)2 – 5 = 4?
What other equations
could you solve?
x = -6
x=0
Day 39: March 26th
Objective: Solve a variety of equations and discuss different methods
for solving them. Also, justify strategies and develop methods for
checking solutions. THEN Use graphs to validate algebraic
solutions and to approximate solutions when no algebraic method is
available, and use two different methods to solve one-variable
equations graphically.
•
•
•
•
•
HW Check Quickly!
Problems 5-3 to 5-5
Problems 5-13 to 5-17
Time? Start Lesson 5.1.2
Conclusion
Homework:
Problems 5-6 to 5-12 AND 5-18 to 5-24
Ch. 4 Individual Test Friday – Start Studying!
[If you know you’re not going to be here due to
extenuating circumstances, you must see me ahead of
time to take the test.]
5-5: Learning Log
Title: Strategies for Solving Equations
• Summarize all of the solving strategies
you saw today.
• Show an example of each strategy.
• Explain the type of equations for which
each strategy works best.
• Make sure to explain Rewriting.
• Make sure to explain Undoing.
• Make sure to explain Looking Inside.
Extraneous Solutions
2x  3  x

2x  3

2
x
2
2x  3  x
2
Calculator & Solving Equations
x  2x 
3
Solve:
 
1
x 2 5
3
Method 1: Intersection
y1  x3  2 x
Enter –
3
1
y2  x 2  5
 
Calculator Function – CALC: intersect
Enter –
y1  x  2 x 
3
 
1
x 2 5
Calculator Function – CALC: zero
3
x-coordinates!
Method 2: X-intercept
Day 40: March 27th
Objective: Use graphs to validate algebraic solutions and to approximate
solutions when no algebraic method is available, and use two different
methods to solve one-variable equations graphically. THEN Solve
systems of linear and non-linear equations using multiple strategies.
Determine the number of solutions for systems and interpret solutions
graphically. THEN Use problem solving to write equations and find
solutions for real-life applications.
•
•
•
•
•
HW Check and Correct (in red) Quickly! & Look at Ch. 4 Team Tests(?)
Problems 5-16 to 5-17
Problems 5-33 to 5-36
Problems 5-44 to 5-47
Conclusion
Homework:
Problems 5-25 to 5-32 AND 5-37 to 5-43
Ch. 4 Individual Test Friday – Start Studying!
[If you know you’re not going to be here due to
extenuating circumstances, you must see me ahead of
time to take the test.]
5-17: Learning Log
Title: The Meaning of Solution, Part 1
• What does the solution to an equation mean?
• Do you have any new ideas about solutions that you did
not have before?
• Do you have any new methods to find solutions?
• How can you use the calculator to solve an equation?
• How can you use the intersection function on the
calculator to find a solution?
• How can you use the zero function on the calculator to
find a solution?
• Why are there equations that we can not solve
algebraically yet?
5-34
x  y  25
2
2
y  x  13
2
Solve the Following Algebraically
x  y  25
2
y  x  13
2
x  y  25
2
2
OR
2
y  13  x 2
x   x  13  25
2
2
2
x   x  13 x  13  25
2
2
2
x  x  26 x  169  25
2
4
2
y  13  y  25
2
Solve the Following Algebraically
x   x  13  25
2
2
2
x 2   x 2  13 x 2  13  25
x 2  x 4  26 x 2  169  25
wx  25 xw  144  0
24
2
 xw  16  xw  9   0
2
2
x  16  0
x 9  0
x  4
x  3
2
2
Day 41: March 28th
Objective: Use problem solving to write equations and find solutions
for real-life applications. THEN Extend what was learned about
solving systems of equations graphically to solving systems of
inequalities.
•
•
•
•
HW Check and Correct (in red) Quickly! & Look at Ch. 4 Team Tests
Problems 5-44 to 5-47
Problems 5-54 to 5-61
Conclusion
Homework:
Problems 5-48 to 5-53 AND 5-62 to 5-67
Ch. 4 Individual Test Friday –Study!
[If you know you’re not going to be here due to
extenuating circumstances, you must see me ahead of
time to take the test.]
5-47: Learning Log
Title: The Meaning of Solution, Part 2
• What does the solution to an equation or a
system of equations mean?
• What does a solution to a one variable
equation look like on a graph?
Algebraically?
• What does the solution to a system of
equations look like on a graph?
Algebraically?
5-54
y  2 x  5x  3
2
y  x  4x  3
2
2 x2  5x  3  x2  4 x  3
-3
2
2 x2  5x  3  x2  4 x  3
