Do Now – 8/24/15
 Find the slope (Simplified!) between the given
points:
a)
1,  3 and  3 ,  1
b)
4 ,  2 and  1,  2
c)
 3 , 2 and 5 ,12
d)
7 , 3 and 7 ,  1
Forms of Linear Equations
 Slope-Intercept Form
 Point-Slope Form
 Standard Form
Writing a Linear Equation
 Write the equation of a line that passes through the
point (-3 , 6) with a slope of -1.
Writing a Linear Equation
 Write the equation of a line that passes through the
point (-3 , 1) with a slope of -2.
Writing a Linear Equation
 Write the equation of a line that passes through the
points (-3 , 1) and (1 , 9).
Parallel and Perpendicular Lines
 Parallel Lines have the same
slope.
 Perpendicular Lines have
opposite reciprocal slopes.
Writing a Linear Equation
 Write the equation of a line that passes through the
point (4 , 3) and is parallel to y  2 x  3 .
 Write the equation of a line that passes through the
point (4 , 3) and is perpendicular to y  2 x  3 .
Writing Linear Equations
a) Write the equation of a line that passes through the
point (-2 , 3) and is perpendicular to y 
1
x7
3
b) Write the equation of a line that passes through the
point (-3 , 1) with a slope of 0.
c) Write the equation of a line that passes through the
points (1 , -4) and (-2 , -10).
Graphing a Linear Equation
 Graph:
y  3 x  5
Graphing a Linear Equation
 Graph:
y 3 
1
x  2
2
Graphing a Linear Equation
 Graph:  2 x  3 y  18
Learning Log Summary
LT 1 – I can graph linear equations in one variable and
write equations based on a graph or given information.
The forms of linear equations are…
They are useful when…
Closure
Homework (record on Learning Log):
LT 1: pg. 116 ~ 1-29 (O)
pg. 121 ~ 3-36 (Every 3rd problem)
Systems of Linear Equations
 Example:
3x  y
4 x  y  70
 What is the objective when solving a system of
equations?
 Methods to solve a system of equations:
1)
2)
3)
Student Work Critique
 For each piece of student work:
-Identify what method they used to solve the system.
-Identify the strengths of that method.
-Identify the weaknesses of that method.
-Look over their work and offer feedback.
Student Work Critique
Student Work Critique
Student Work Critique
Student Work Critique
Possible Solutions to a
System of Equations
Intersecting Lines
1 Solution
Parallel Lines
No Solution
Same Line
Infinite Solutions
Solving a System of Equations
 Solve:
4x  3y  5
2 x  5 y  17
 1, 3
Solving a System of Equations
 Solve:
3 x  4 y  14
4 x  10 y  11
1

