A Matrix is named by the Definition: Matrix A · A soil analysis of Hector’s lawn determined that it ne Step 1: Write the Let Statements defining the variables Step 2: Make a Table

Date:___________ Created by: Loren L. Spencer

1

Main Idea

Naming Matrices

Note: A Matrix is named by the

number of rows and columns it has.

Rule: The number of rows comes

first followed by the number of

columns (Rows x Columns)

Solving Systems of Equations

Step 1: Write the system as a

matrix. The coefficients are the

elements

Step 2: Use the commands below

to enter into the calculator

Press 2ND (blue button)

Press X-1 Matrix (blue)

Press (to highlight edit)

Press Enter

Enter the matrix size

Press (2) Enter

Press (3) Enter

Now Enter the Elements (Press enter after each element)

Use (-) for negative numbers


Press MODE QUIT (blue)

Press CLEAR



Press (to highlight math)

Press (to highlight B:rref)

Press Enter



Press Enter (TWICE)

Step 3: Write the solution

Through

Matrices Solving Systems of Equations

Definition: Matrix (plural is matrices)-A

rectangular array of numbers, symbols, or expressions,

arranged in rows and columns. The individual items in a

matrix are called its elements or entries.

Example: 3 x 4 Matrix with 12 elements

Example: 2y – 3x = 19

3y + 3x = -9

Note: The solution is the last column of the matrix

and follows the order of the variable in the system.

Caution 1: The rest of the matrix must have ones on

the diagonal and zeroes everywhere else.

Colum

ns

Colum

ns

Colum

ns

Colum

ns

Rows

Rows

Rows

A

E

A I

B

F

J

C

G

K

D

H

L Colum

ns

Colum

ns

Solut

ion

Colum

ns

Rows

Rows

Rows

1

0

A 0

0

1

0

0

C

0

1

3

7

8


2

Main Idea

Matrices


matrix. The coefficients are the

elements



Solving Systems of Equations




Press Enter


Press (2) Enter

Press (3) Enter

Now Enter the Elements

(Press enter after each element)




Press CLEAR





Press Enter



Press Enter (TWICE)

Step 3: Write the solution

Through


Definition: Matrix (plural is matrices)- A

rectangular array of numbers, symbols, or expressions,

arranged in rows and columns. The individual items in a

matrix are called its elements or entries.

Example: 3x + y = 10

-2x - y = 5

Example: r + s = 8

r - s = 9

Example: a = -2

4b – 3a = 18

Note: The solution is the last column of the matrix

and follows the order of the variable in the system.

Caution 1: The rest of the matrix must have ones on

the diagonal and zeroes everywhere else.

Colum

ns

Colum

ns

Solut

ion

Colum

ns

Rows

Rows

Rows

1

0

A 0

0

1

0

0

C

0

1

3

7

8


3

Main Idea

Matrices


matrix. The coefficients &

constants are the elements

Step 2: Place the variables in

columns on the left side of =

Step 3: Place Constants on the

right side of =

Remember: The sign changes for

all variables and constants that

swap sides of =.






Press Enter


Press 2 Enter

Press 3 Enter





Press CLEAR





Press Enter



Press Enter (TWICE)

Through


Example: Solve the following system of equations

by Matrices. 7d = - 9y + 14

– 6y = d - 2


by Matrices. – 5c = 7 - 2d

-17 - 3d = -2c


4

Main Idea

Matrices


matrix. The coefficients &

constants are the elements

Step 2: Place the variables in

columns on the left side of =

Step 3: Place Constants on the

right side of =

Remember: The sign changes for

all variables and constants that

swap sides of =.






Press Enter


Press 2 Enter

Press 3 Enter





Press CLEAR





Press Enter



Press Enter (TWICE)

Through



by Matrices. 6x − 12y = 24 −x − 6y = 4

Example: −16 + 20x − 8y = 0

36 = −18y − 22x

Example: −9 + 5y = −4x

−11x = −20 + 9y

Example: −25 = 5y

5x + 20 = −4y


5

Main Idea

Mixture Problems

Step 1: Write the Let

Statements defining the variables

Step 2: Make a Table

(Organize the information)

Step 3: Write your equations

Hint 1: Write the equation

referring to the amounts first.

(Looks like x+y=#)

Hint 2: The 2nd equation will have

the coefficients.