Day 42: March 29th
Objective: Extend what was learned about solving systems of
equations graphically to solving systems of inequalities. THEN
Apply linear inequalities to solve a problem.
•
•
•
•
HW Check and Correct (in red) Quickly!
Problems 5-57 to 5-61
Problems 5-75 to 5-77
Conclusion
Homework:
Problems 5-68 to 5-74
Ch. 4 Individual Test Tomorrow –STUDY!
Solving a 1 Variable Inequality
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
2 x  5x  3  x  4 x  3
2
2
Find the Boundary
Test Every Region
x
Change inequality to equality
2 x2  5x  3  x2  4 x  3
Solve
Pick a point in
each region
x  x  6  0 Substitute
 x  3 x  2  0 into Original 9 ≤ 3
2
x  3 or x  2
Plot Boundary Point(s)
0
x = -4
x=3
x=0
2  4   5  4   3   4   4  4   3
2  3  5  3  3   3  4  3   3
2
2
2  0  5 0  3  0  4 0  3
2
2
False
Shade True
Region(s)
2
-3 ≤ 3
True
2
30 ≤ 24
False
3  x  2
Write
Inequality
Solving a System of Inequalities
Graphically represent the solutions to the following
system of inequalities: y  2 x 2  5 x  3
Solid or
Dashed?
y  x2  4x  3
Find the Boundaries
Plot points for the equalities one at a time
y  2 x2  5x  3
y  x2  4 x  3
0   2 x  1 x  3 0   x  1 x  3
Test Every Region
Find which side to shade for each inequality
(0 ,0)
0   0  5 0  3
2
Shade the Feasible Region
0 ≥ -3
True
(0 ,0)
0   0  4 0  3
2
0<3
True
5-60
5-61: Learning Log
Title: The Meaning of Solution, Part 3
• What does the solution to an equation or a system of
equations mean?
• What does a solution to a one variable equation look like on a
graph? Algebraically?
• What does the solution to a system of equations look like on a
graph? Algebraically?
• What does the solution to an inequality or a system of
inequalities mean?
• What does a solution to a one variable inequality look like on
a graph? Algebraically?
• What does the solution to a system of inequalities look like on
a graph? Algebraically?
Day 43: March 30th
Objective: Assess Chapter 4 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Calculator Portion…put calculator away when finished
• Second: Non-Calculator Portion (ask for it)
• Third: Check your work & hand the test in to Ms. Katz
• Fourth: Correct last night’s homework
Homework:
Problems 5-79 to 5-86
Enjoy your week away from school!
Day 44: April 10th
Objective: Apply linear inequalities to solve a problem.
•
•
•
•
HW Check and Correct (in red) Quickly!
Problem 5-77
Start Problems 5-87 to 5-88 [Graded Teamwork]
Conclusion
Homework:
Problems 5-89 to 5-95
You will have another ½ hour tomorrow before you
must hand in the good copy of your graded
teamwork – you may want to work on it a little
tonight
Midterm (Ch. 5 Individual Test) Friday
Define the Variables
x: Number of cars built
y: Number of trucks built
Does it matter?
Cover-Up Method
Plot :
-2x + 5y = -10
Find the
intercepts
X
0
5
Y
-2
0
Vertices of the Boundary
Constraints and Feasible Region
Cars: x-axis
Trucks: y-axis
Constraints
Wheels: 4c  6t  36
Seats: 2c  t  14
 0, 0   7, 0  Gas Tanks: c  3t  15
Profit Equation:
6,
2
3,
4
   
P = c + 2t
 0,5
NonNegative
c0
t0
Graph the System:
True
0

36
Test Point
True
0

14
(0,0)
0  15 True
Feasible Region
Critical Points and Conclusion
Test Every Critical Point in the Profit Equation:
P = c + 2t
3, 4
 6, 2 
 0, 0 
3  2  4  11
6  2  2  10
0  2  0  0
 0,5
 7, 0 
0  2  5  10
7  2  0  7
CONCLUSION:
Otto should build 3 cars and 4 trucks for $11.
Day 45: April 11th
Objective: Apply linear inequalities to solve a problem. THEN
Learn how to simplify algebraic fractions.
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Review Chapter 4 Test
Work on Graded Teamwork (1/2 hour)
Notes: Simplify Rational Expressions
Rational Expressions 1 (Odds)
Conclusion
Homework:
Problems 5-97 to 5-102
Do EVENS from Classwork Worksheet
Finish team project – ready to hand in at 11:27 am
Midterm (Ch. 5 Individual Test) Friday
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
3
x 2
x 2

x 5
x 1
x
3
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 MULTIPLICATION!
Simplifying Rational Expressions
Simplify:
2 x  3 x  20
4 x3  64 x
2
2 x  5  x  4 