4,  
2

Solving a System of Equations
x  2y 1
 Solve:
1
y  x4
2
 
Solving a System of Equations
 Solve:
 3 x  5 y  6
6 x  10 y  12
6

 0 ,   , 2 , 0 , 7 , 3
5

Practice: Solving a System of Equations
 Solve:
 Solve:
 Solve:
2 x  y  2
2 x  3 y  15
2x  y  2  x
x  2y  2  y
y  5x
x  y  10
Learning Log Summary
LT 2 – I can solve a system of equations in two variables.
The methods I can use to solve a system
are…
The types of solutions to a system are…
Common mistakes when solving a system
of equations are…
Closure
Homework (record on Learning Log):
LT 1: pg. 129 ~ 13-27 (O)
Linear Inequalities
A linear inequality in two variables makes a
comparison (<, >, etc.) of two expressions.
A solution is an ordered pair that must
satisfy the inequality.
Are these points solutions to the
statement y   x  3 ?
 Point 1: (2, -5)
 Point 2: (5, -5)
 Point 3: (-3, 0)
 Point 4: (-8, 2)
 Point 5: (-4, 5)
 Point 6: (4,-5)
 Point 7: (-1, 10)
 Point 8: (1, 0)
Is there a way to represent ALL
solutions to: y   x  3 ?
Represent all solutions to:
y  2x  3
Associated
Equation:
Line Type:
Test point:
y  2x  3
Represent all solutions to:
3x  2 y  4  0
Associated
Equation:
Line Type:
Test point:
Represent all solutions to:
x  2
Associated
Equation:
Line Type:
Test point:
y5
Represent all solutions to:
2
y  3   x  1
3
Associated
Equation:
Line Type:
Test point:
Learning Log Summary
LT 3 (pt 1) – I can graph linear inequalities in two variables.
Steps to graph a linear inequality…
A method to know where to shade is…
Closure
Homework (record on Learning Log):
LT 3: pg. 138 ~ Oral Exercises 13-16
Written Exercises 1-17 (O)
System of Linear Inequalities
How do you know if (-1 , 3) is a solution to:
x  2 y  1
How do you know if (-1 , 3) is a solution to:
x  2 y  1
2x  y  0
System of Linear Inequalities
Graph the solution to:
y  x  2
y  5 x  2
System of Linear Inequalities
x  3
Graph the solution to:
5
y  x2
3
System of Linear Inequalities
Graph the solution to:
3 x  2 y  2
x  2y  2
Learning Log Summary
LT 3 – I can graph linear inequalities in two variables and
graph systems of linear inequalities.
Steps to graph a system of linear
inequalities…
A method to know where to shade is…
Closure
Homework (record on Learning Log):
LT 3 (pt 2): System of Inequalities Wksht
Functions
Assign each number to the number of letters in
its name.
D
R
0
1
3
2
3
4
4
5
5
Functions
A function is a correspondence between two
sets, D and R, that assigns each member of D to
exactly one member of R.
D
R
0
1
3
2
3
4
4
5
5
Functions
Functions are named by letters:
g : x  2x
“g, the function that assigns x to 2x”
D
0
1
2
3
4
5
R
Functions
Given:
f : x  2x  x
2
a) Find the range of f.
b) Graph f.
, with domain D  1, 2 , 3 , 4 , 5
Functions
The value of a function are the numbers in the range
that correspond to numbers in the domain.
To evaluate a function, we find the output for a certain
input.
If
f x    x  3x
, then
f 2   ?
If
f x    x 2  3x
, then
f  3  ?
2
Functions
Given
f x   2 x  3
a)
f  1
b)
g 3
c)
f  g 3
d)
g  f  1
and
g  x   x 2  1 , find:
Functions
Given:
f x   x  3
, with
a) Find the range of f.
b) Graph f(x)
D   4 ,  3 ,  2 ,  1, 0 ,1, 2
Functions
Sometimes the domain of a function is not given. The
implied domain of a function is the set of x-values
(inputs) that produce real number outputs.
Ex) What is the domain of
f x   x  3 ?
Functions
Find the implied domain of each function:
a)
f x   5  x
b)
g x   x 2  3
c)
1
h x  
x3
Learning Log Summary
LT 4 – I can define a function and use it to evaluate and
graph.
A function is…
To evaluate a function…
f(x) means…
Closure
Homework (record on Learning Log):
LT 4: pg. 143 ~ O.E. 2-16 (even)
W.E. 3-39 (every 3rd)
Functions
A function is a correspondence between two
sets, D and R, that assigns each member of D to
exactly one member of R.
D
R
Aaron
Beth
LCHS
Carl
Deb
Chipotle
Edith
Frank
Home
Relations
A relation is any set of ordered pairs. The set of first
coordinates is the domain of the relation, and the
set of the second coordinates is the range.
Relations
Ex) Is the relation shown below a function?
D
R
0
1
3
2
3
4
4
5
5
Relations
Ex) Is the relation shown below a function?
 2 , 0  1,1
0 ,1
1, 2
2 , 4
Relations
Ex) Is the relation shown below a function?
0 , 4
1, 3
2 , 3
4 , 4
4 , 5
Relations
Ex) Is the relation shown below a function?
Relations
Ex) Is the relation shown below a function?
Relations
The Vertical Line Test – A relation is a function if and only
if no vertical line intersects its graph more than once.
Domain and Range
The domain is a collection of the possible x-values.
The range is a collection of possible y-values.
Domain and Range
The domain is a collection of the possible x-values.
The range is a collection of possible y-values.
Learning Log Summary
LT 5 – I can graph functions and determine when
relations are also functions.
A relation is…
A relation is different than a function…
Closure
Remind your parents about the location of
this class for Back to School Night tomorrow.
Room # and Period #!
On the index card, write one thing you think I
should share with them tomorrow night.
Closure
Homework (record on Learning Log):
LT 4: pg. 155 ~ O.E. 1-19
W.E. 1-4
Warm-Up – 9/8/15
 Multiply the following:
a)
 
x 2 x 3 2 x 
b)

x 2 x3  2 x

c)
x
2

 1 x3  2 x
What is the different between (a) and (b)? How does that change the
multiplication?
What is the difference between (b) and (c)? How does that change the
multiplication?

Multiplying Polynomials
 Multiply:
2827 
“Traditional Model”:
28
27
Area Model:
20  820  7 
Multiplying Polynomials
 Multiply:
x  8x  7 
Could we use the distributive property differently?
x  8x  7 
Multiplying Polynomials
 Multiply:
 Multiply:
x
2
x

 3x  1 x 2  5 x  2
2

 3x  1 x 2  2


Multiplying Polynomials: Special Patterns
 Expand and multiply:
1.
x+1
2
2.
x−5
2
3.
2x + 1
4.
2m − 3n
2
2
MAKE A GENERALIZATION: “When multiplying 𝒂 ± 𝒃 𝟐 , the product is..
__________±_________+_________
Multiplying Polynomials: Special Patterns
 Expand and multiply:
1. (𝑥 + 1)(𝑥 − 1)
2. (𝑥 − 5)(𝑥 + 5)
3. (2𝑥 + 1)(2𝑥 − 1)
4. (2𝑚 − 3𝑛)(2𝑚 + 3𝑛)
5. MAKE A GENERALIZATION: “When multiplying (𝒂 + 𝒃)(𝒂 − 𝒃), the product is..
Multiplying Polynomials
 Multiply:
x  1x  32 x  1
Extending our Understanding
 Expand and multiply:
1.
x
2.
x
3.
4.
2








 y2 x2  y2
3
 y 3 x3  y 3
x
5
 y 5 x5  y 5
x
n
 yn xn  yn

MAKE A GENERALIZATION: “When multiplying x  y
n
n
x
n

 y n the product is..
Practice
 Multiply:
1.
a  b 3
2.
r
3.
p
4.
a  b a 2  2ab  b 2 
n