Step 4: Solve using Matrix

Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through

Writing Systems of Equations

Example: The owner of Sprouts wants to mix raisins

that sell at $5.75 per pound with nuts which sell for

$4.00 per pound. How many pounds of raisins and how

many pounds of nuts must be used if he wants to make

a 100 pound mix that sells for $4.70 per pound?

Let

Let

Labels for

each Variable

1st Equation Rates/Percents

amounts/dollars

2nd Equation

Total/Mixture


6

Main Idea

Mixture Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: A soil analysis of Hector’s lawn determined

that it needed 50 kilograms of a fertilizer containing

20% nitrogen. How can this mixture be made from two

different fertilizers, one containing 25% nitrogen and

the other containing 15% nitrogen?

Let

Let

Labels for

each Variable


amounts/dollars

2nd Equation

Total/Mixture


7

Main Idea

Mixture Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: A chemist has two acid solutions. One is

45% pure and the other is 90% pure. The chemist has

an order for 10 grams of an acid solution that is 72%

pure. How many grams of each acid solution is needed?

Let

Let

Labels for

each Variable


amounts/dollars

2nd Equation

Total/Mixture


8

Main Idea

Mixture Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: A pharmacist determines that 20 liters of

an antiseptic containing 15% peroxide is needed. How

can this be made from two antiseptics, one containing

12% peroxide and the other containing 17% peroxide?

Let

Let

Labels for

each Variable


amounts/dollars

2nd Equation

Total/Mixture


9

Main Idea

Mixture Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: A restaurant needed to buy 24 new tables,

but only had $3080 dollars to spend. If the larger

tables cost $145 each and the smaller tables cost $120

each, how many tables of each sized did she buy?

Let

Let

Labels for

each Variable


amounts/dollars

2nd Equation

Total/Mixture


10

Main Idea

Money Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: An auditorium seats 2,500 people. How

many balcony tickets must be sold for $4.50 each and

how many symphony tickets must be sold for $5.25

each in order to receive total receipts of $12,675 each

time the auditorium is full?

Let

Let

Labels for

each

Variable

1st

Equation

Rates/ Dollar

amounts/percent

2nd

Equation

Total


11

Main Idea

Money Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: Gina worked a total of 57 hours at her two

jobs. Her job at the restaurant paid $9.50 an hour and

her job at the beauty salon paid $10.75 an hour. Her

total earned for the 57 hours was $572.75. How many

hours did she work at each job?

Let

Let

Labels for

each

Variable

1st

Equation

Rates/ Dollar

amounts/percent

2nd

Equation

Total


12

Main Idea

Money Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: Daniel has 12 coins in dimes and quarters.

The total value of the coins is $1.95. How many

quarters and how many dimes does Daniel have

Let

Let

Labels for

each

Variable

1st

Equation

Rates/ Dollar

amounts/percent

2nd

Equation

Total


13

Main Idea

Money Problems








(Looks like x+y=#)


the coefficients.


Operations

Key Words for Rate:

Rate

Per

Percent

Each

For Each

Every

For every

Through


Example: Cora invested $5,000. She invested part

at a rate of 9% and the rest at a rate of 8.5%. The

total interest earned for the year was $442.50. Find

the amount invested at each rate.

Remember: I=prt

Intereret=amount invested × rate × time

Let

Let

Labels for

each

Variable

1st

Equation

Rates/ Dollar

amounts/percent

2nd

Equation

Total


14

Main Idea

Finding the Rule

or finding the nth term.

Step 1: Make a chart of the

values in the sequence.

Step 2: Number each value

beginning with 1. (always start with

the left hand number)

Step 3: Write ___ x N ___

Step 4: Find the Common

Difference and place it in the blank

before the N

Step 5: Find the zero term and

place it in the blank after the N.

Remember: If the zero term is

negative you must have a minus sign

and you must have a plus sign if the

zero term is positive.

Remember:

___ x N ___

Through

Sequences

Definitions:

1. Sequence- a list of numbers that follows a

pattern

2. Common Difference- the difference between

any two terms in a sequence.

Example: Find the rule for the pattern 4,7,10,13…




Com

mon

Diffe

rence

Zero

Term

CD Zero

term


15

Main Idea

Finding the Rule or finding the nth term.

Step 1: Number each picture

going from left to right beginning

with one.

Step 2: Count and record the

objects in each picture.

Step 3: Make a table containing

the information from step 1 and 2.




before the N







Remember:

___ x N ___

Through

Sequences

Example: Find the rule for the pattern.