4 x  x 2  16 
Can NOT reduce since
everything does not have a
common factor and it’s not in
factored form
Factor Completely
2 x  5  x  4  CAN reduce since the

top and bottom have a

4 x  x  4  x  4  common factor
2x  5
2x  5

or
2
4x  x  4
4 x  16 x
Day 46: April 12th
Objective: Understand how to multiply and divide rational
expressions. THEN Understand how to add and subtract rational
expressions and continue to learn how to simplify rational
expressions.
•
•
•
•
•
•
HW Check and Correct (in red) Quickly! & Hand-in Project
Notes: Multiply/Divide Rational Expressions
Rational Expressions 2 (Odds)
Notes: Add/Subtract Rational Expressions
Rational Expressions 3 (Odds)
Chapter 5 Closure
Homework:
EVENS from Rational Expressions 2 & 3
(You can show me this HW tomorrow or Monday…but
you will have another worksheet over the weekend)
Midterm (Ch. 5 Individual Test) TOMORROW
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
Simplifying 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  3 x  8 


 x  3 x  2 
x  2  x  7  3x  8


x3
Combine
Rewrite
Reduce
Simplifying Rational Expressions
3x  15 3x  15 x  18
 2
2
25  x
x  3x  10
2
Simplify:
Turn it into a
3x  15 x  3x  10
multiplication problem

 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
Reduce
1

x3
Factor Completely
Adding and Subtracting Fractions
Subtraction:
Addition:
2
3
2
3

1
5
  
5
5
10
15
1
5

13
15
3
15
3
3
Common
Denominator
Add the
Numerators
7
4
7
4

3
10
  
5
5
35
20

29
20
Least
Common
Denominator
(if you can
find it)
3 2
10 2
6
20 Subtract the
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 


Same denominator!
Half the work is done!
CAREFUL with
subtraction!
x 2  2 x  15
2x 1 x  4
 2
x  2 x  15
x 5
Combine Like
 2
Terms
x  2 x  15
x 5
Make sure it can’t be

 x  5 x  3 simplified more
1

x3
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

2x2
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
3 x  6    5 x  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  4x