1
2
n

 s n r n  2s n

Learning Log Summary
LT 8 – I can simplify the product of polynomials,
including special cases.
Multiplying polynomials is like the
distributive property because…
If the factors are the sides of a rectangle,
the area is…
Closure
Homework (record on Learning Log):
LT 8: pg. 175 ~ 1-31 (O)
Multiplying Polynomials: Special Patterns
 Expand and multiply:
1. (𝑥 + 1)(𝑥 − 1)
2. (𝑥 − 5)(𝑥 + 5)
3. (2𝑥 + 1)(2𝑥 − 1)
4. (2𝑚 − 3𝑛)(2𝑚 + 3𝑛)
5. MAKE A GENERALIZATION: “When multiplying (𝒂 + 𝒃)(𝒂 − 𝒃), the product is..
Multiplying Polynomials: Special Patterns
(Refresher)
 Expand and multiply:
1.
x+1
2
2.
x−5
2
3.
2x + 1
4.
2m − 3n
2
2
MAKE A GENERALIZATION: “When multiplying 𝒂 ± 𝒃 𝟐 , the product is..
__________±_________+_________
Multiplying Polynomials
 Multiply:


3x 2 2 x 2  5 x  1
Multiplying Polynomials


3x 2 2 x 2  5 x  1
6 x 4  15 x 3  3 x 2
Factoring Polynomials
6 x 4  15 x 3  3 x 2


3x 2 2 x 2  5 x  1
Factoring a polynomial means to re-write it using multiplication.
Factoring Techniques (so far…)
• Factor out a GCF
Ex)


3 x 4  15 x 3  3 x 2  3 x 2 x 2  5 x  1
Review from Wednesday…
 Factor out the GCF of:
2 x 3  12 x 2  18 x
Can we factor further?
Factoring Techniques (so far…)
• Special Pattern: Perfect Square Trinomial
•
Ex)
a 2  2ab  b 2  a  b 
2
x 2  6 x  9   x  3
2
Review from Wednesday…
 Factor out the GCF of:
2 x 2  18
Can we factor further?
Factoring Techniques (so far…)
• Special Pattern: Difference of Squares
•
Ex)
a 2  b 2  a  b a  b 
x 2  9   x  3 x  3
Review from Wednesday…
 Factor out the GCF of: 2 x 4
 54 x
Can we factor further?
Factoring Techniques (so far…)
• Special Pattern: Sum/Difference of Cubes
•

a 3  b 3  a  b  a 2  ab  b 2
S
•
Ex)

O
AP
a 3  b 3  a  b  a 2  ab  b 2

x 3  27   x  3 x 2  3 x  9



Learning Log Summary
LT 10 – I can factor polynomials by using the GCF, by
recognizing patterns, and by grouping.
To factor means…
Special cases of factoring include…
Closure
Homework (record on Learning Log):
LT 10: pg. 185 ~ 1-22 (All)
Factoring Polynomials
 Multiply:
x  8x  7 
Factoring Polynomials
 Factor:
x 2  15 x  56
Factoring Polynomials
 Factor:
x 2  8 x  15
Factoring Polynomials
 Factor:
x 2  2 x  15
 Factor:
x 2  2 x  15
Factoring Polynomials
 Factor:
3x 2  8 x  5
Factoring Polynomials
 Factor:
4r 2  8r  3
Factoring Polynomials
 Factor:
6 s 2  st  5t 2
Learning Log Summary
LT 11 – I can factor quadratic polynomials and identify
prime polynomials.
To factor means…
A strategy to factor polynomials is…
Closure
Homework (record on Learning Log):
LT 10: pg. 191 ~ 1-29 (Odds)
Warm-Up
Factor completely:

x 2  3 x  10

3 x 6  48 x 2
Solving Polynomial Equations
Find a solution to the equation by guess-and-check:

x 2  3 x  10  0
Zero Product Property
If a b  0 , then either a=0 or b=0.
Product
x 2  3 x  10  0
Can we re-write as a product?
Solving Polynomial Equations
Solve:

x 2  x  30
Solving Polynomial Equations
Solve:
 3x
3
 4 x2 x  1
Solving Polynomial Equations
The solutions to a polynomial equation can also be
called “roots” or “zeros” since they are the values of
the equation that produce an output of 0.
 Find the roots of:
3




f x  x  4  43 x  16 
Learning Log Summary
LT 12 – I can solve polynomial equations using factoring
and the Zero Product Property.
To factor means…
A strategy to factor polynomials is…
Closure
Homework (record on Learning Log):
LT 12: pg. 196 ~ 1-20 (All)