Example: What rule describes the pattern for the

perimeter in the figures below?

Com

mon

Diffe

rence

Zero

Term

CD Zero

term

1 1 1

1 1 1

1 1

1 1

1 1

1 1 1 1

1 1

1 1

1 1

1 1 1

1

1 1

1 1

1 1

1 1

1 1 1


16

Main Idea

Finding the Rule










before the N




negative, you must have a minus

sign. You must have a plus sign if

the zero term is positive.

Remember:

___ x N ___

Through

Sequences

Definitions:

1. Sequence- a list of numbers that follows a

pattern

2. Term-a number or element in a sequence

3. Arithmetic Sequence- a sequence in which the

terms change by the same amount each time.

4. Common Difference- the difference between

any two terms in a sequence.

5. Zero Term-the n or x term is zero. The term

where the equation crosses the y-axis.

6. Rule-The expression that describes a sequence

Find the rule for the following

Example: 41, 35, 29, 23…

Example:

Example: 1.0, 1.5, 2.0, 2.5…

Example: -.15, -.95, -1.75, -2.55, -3.35

𝟏

𝟖

𝟑

𝟒 1

𝟑

𝟖 2 2

𝟓

𝟖

Com

mon

Diffe

rence

Zero

Term

CD Zero

term


17

Main Idea

Finding Any term of a sequence









before the N



Step 6: Substitute in for N.

Step 7: Solve.





Through

Sequences

Example: Write the rule and find the 50th term for

the pattern 5,10,15,20…


the pattern 13, 20, 27, 34……


the pattern 5,12,19,26…


the pattern 29, 33, 37, 41…


18

Main Idea

Finding the Rule










before the N







Remember: Example: 4, 6, 8, 10,

___ x N ___

Through

Sequences

Example: 3, 6, 9, 12

12

11

10

9

8

7

6

5

4

3

2

1

- 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13

- 1

-2

- 3

Is this sequence proportional?

Example: 0, 3, 6, 9, 12 …

12

11

10

9

8

7

6

5

4

3

2

1

- 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13

- 1

-2

- 3

Is this sequence proportional?

Com

mon

Diffe

rence

Zero

Term

CD Zero

term


19

Main Idea

Finding the Rule










before the N







Remember: Example: 4, 6, 8, 10,

___ x N ___

Through

Sequences

Example: 2, 5, 8, 11

12

11

10

9

8

7

6

5

4

3

2

1

- 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13

- 1

-2

- 3

Example: 5, 8, 11, 14 …

12

11

10

9

8

7

6

5

4

3

2

1

- 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13

- 1

-2

- 3

Com

mon

Diffe

rence

Zero

Term

CD Zero

term


20

Main Idea

Graphing Sequences









before the N



Step 6: Graph the points

Hint 1: The Table values are your

ordered pairs.

Hint 2: The 0, 1, 2, 3,… are your x

values

Through

Graphing Sequences

Graph the following sequence: 1, 3, 5, 7

10

9

8

7

6

5

4

3

2

1

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-1

-2

-3

-4

-5

-6

-7

-8

-9

Graph the following sequence on the

Cartesian plane above 8, 5, 2, -1.


21

Main Idea

Finding the equation of a line given a graph

Step 1: Mark points on the graph

that lie on the grid.

Step 2: Place the coordinates in

marked in step 1 in a table.




before the N



Step 6: Graph the points


ordered pairs.


values

Question: What is the graph

doing at the Zero Term?

Through

Equation of the Line

Example: Find the equation for Line A.

10

9

8

7

6

5

4

3

2

1

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-1

-2

-3

-4

-5

-6

-7

-8

-9

Example: Find the equation for Line B.

Line B Line A


22

Main Idea

Graphing Sequences or finding the nth term.



with one.






ordered pairs.


values

Hint 3: The zero term is the

constant

Identify the constant and the

rate of change for the pattern

graphed.

Through

Graphing a Sequence

Definitions:

1. Rate of Change-the common difference or

the difference between any two terms in a

sequence.

2. Constant: the part of an equation or pattern

that does not change or vary—the zero term

Example: Graph the rule for the pattern

10

9

8

7

6

5

4

3

2

1

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-1

-2

-3

-4

-5

-6

-7

-8

-9


23

Main Idea

Graphing Sequences or finding the nth term.



with one.






ordered pairs.


values

Hint 3: The zero term is the

constant

Identify the constant and the

rate of change for the pattern

graphed.