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 47: April 13th
Objective: Assess Chapters 1-5 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Calculator Portion…put calculator away when finished
• Second: Non-Calculator Portion (ask for it)
• Third: Check your work & hand the test in to Ms. Katz
• Fourth: Correct last night’s homework
Homework:
Rational Expressions 4 (ALL)
(And finish 2&3 Evens if you haven’t done so already)
Day 48: April 16th
Objective: Learn to find rules that “undo” functions, and develop
strategies to justify that each rule undoes the other. Graph
functions along with their inverses and make observations about
the relationships between the graphs. THEN Introduce the term
“inverse” to describe undo rules. Also, graph the inverse of a
function by reflecting it across the line of symmetry, and write
equations for inverses.
•
•
•
•
HW Check and Correct (in red) Quickly!
Answer any questions about Rational Expressions
Problems 6-1 to 6-6
Start Problems 6-16 to 6-25
Homework:
Problems 6-7 to 6-15 AND 6-26 to 6-32
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
“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!
Day 49: April 17th
Objective: Introduce the term “inverse” to describe undo rules. Also,
graph the inverse of a function by reflecting it across the line of
symmetry, and write equations for inverses.
•
•
•
•
HW Check and Correct (in red) Quickly!
Wrap-Up Problems 6-5 to 6-6
Problems 6-16 to 6-25
Conclusion
Homework:
Problems 6-33 to 6-37
Tables and Graphs of Inverses
Switch
x and y
Original
(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 50: April 18th
Objective: Introduce the term “inverse” to describe undo rules. Also,
graph the inverse of a function by reflecting it across the line of
symmetry, and write equations for inverses. THEN Use ideas 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.
•
•
•
•
HW Check and Correct (in red) Quickly!
Problems 6-22 to 6-25 and Slides
Problems 6-38 to 6-42
Conclusion
Homework:
Problems 6-44 to 6-53
Inverse Notation
f
f  x
Original
function
 x
Inverse
function
1
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
Algebraically Finding an Inverse
Find the inverse of the following:
Switch x and y
x  6 y  11
x  11  6 y
y  6 x  11
Solve for y
x 11
6
y
y
x 11
6
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
Solve for y
4
4 x   y  10 
3
3
Really y =
3
3
e
1
 x 
3
4 x  10
4 x  y  10
4 x  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.
Day 51: April 19th
Objective: Use ideas 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.
THEN Apply strategies for finding inverses to parent graph
equations.
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problems 6-38 to 6-42
Problems 6-54 to 6-58
Conclusion
Homework:
Problems 6-59 to 6-66
Ch. 6 Team Test Thursday
Composition of Functions
Substituting a function or its 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   2 x  3 and g  x    x 2  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   2 x  3 and g  x    x 2  5 . Find:
g  f  x 
Plug the result into g(x) last
g  2 x  3    2 x  3  5
g  2x  3    2 x  3 2 x  3  5
g  2 x  3    4 x 2  12 x  9   5
g  2 x  3  4 x2  12 x  9  5
2
g  2x  3  4 x 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 52: April 20th
Objective: Apply strategies for finding inverses to parent graph
equations. 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.”
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problems 6-56 to 6-58
Problems 6-67 to 6-71
Conclusion
Homework:
Problems 6-72 to 6-80
Ch. 6 Team Test Thursday
Silent Board Game
1
2
x 8 32 1 16 4 3 64
  3 5 1 0 4 2
6
1
x 2 0 0.25 1 2 0.2 8
1
  1  2 
3
2
g x
g x
1.6
~ 2.3
g  x   log 2  x 
Silent Board Game
x 1 0 0.2
1 2
1
    3
2 1 0 2
x 2 3 4 8 16 32 64
  1
2 3 4 5 6
1
0.25
2
1
8
~ 2.3
g x
g x
1.6
g  x   log 2  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
Day 53: April 23rd
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. Extend
knowledge of general equations for parent functions to transform
the graph of y = log(x).
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Check Problems 6-70 to 6-71
Logarithms and Graphs Packet (Extra Visual)
Problems 6-93 and 6-95
Conclusion
Homework:
Problems 6-84 to 6-92 AND 6-96 to 6-105
Ch. 6 Team Test Thursday
Ch. 6 Individual Test Tuesday
Inverse of an Exponential Equation
Original
Inverse
y2
x2
x
OR
y
y  log 2  x 
Logs 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 54: April 25th
Objective: Look into the base of the log key on the calculator.
Extend knowledge of general equations for parent functions to
transform the graph of y = log(x).
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Wrap-up/Recap Logs and Graphs exploration
Problems 6-93 and 6-95
Review Midterm
Introduction to Chapter 7
Conclusion
Homework:
Problems 6-113 to 6-120 (Skip 116, 118)
Change 113 to the square root of 7-x
Ch. 6 Team Test Tomorrow
Ch. 6 Individual Test Tuesday
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: Logarithmic
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
Day 55: April 26th
Objective: Assess Chapter 6 in a team setting. THEN Create and
use a model to locate points in 3-D space, and plot points in 3-D
on isometric paper.
• HW Check and Correct (in red) Quickly!
• Problem 6-113 from HW should not have had a “square” on the 7
minus x…
• Chapter 6 Team Test
• Introduction to Chapter 7
• Start Problems 7-1 to 7-7
• Conclusion
Homework:
Problems CL6-121 to 6-130 AND 7-8 to 7-15
[Check this assignment w/Ms. Katz
before leaving today]
Ch. 6 Individual Test Tuesday
Day 56: April 27th
Objective: Create and use a model to locate points in 3-D space,
and plot points in 3-D on isometric paper. THEN Graph
planes.
*NEW SEATS*
• HW Check and Correct (in red) Quickly!
• Problems 7-1 to 7-7
• Problems 7-16 to 7-20
• Conclusion
Homework:
Problems 7-8 to 7-15 AND 7-21 to 7-28
Ch. 6 Individual Test Tuesday
Start evaluating your textbook…if your cover is torn/missing or
there is other significant damage, you owe $19 to replace it.
Please do not make a mess of it with tape – if you think it can
be repaired, see Ms. Katz. Otherwise, bring cash or check by
the time we finish Chapter 7.
Plotting Points in xyz-Space
(x,y,z)
z
(2,3,5)
x
y
Link
Plotting Planes in xyz-Space
z
2x + 3y + z =6
x
y
Day 57: April 30th
Objective: Graph planes. THEN Investigate the graphs of
systems of equations with three variables. Find the points that
lie on two planes simultaneously.
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Review Ch. 5 Project and Ch. 6 Team Test
Finish Problems 7-18 to 7-20
Problems 7-29 to 7-33
Start Problems 7-43 to 7-48 and 7-49
Conclusion
Homework:
Problems 7-39 to 7-42 & STUDY!
Ch. 6 Individual Test Tomorrow
Start evaluating your textbook…if your cover is torn/missing or there is
other significant damage, you owe $19 to replace it. Please do not
make a mess of it with tape – if you think it can be repaired, see Ms.
Katz. Otherwise, bring cash or check by the time we finish Chapter 7.
7-20: x=4 in Different Dimensions
One Dimension
Two Dimensions Three Dimensions
Point
Line
Plane
Day 58: May 1st
Objective: Assess Chapter 6 in an individual setting.
• Silence your cell phone and put it in your school bag (not
your pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Calculator Portion…put calculator away when finished
• Second: Non-Calculator Portion (ask for it)
• Third: Check your work & hand the test in to Ms. Katz
• Fourth: Correct last night’s homework
Homework: Problems 7-34 to 7-38 AND 7-50 to 7-59
Start evaluating your textbook…if your cover is torn/missing or there is
other significant damage, you owe $19 to replace it. Please do not
make a mess of it with tape – if you think it can be repaired, see Ms.
Katz. Otherwise, bring cash or check by the time we finish Chapter 7.
Day 59: May 2nd
Objective: Develop an algebraic strategy to solve systems of three
equations with three variables. Also, determine the different
ways three planes can intersect, and investigate the graphs of
3-D systems. THEN Find the equation of a quadratic function
y=ax2+bx+c that passes through three given points when
graphed.
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problems 7-43 to 7-48 and 7-49
Problems 7-60 to 7-68
Conclusion
Homework:
Problems 7-71 to 7-86
Start evaluating your textbook…if your cover is torn/missing or there is
other significant damage, you owe $19 to replace it. Please do not
make a mess of it with tape – if you think it can be repaired, see Ms.
Katz. Otherwise, bring cash or check by the time we finish Chapter 7.
Solving a 3 Variable System
Solve the system:
x  y  3z  3
2x  y  6z  2
2 x  y  3 z  7
2. Solve the system
+
12 x  24 z  16 3x  6  13   4
3x  2  4
+ + 12 x  27 z  15
3x  6
x  2
3z  1
z  13
1. Use Elimination to write a
3. Solve for the 3rd Variable
2-Variable System Must be the same 2
2  y  3 13  3
 