Through

Graphing a Sequence

Definitions:

1. Rate of Change-the common difference or

the difference between any two terms in a

sequence.

2. Constant: the part of an equation or pattern

that does not change or vary—the zero term

Example: Graph the rule for the pattern.

10

9

8

7

6

5

4

3

2

1

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-1

-2

-3

-4

-5

-6

-7

-8

-9


24

Main Idea

Graphing Linear Inequalities

Definitions:

Linear Inequality-A linear

expression that divides a plane into

2 parts using the symbols (<,>,≤

or≥) such that one part of the

plane is the solution

Less than (<): The solution is

below the equation of the line and

does not include the line. Use a dashed line

Step 1: Write the equation in

slope intercept form.

Step 2: Make certain the slope is

expressed as a fraction.

Step 3: Locate the y-intercept on

the graph.

Step 4: With your pencil on the y-

intercept, count out the change in

the y-values—the rise (do not lift your pencil)

Step 5: Now count out the change

in the x-values—the run.

Step 6: Connect the points with a

dashed line

Hint: For a (<) shade below the

line.—If the line falls down, it will

fall on the shaded portion.

Through

Linear Inequalities

Slope intercept form y=mx+b

y= m x + b

Note: the “y” and “x” form an ordered pair (x,y)

Examples: Graph the linear inequality y<2x-4

a) Identify the y-intercept

b) Identify the slope

6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5

Where to shade: Place your pencil on the y-

intercept.

1. If the symbol is < or ≤ shade below the line.

2. If the symbol is > or ≥ shade above the line.

slope

y-interce

pt

y-coordinate

x-coord

inate


25

Main Idea


Definitions:






Greater than (>): The solution is

above the equation of the line and







the graph.







dashed line

Hint: For a (>) shade above the


not fall on the shaded portion.

Through

Linear Inequalities


y= m x + b


Examples: Graph the linear inequality y>-½x-5



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


intercept.



slope

y-interce

pt

y-coordinate

x-coord

inate


26

Main Idea


Definitions:






Greater than or equal to (≥):

The solution is above the equation

of the line and includes the line.

Use a solid line






the graph.







dashed line

Hint: For a (≥) shade above the


not fall on the shaded portion.

Through

Linear Inequalities


y= m x + b


Examples: Graph the linear inequality y≥-x



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


intercept.



slope

y-interce

pt

y-coordinate

x-coord

inate


27

Main Idea


Definitions:






Less than or equal to (≤): The

solution is below the equation of

the line and includes the line.

Use a solid line






the graph.







dashed line

Hint: For a (≤) shade below the


fall on the shaded portion.

Through

Linear Inequalities


y= m x + b


Examples: Graph the linear inequality y≤-¾x-6



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


intercept.



slope

y-interce

pt

y-coordinate

x-coord

inate


28

Main Idea

Graphing Horizontal Inequalities

Horizontal inequality-has a slope

of zero and all y values are the

same regardless of what the value

of x is.




Use a solid line

Step 1: Write the equation in the

form y≤ some number

Step 2: The number represents

the y intercept.

Step 3: Plot the y- intercept and

draw a solid horizontal line through

it.

Step 4: Shade below the line


The solution is below the equation


Use a solid line


form y≥ some number


the y intercept.


draw a solid horizontal line through

it.

Step 4: Shade above the line

Through

Linear Inequalities

Examples: Graph the linear inequality y≤-4



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5

Examples: Graph the linear inequality y≥3



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


29

Main Idea

Graphing Horizontal Inequalities

Horizontal inequality-has a slope

of zero and all y values are the

same regardless of what the value

of x is.





form y< some number


the y intercept.


draw a dashed horizontal line

through it.






form y> some number


the y intercept.


draw a dashed horizontal line

through it.

Step 4: Shade above the line

Through

Linear Inequalities

Examples: Graph the linear inequality y<-4



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5

Examples: Graph the linear inequality y>0



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


30

Main Idea

Graphing Vertical Inequalities

Vertical inequality-has an

undefined slope and all x values

are the same regardless of what

the value of y is.


solution is left of the equation of


Use a solid line


form x ≤ some number


the x intercept.

Step 3: Plot the x- intercept and

draw a solid vertical line through it.

Step 4: Shade left of the line




Use a solid line


form x ≥ some number


the y intercept.


draw a solid vertical line through it.