3x  6z  4  4
4x  9z  5  3
variables!
4. Solution:
2  y  1  3
y 1  3
y4
1

2,
4,

3
Solving a 3 Variable System
Solve the system:
2. Solve the new system
1
5 x  4 y  6 z  19 You must
2
2 x  2 y  z  5 multiply to
eliminate
3
3 x  6 y  5 z  16
3x  12 z  27
+ 3x  2 z  1
1. Use Elimination to write a
2-Variable System
Multipliy 2nd by 2
1
5 x  4 y  6 z  19
Multiply 2nd by 3
2
6 x  6 y  3z  15
x  4  2   9
x  8  9
14z  28
x  1
z2
3. Solve for the 3rd Variable
2  1  2 y   2  5
+4 x  4 y  2 z  10 +
2  2y  2  5
x  4z  9
3x  2z  1
2y  4  5
2
y

1

3
x  4 z  9
y  12
4. Solution:
2
3 3x  6 y  5 z  16
3x  2 z  1
1

1,
 2 , 2
Two Forms of a Quadratic
y = ax2 + bx + c
•Standard Form
•Parabola
•a is the stretch factor
•a tells whether it opens
up/down
•Can be put into factored
form
•Use the quadratic formula
•c is the y-intercept
y = a(x – h)2 + k
•Graphing/General Form
•Parabola
•(h,k) is the vertex
•a is the stretch factor
•a tells whether it opens
up/down
Writing a Contextual 3 Variable System
Suppose the graph of a quadratic function passes
through the points (1,0), (2,5), and (3,12). Algebraically
find the quadratic equation.
1. Use the Standard Quadratic Form: y  ax  bx  c
2
x
y
0  a 1  b 1  c
2
2. Substitute each Point into the Equation: 5  a  2  2  b  2   c
12  a  3  b  3  c
2
0  abc
3. Simplify the Equations: 5  4a  2b  c
12  9a  3b  c
Solving a Contextual 3 Variable System
Solve the system:
1
 0  a  b  c   1 Eliminate
2
the “c”
5  4a  2b  c
first!
3
12  9a  3b  c
1. Use Elimination to write a
2-Variable System
1
2. Solve the new system
10  6a  2b
+ 12  8a  2b
2  2a
1 a
0  a  b  c
0  a  b  c
2
3
5