Step 4: Shade right of the line

Through

Linear Inequalities

Examples: Graph the linear inequality x ≤-4

a) Identify the x-intercept


6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5

Examples: Graph the linear inequality x≥3



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


31

Main Idea

Graphing Vertical Inequalities

Vertical inequality-has an

undefined slope and all x values

are the same regardless of what

the value of y is.

Less than (<): The solution is to

the left of the equation of the line

and does not include the line. Use a dashed line


form x< some number


the y intercept.


draw a dashed vertical line through

it.



to the right of the equation of the

line and does not include the line.

Use a dashed line


form y> some number


the x intercept.


draw a dashed vertical line through

it.

Step 4: Shade right of the line

Through

Linear Inequalities

Examples: Graph the linear inequality x <-4



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5

Examples: Graph the linear inequality x>0



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5


32

Main Idea







the graph.







dashed or solid line and shade







Use a solid line




Use a solid line




Through

Linear Inequalities

Note: Flip the inequality sign when multiplying or

dividing by a negative.

Hint: Re-write the inequality with the y on the

left.

Remember: Flip the sign if you flip the equation

Examples: Graph the linear inequality 4y + 3x < -6

a) y-intercept c) solid or dashed

b) slope d) above or below

6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5

Where to shade: Place your pencil on the y-intercept.




33

Main Idea







the graph.














Use a solid line




Use a solid line




Through

Linear Inequalities




left.


Examples: Graph the linear inequality -y + x ≤ -2



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





34

Main Idea







the graph.














Use a solid line




Use a solid line




Through

Linear Inequalities




left.


Examples: Graph the linear inequality 5x – 3y ≥-15



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





35

Main Idea







the graph.














Use a solid line




Use a solid line




Through

Linear Inequalities




left.


Examples: Graph the linear inequality 5x <-5y



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





36

Main Idea







the graph.














Use a solid line




Use a solid line




Through

Linear Inequalities




left.


Examples: Graph the linear inequality 2x >-8 – 4y



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





37

Main Idea


Step 1: Write each equation in


Step 2: Make certain the slopes

are expressed as a fraction.

Step 3: Graph the 1st equation.

Note: Locate the y-intercept on

the graph and then count out the

slope


dashed or solid line and shade.

Step 5: Graph the 2nd equation

and shade.

Hint: The solution area is where

the shading overlaps.

Remember: (<) & (>)

Use a dashed line

Remember (≤) & (≥)

Use a solid line

Remember: Flip the inequality sign when multiplying or dividing

by a negative.

Hint: Re-write the inequality

with the y on the left.

Through

Systems of Linear Inequalities

Example: Graph the System of inequalities

x + y > 10

y < x + 4

Line 1 Line 2

a) y-intercept a) y-intercept

b) slope b) slope

c) solid or dashed c) solid or dashed

d) shade above or below d) shade above or below

6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





38

Main Idea









slope




and shade.



Remember: (<) & (>)

Use a dashed line


Use a solid line


by a negative.



Through



x - 2y ≥ 6

3x + 3y ≥ 9

Line 1 Line 2


b) slope b) slope



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





39

Main Idea









slope




and shade.



Remember: (<) & (>)

Use a dashed line


Use a solid line


by a negative.



Through



y ≤ 4x - 1

y >2x + 1

Line 1 Line 2


b) slope b) slope



6

5

4

3

2

1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-1

-2

-3

-4

-5





40

Main Idea

Parts of a Parabola

The graph of a quadratic function

is a parabola.

Vertex: The point (ordered pair)

where the parabola changes

direction

Maximum (Max)–when the

parabola opens Down

“a” is negative”

Minimum (Min)-when the

parabola opens up

“a” is positive”

Roots: The point(s) where the

graph crosses the x-axis

Hint 1: The y value will always be

“Zero” for the root (#, 0)

Note 1: root, solution, factors,

x-intercept are used to mean the

same thing.

Note 2: there may be 1 root,

2 roots or no roots

Line of Symmetry-a vertical line

which passes through the

x-coordinate of the vertex.

Remember: x =

Hint 1: If the x value is not

obvious you can find it by adding

the roots and dividing by 2.

Note: “a” also effects how wide

or narrow a parabola is

Through

Quadratic Functions Definition: Quadratic Function- a function which is

defined by the equation y=ax2+bx+c where a, b & c are

real numbers and a≠0

Identify the Characteristics:

Vertex ( , )

Up or Down

“a” positive or negative

Max or Min

Roots 0, 1, 2

Line of symmetry

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1


41

Main Idea

Parts of a Parabola


is a parabola.



direction


parabola opens Down



parabola opens up








same thing.