4
a

2
b

c
+
+12  9a  3b  c
5  3a  b
12  8a  2b
1
 5  3a  b  2
12  8a  2b
5  3 1  b
5  3 b
2b
3. Solve for the 3rd Variable
0  1 2  c
0  3 c
3  c
4. Subsititue into the
Standard Form:
y  x  2x  3
2
Day 60: May 3rd
Objective: Develop the Power Property of Logs and use it to
develop an efficient method to solve exponential equations in
ax=b form. THEN Learn the Product and Quotient Properties of
logs and how to rewrite equations with different bases.
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Wrap-Up Problems 7-64 to 7-68
Problems 7-87 to 7-93
Start Problems 7-103 to 7-109
Conclusion
Homework:
Problems 7-94 to 7-102
Day 61: May 4th
Objective: Learn the Product and Quotient Properties of logs and
how to rewrite equations with different bases. THEN Develop
strategies for finding the equation of an exponential function
given two points and an asymptote.
•
•
•
•
HW Check and Correct (in red) Quickly!
Problems 7-103 to 7-110
Problems 7-123 to 7-126
Conclusion
Homework:
Problems 7-111 to 7-122 AND 7-127 to 7-136
(These are long…leave yourself enough time!)
Power Property of Logs
logm a   n  logm  a
n
Solving Equations with the Power
Property of Logs
3.46  454
x
log346
.   log 454
x
x  log 346
.   log 454
x  log3.46  454 
log 454
x
log 3.46
x  4.9289
The Change of Base Formula
log  a 
log b  a  
log  b 
For a and b greater than 0 AND b≠1.
Properties of Logarithms
Power Property:
   n  log  a 
log m a
n
m
Product Property:
logm  a  b   logm  a   log m  b 
Quotient Property:
logm  ba   log m  a   log m  b 
Day 62: May 7th
Objective: Develop strategies for finding the equation of an
exponential function given two points and an asymptote. THEN
Apply knowledge of exponential functions to solve a murder
mystery. THEN Add, subtract, and start to multiply matrices.
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Review Chapter 6 Individual Test
Problems 7-123 to 7-126
Problem 7-137
Notes: Matrices
Conclusion
Homework:
Problems 7-138 to 7-147 AND 7-148 to 7-149 (CW)
System of Exponential Equations
Find an exponential function that passes through (3,12.5)
and (4,11.25) and has a horizontal asymptote of y = 10.
Substitute into twice:
11.25  ab  10
4
– 10
– 10
12.5  ab  10
– 10
Divide #s
÷
3
– 10
1.25  ab 4
2.5  ab3
0.5  b
y  ab  c
x
Larger
exponent
first
Rewrite into
y=abx
Subtract
Exponents
Asymptote
c=10
Substitute into either
equation to find a
3
12.5  a  0.5   10
12.5  0.125a 10
2.5  0.125a
20  a
y  20  0.5   10
x
Warning: This is not addressed a lot in the homework but will be assessed.
Day 63: May 8th
Objective: Apply knowledge of exponential functions to solve a
murder mystery. THEN Add, subtract, and start to multiply
matrices. THEN Use matrix multiplication to solve problems.
•
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problem 7-137
Notes: Matrices
Problems 7-148 to 7-154
Start Problems 7-165 to 7-169
Conclusion
Homework:
Problems 7-155 to 7-164 AND 7-171 to 7-174
(Do more if you have time)
Chapter 7 Team Test Thursday
Is your book cover torn? Is your book in poor condition? Bring
cash/check ($19) so that I can replace it. Please DO NOT attempt to
tape it with white tape! See me if you think it can be repaired.
Matrix
A matrix M
A is an array of cell entries
(m
arow,column
row,column) and it must have rectangular
dimensions (Rows x Columns).
Example:
4
M
A 3
 5
 5

 r  t
Dimensions:
r
17  2
0
2
g
6
21
20 

15 x 
10
3x4 ma2,4 : 15x
Scalar Multiplication
Every entry in the matrix is multiplied by
the number outside the matrix (scalar).
Example:
3 2 5
4


8 3 1 
12 8 20 
32 12 4 


Matrix Addition/Subtraction
IF the matrices have the same dimensions, add
or subtract corresponding cell entries.
Examples:
a b
d e

c  g


f j
h ci 
h i   a  g bb+h
 


k l d  j e  k f  l
 5   3
12    0  
   
 4 10 
 5  3 
 12  0  


 4  10
 8 
 12 


 14 
Matrix Addition/Subtraction
Perform the indicated operation:
 z 3 0.4  w 0 


8 7 4   18 2 

 

The matrices
MUST have the
same
dimensions!
Matrix Multiplication
1Multiply
the elements of each row of the first matrix by
the elements of each column in the second matrix. 2Add
the products. 3The answer goes into arow of 1st, column of 2nd.
3x2
a1,2
a1,1
2x3
 4 5 

2

5

1

3

3

1
2

4

1

1

3

2
1  2 1 3 

A

1
3
a
a

2,2
2,1




2  4 2 1
4  4  2 1  1  2 4  5  2  3  11
 2 1 

1 2
15 16

2x2

20 27
Matrix Multiplication
Can we multiply these…
2x3
2x2
 2 1 3  8 7 
 4 2 1   5 2   No

 

?
# of columns in
1st MUST be the
same as # of
rows in 2nd!
3x4
5x1
8
.1
2 5  2 
8
 
0

1 52 2    0  

 
 8 17 5 5   9 
 4 
3x2
 4 5
 1 3 
7
2
.75


 
 2 1
1x3
Yes
No
Day 64: May 9th
Objective: Use matrix multiplication to solve problems. THEN Use a
graphing calculator to perform operations with matrices.
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Problem 7-154
Problems 7-165 to 7-169
Start Problems 7-179 to 7-184
Conclusion
Homework:
Change
the W’s to
E’s!
Problems 7-175 to 7-178 AND 7-185 to 7-193
Chapter 7 Team Test Tomorrow
Is your book cover torn? Is your book in poor condition? Bring
cash/check ($19) so that I can replace it. Please DO NOT
attempt to tape it with white tape! See me if you think it can be
repaired.
Matrix Multiplication with a Context
Cars
Bull’s Eye
Order
JC Nickels Department
Store Order
Trucks
Wheels Seats
Gas Tanks
 20 25  4 2 1
B
A

15 30  6 1 3

 

Cars
Trucks
20  4  25  6 20  2  25 1 20 1  25  3
15  4  30  6 15  2  30 1 15 1  30  3


Wheels
Bull’s Eye
Total Order
JC Nickels Department
Store Total Order
Seats
Gas Tanks
 230 65 95 
 240 60 105


Matrices from 7-171 and 7-172
6 4 7 


E  4 8 5
 5 6 6 
 5 4 3


B   4 3 3
 4 6 6
Day 65: May 10th
Objective: Assess Chapter 7 in a team setting. THEN Use a
graphing calculator to perform operations with matrices.
•
•
•
•
HW Check and Correct (in red) Quickly!
Chapter 7 Team Test
Finish Problems 7-179 to 7-184
Conclusion
Homework:
Finish Problems 7-180 to 7-184 AND 7-205 to 7-208
Chapter 7 Individual Test next week (Thursday?)
Is your book cover torn? Is your book in poor condition? Bring
cash/check ($19) so that I can replace it. Please DO NOT
attempt to tape it with white tape! See me if you think it can be
repaired.
Day 66: May 11th
Objective: Use a graphing calculator to perform operations with
matrices. THEN Write systems of equations as matrix equations.
Find the identity element for a matrix and consider inverses for
matrices.
•
•
•
•
HW Check and Correct (in red) Quickly!
Wrap-Up Problems 7-179 to 7-184
Problems 7-194 to 7-199
Conclusion
Homework:
Finish Problems 7-200 to 7-204 AND 7-221 to 7-226
Chapter 7 Individual Test next week (Thursday?)
Is your book cover torn? Is your book in poor condition? Bring
cash/check ($19) so that I can replace it. We will be trading
books TUESDAY – bring yours! Please DO NOT attempt to
tape it with white tape! See me if you think it can be repaired.
Order in Matrix Multiplication Matters
E  B
 6 4 7  5 4 3 74 78 72
 4 8 5   4 3 3  72 70 66 


 

 5 6 6   4 6 6  73 74 69
B  E
5 4 3 6 4 7   61 70 73
 4 3 3  4 8 5    51 58 61


 

 4 6 6 5 6 6  78 100 94 

 6 4 7  5 4 3 74 78 72
 4 8 5   4 3 3  72 70 66 
3x3   3x3
3 3 
3   3x3
 3
 5 6 6   4 6 6  73 74 69
The dimensions of a product of
matrices are the # of rows of the
first matrix by the # of columns of
the second matrix.
6 4 7 
 4 8 5   15 18 18
1
1
11

1x3  3x3
3  1
1x3
3 
5 6 6 
 2 3 1 2 11 16 
 2x2
  2x2
2x2



2
2
2
2
4
5
3
4
19
28

 
 

183 (b)
182
180 (a)
179 (b)
Matrix Multiplication
5 4 3 0.30  5.10 
 4 3 3  0.45   4.35
3
3x1
1  3
1
 3x3
  3x1
 4 6 6 0.60 7.50
The Race
8
 2 3     4
 4 
1 0 2 
8 0 1 5 
7

8
1

3
 6 7 8 3 1  


 15 3 2  
 0 10 5 16 3  


 10 8 1 
 22 4 7 6 2  
 0 4 7 
 81 4 44 
100 3 27 


155 115 57 


 219 39 38 
Day 67: May 14th
Objective: Write systems of equations as matrix equations. Find the
identity element for a matrix and consider inverses for matrices.
THEN Solve systems of equations using matrices and graphing
calculators. THEN Introduce a simplified method of finding the vertex
of a quadratic function.
•
•
•
•
•
HW Check and Correct (in red) Quickly!
Finish Problems 7-197 to 7-199
Problems 7-209 to 7-216
Vertex Simplified Notes
Conclusion – Recap Chapter 7 Topics
Homework:
Problems 7-218 to 7-220 AND CL7-228 to 7-238
BRING TEXTBOOK (OR $19) FROM HOME!
Chapter 7 Individual Test (Thursday?)
Is your book in poor condition? Bring cash/check ($19) so that I
can replace it. We will be trading books TUESDAY – bring
yours! Please DO NOT attempt to tape it with white tape!
Identity Matrix
The product of a square matrix A and its
identity matrix I, on the left or the right, is A.
AI = IA =A
General Form:
1
0