2 roots or no roots




Remember: x =






Through





Vertex ( , )

Up or Down


Max or Min

Roots 0, 1, 2

Line of symmetry

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2


42

Main Idea

Parts of a Parabola


is a parabola.



direction


parabola opens Down



parabola opens up








same thing.


2 roots or no roots




Remember: x =






Through





Vertex ( , )

Up or Down


Max or Min

Roots 0, 1, 2

Line of symmetry

Quadratic Parent Function: y=x2

a=1, b=0, c=0

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

Q


43

Main Idea

Finding the Equation given

the Graph

“a” is the factor of dilation for

a quadratic

Rule 1: If is < 1 the parabola

is wider than the parent function.

Rule 2: If is > 1 the parabola

is narrower than the parent

function.

Rule 3: The Vertex is (h, k)

Step 1: Write the equation form

y = a (x – h)2 + k

Step 2: Write the vertex

ordered pair ( h, k ) & plug into the

equation.

Caution: h lies & k tells the truth

Step 3: Plug in an (x,y) orderd

pair into the equation.

Step 4: Solve for “a”

Step 5: Plug “a” and the vertex

( h, k ) into the equation


Remember: a is positive -if the

parabola opens up

Remember: a is negative -if the

parabola opens down

Through

Quadratic Functions

Vertex Form of the Quadratic

y = a (x – h)2 + k

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

a

a Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1

Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1


44

Main Idea


the Graph


a quadratic





function.



y = a (x – h)2 + k



equation.









parabola opens up


parabola opens down

Through

Quadratic Functions


y = a (x – h)2 + k

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1

a

a

Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1


45

Main Idea


the Graph


a quadratic





function.



y = a (x – h)2 + k



equation.









parabola opens up


parabola opens down

Through

Quadratic Functions


y = a (x – h)2 + k

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

a

a Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1

Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1


46

Main Idea


the Graph


a quadratic





function.



y = a (x – h)2 + k



equation.









parabola opens up


parabola opens down

Through

Quadratic Functions


y = a (x – h)2 + k

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

a

a Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1

Vertex

( , )

Line of symmetry

x =

Roots: 0, 1, 2

a=

Open:

Up or down

Max or Min

Wider / Narrower

a

-1


47

Main Idea

Graphing the Vertex Form of

the Equation

y = a (x – h)2 + k

Step 1: Write the ordered pair

for the vertex (h, k) & graph.


Step 2: Identify & write “a”

(Write “a” as a fraction)

Step 3: Finding x-values

Place your pencil on the vertex &

count out the denominator of “a”

The denominator is the x-

direction

Plug the resulting x-value into

the equation and solve for y

Plot and record the resulting

ordered pair

Hint: If the denominator of “a”

is even, place your pencil on the

vertex & count ½ of the

denominator of “a”

Step 4: Graph a point equal distant

from the line of symmetry.

Note: y-values will be the same.

x-values will be different.

Step 5: Connect points with a curve

Through

Quadratic Functions

Vertex Form of the Quadratic: y = a (x – h)2 + k

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

-1

Y = (x + 1)2 - 8 2 7


48

Main Idea


the Equation

y = a (x – h)2 + k










direction




ordered pair










Through

Quadratic Functions


10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

Y = - (x - 3)2 + 9 2 5

-1

a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower


49

Main Idea


the Equation

y = a (x – h)2 + k










direction




ordered pair










Through

Quadratic Functions


10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

Y = - (x + 1)2 + 9 3 8

-1

a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower


50

Main Idea


the Equation

y = a (x – h)2 + k










direction




ordered pair










Through

Quadratic Functions


10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

-1

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

Y = (x - 2)2 - 10 5 4


51

Main Idea


the Equation

y = a (x – h)2 + k










direction




ordered pair










Through

Quadratic Functions


y=4(x+5)2-6

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

-1

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower


52

Main Idea


the Equation

y = a (x – h)2 + k










direction




ordered pair










Through

Quadratic Functions


y=-2(x-3)2-1

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

-1

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower


53

Main Idea

Graphing Intercept Form Of a Quadratic Function

Step 1: Change the sign of the

values inside the parenthesis and

record them as ordered pairs with

y=0 (these are your roots/x-intercepts)

Step 2: Graph the x-intercepts

Step 3: Add the X-values

together and divide the total by 2.