0
0
0
1
0
0
0
0
0
0
0 0
0 0 


1 0
0 1 
I Must be a
square matrix
Identity Matrix Example
Must be
a square
matrix
The identity matrix
must be the same
dimensions with 0’s in
every cell except for
1’s in the main
diagonal
 5 8 1  1 0 0 
 5 2 8  0 1 0 

 

 0 7 15  0 0 1 
The
same!
5 8 1
 5 2 8


 0 7 15 
Inverse Matrix
The product of a square matrix A and its
inverse matrix A-1, on the left or the right, is the
identity matrix I.
(A Must be a
AA-1= A-1A =I square matrix)
How do we find the Inverse Matrix:
2 1
1 0 

?



 4 0
0 1 




Converting a System of Equations to a
Matrix Equation
Make sure
Identify all of
99xx33yy1z  7
the
the
equations
coefficients
1x 
x 1yy1z  3
are in
to the
alphabetical
variables
1
6
x

4
y

1
z

2
1
16
x

4
y

21
order
Coefficient
Matrix
Variable
Matrix
Constant
Matrix
 9 3 1  x   7 
 1 1 1   y    3

    
16 4 1  z   21
Solving a System of Equations with
Matrices
Solve:
Identify all of
4 x  7 y  12 z  3.8
the
Make sure the
equations are in
coefficients
5
x

8
5
x
y


8
0
y
z


14.8
alphabetical
to the
order and that
variables
1
x

4
y

9
z

7.6
7.
6
every variable is
in each equation
Coefficient
Matrix “A”
Variable
Matrix “X”
Constant
Matrix “B”
 4 7 12  x   3.8 
 5 8 0    y    14.8

   

 1 4 9   z   7.6 
Solving a Systems of Equations with
Matrices
Continued…
Multiply by the
inverse of A to
isolate the
variable matrix
A
X
 4 7 12  x   3.8 
 5 8 0   y    14.8

  

 1 4 9   z   7.6 
A-1
A
X
A-1
A
X
Which Order
is Correct?
B
3x3.3x1
A-1
B
 8 13  5 39  32 39   4 7 12   x   8 13  5 39  32 39   3.8 
  5 13  8 39  20 39   5 8 0   y     5 13  8 39  20 39   14.8


  


  4 39 1 13 1 39   1 4 9   z    4 39 1 13 1 39   7.6 
OR
3x1.3x3
B
A-1
 8 13  5 39  32 39   4 7 12   x   3.8   8 13  5 39  32 39 
  5 13  8 39  20 39   5 8 0   y    14.8   5 13  8 39  20 39 


  


  4 39 1 13 1 39   1 4 9   z   7.6    4 39 1 13 1 39 
Solving a System of Equations with
Matrices
Continued…
Multiply by the
inverse of A to
isolate the
variable matrix
A-1
A
X
B
Step 1: Store
Matrix A and
B in your
calculator
 4 7 12  x   3.8 
 5 8 0   y    14.8

  

 1 4 9   z   7.6 
A
X
A-1
B
 8 13  5 39  32 39   4 7 12   x   8 13  5 39  32 39   3.8 
  5 13  8 39  20 39   5 8 0   y     5 13  8 39  20 39   14.8


  


  4 39 1 13 1 39   1 4 9   z    4 39 1 13 1 39   7.6 
Step 2: Enter
x
THUS:
 2 
You do not
this in your  
If…


need to
calculator to  
y

3
5
calculate the
solve the




Then…
inverse
system
1
 z   4 3
matrix!
AX  B
XA B
Finding a Vertex
Use the following equation to answer the questions below:
y = -3x + 6x – 7
2
1. Find the coordinates of the vertex.
2. Write the equation in graphing form.
Vertex Simplified
2
If f(x) = ax + bx + c, then the vertex is:
x
 b
 b  
,
f


 2a
2
a



Opposite
of b
y is plug and chug
with x
Example
Use the following equation to answer the questions below:
y = -3x2 + 6x – 7
1. Find the coordinates of the vertex.
x  263 
6
6
1
a  3
b 6
c  7
y  3 1  6 1  7  3 1  6 1  7  3  6  7  4
2
1, 4
2. Write the equation in graphing form.
y  3  x  1  4
2