Step 4: Plot the value found in

step 3 on the x-axis and draw a

vertical line through it. x = # (this is your line of symmetry)

Step 5: Plug the resulting x-value

found in step 3 into the equation

and solve for y

Step 6: Create an ordered pair

using the x-value from step 3 and

the y-value from step 5. Graph it. This is the vertex


curve.





function.

Through

Quadratic Functions Quadratic Intercept Form

y=a(x-p)(x-h)

or

y=a(x-r1)(x-r2)

Example: Graph the function y=-2(x+3)(x-1)

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1

Example: Graph the function y=¼(x+3)(x-1)

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

a

Wider or Narrower

a

Q

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

( , ) ( , )

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

( , ) ( , )

Line of symmetry

x =

Vertex

( , )

a


54

Main Idea

Graphing Intercept Form Of a Quadratic Function with only (1) Root




y=0 (this is the root/ the x-intercept

and the vertex)

Step 2: Graph the

x-intercept/vertex

Step 3: Draw the line of symmetry

Step 4: From the line of symmetry,

count over the denominator of “a”

Hint: If “a” is an even fraction

divide the denominator of “a” by 2

and count that distance from both

sides of the line of symmetry.

Step 5: Plug the resulting x-values

found in step 4 into the equation

and solve for y


using the x-values from step 4 and

the y-values from step 5 & graph.


curve.





function.

Through

Quadratic Functions Quadratic Intercept Form

y=a(x-p)(x-h)

or

y=a(x-r1)(x-r2)

Example: Graph the function y=3(x+2)2

Rewrite

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

Example: Graph the function y=-⅛(x-1)2

Rewrite

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

-4

-5

-6

-7

-8

-9

a

Wider or Narrower

a

Q

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

( , ) ( , )

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

( , ) ( , )

Line of symmetry

x =

Vertex

( , )

a


55

Main Idea

Graphing Intercept Form Of a Quadratic Function with only (1) Root




y=0 (this is the root/ the x-intercept

and the vertex)

Step 2: Graph the

x-intercept/vertex

Step 3: Draw the line of symmetry

Step 4: From the line of symmetry,

count over the denominator of “a”

Hint: If “a” is an even fraction

divide the denominator of “a” by 2

and count that distance from both

sides of the line of symmetry.

Step 5: Plug the x-values found in

step 4 into the equation and solve

for y


using the x-values from step 4 and

the y-values from step 5 & graph.


curve.





function.

Through

Quadratic Functions

Quadratic Intercept Form

y=a(x-p)(x-h)

or

y=a(x-r1)(x-r2)

Example: Graph the function y=-¾(x+2)2

Rewrite

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

-4

-5

-6

-7

-8

-9

Example: Graph the function y=⅝(x-3)2

Rewrite

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

a

Wider or Narrower

Q

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

( , ) ( , )

Line of symmetry

x =

Vertex

( , )

Q

a=

Open:

Up or down

Max or Min

Roots: 0, 1, 2

( , ) ( , )

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

a

a


56

Main Idea


the Graph

Quadratic Intercept Form y=a(x-p)(x-h)

or

y=a(x-r1)(x-r2)

or for 1 root

y=a(x-r1)2

Step 1: Determine which form of

the intercept equation to use by

counting the roots. 1- root


y=a(x-r1)2

Step 3: Write the roots ordered

pair & plug them into the equation

(CAUTION: insiders lie)




Step 6: Plug “a” and the “x” –

value of the root into the equation



parabola opens up


parabola opens down

Through

Quadratic Functions

Definition: Dilation-a transformation that reduces

or enlarges a figure.

“a” is the factor of dilation for a quadratic

Rule 1: If is < 1 the parabola is wider than the

parent function.

Rule 2: If is > 1 the parabola is narrower than

the parent function.

y=a(x-r1)(x-r2)

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

a

a


57

Main Idea


the Graph


or

y=a(x-r1)(x-r2)

or for 1 root

y=a(x-r1)2



counting the roots. 1- root


y=a(x-r1)2








value of the root into the equation



parabola opens up


parabola opens down

Through

Quadratic Functions

Dilation: a transformation that reduces or enlarges a

figure. “a” is the factor of dilation for a quadratic

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12


58

Main Idea


the Graph


or

y=a(x-r1)(x-r2)

or for 1 root

y=a(x-r1)2



counting the roots. 2- roots


y=a(x-r1)(x-r2)








values of the roots into the

equation (CAUTION: insiders lie)


parabola opens up


parabola opens down

Through

Quadratic Functions

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1


59

Main Idea


the Graph


or

y=a(x-r1)(x-r2)

or for 1 root

y=a(x-r1)2



counting the roots. 2- roots


y=a(x-r1)(x-r2)








values of the roots into the

equation (CAUTION: insiders lie)


parabola opens up


parabola opens down

Through

Quadratic Functions

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12


60

Main Idea

Graphing the Standard Form

of a Quadratic Equation

Step 1: Identify a, b, & c

Step 2: Graph the line of symmetry

To find “x =”, plug in a & b into

the line of symmetry equation

Step 3: Graphing the vertex

To find “y”, plug the x-value found

in step 2 into the original equation

and solve

Step 4: Finding more points




direction




ordered pair





Step 5: Graph a point equal-

distant from the line of symmetry.



Step 6: If possible, plot the y-

intercept and a point equal-distant

from the line of symmetry

Through

Quadratic Functions

Standard Form of the Quadratic: y = ax2 + bx + c

a = dilation and direction

c = y-intercept

line of symmetry Passes through the vertex

y = 2x2+ 4x - 3

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

a=

Open:

Up or d-1

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , ) a

Wider or Narrower

x = 2a -b

x = 2a -b

-1


61

Main Idea










and solve





direction




ordered pair












Through

Quadratic Functions



c = y-intercept


10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

Y = - x2+6x-4 3 4

-1

a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

x = 2a -b

x = 2a -b


62

Main Idea










and solve





direction




ordered pair












Through

Quadratic Functions



c = y-intercept


y=-4x2-16x-10

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

x = 2a -b

x = 2a -b a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

-1


63

Main Idea










and solve





direction




ordered pair












Through

Quadratic Functions



c = y-intercept


10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

Y = x2 – 5x - 5 5

4

-1

a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

x = 2a -b

x = 2a -b


64

Main Idea










and solve





direction




ordered pair












Through

Quadratic Functions



c = y-intercept


10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

Y = x2 – 4 1

8

-1

a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

x = 2a -b

x = 2a -b


65

Main Idea










and solve





direction




ordered pair












Through

Quadratic Functions



c = y-intercept


y=-3x2

10

9

8

7

6

5

4

3

2

1

-10 -9 -8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8 9 10

-2

-3

-4

-5

-6

-7

-8

-9

-10

x = 2a -b

x = 2a -b a=

Open:

Up or down

Max or Min

Roots : 0, 1, 2

Line of symmetry

x =

Vertex

( , )

a

Wider or Narrower

-1


66

Main Idea

Distributive Property

Memorize: a(b+c)=ab+ac

Note 1: You must follow all order

of operations and integer rules.

Note 2: If a variable does not

have a number in front of it, the

coefficient is 1.

Step 1: Circle the number that is

outside the parenthesis with its

sign.

Step 2: Circle each term inside

of the parenthesis with its sign.

Step 3: Draw an arrow from the

number on the outside of the

parenthesis to all of the circled

terms inside the parenthesis.

Step 4: Multiply each term inside

the parenthesis by the number

outside of the parenthesis.

Step 5: Add the exponents of

the variables

Hint: The same rules apply for

division.

Through


Variable: a symbol for a number that we don’t

know yet. (Usually represented by a letter.)

Constant: A number that is by itself or is

separated by either a (+) or (-) sign.

Coefficient: a number that is used to multiply or

divide a variable.

Term: a constant or variable which is separated by

either a (+) or (-).

***Distributive Property: a number multiplied by

something in parenthesis is equal to that number

multiplied by each term inside of the parenthesis.

Expressed as: a(b+c)=ab+ac

Example: -10(20-3)

Example: -15b(-3b+2)

Example: 2c(-4c-8c3)


67

Main Idea


Memorize: a(b+c)=ab+ac

Step 1: Circle the terms with

their signs inside of both sets of

parenthesis.


first term inside of the

parenthesis to every term inside of

the second parenthesis.


second term inside of the



Step 4: Multiply as shown by the

arrows remembering to add

exponents as necessary.

Step 5: Combine like terms.

Step 6: Place each term in

descending order based on the

exponents.





coefficient is 1.


division.

Through

Multiplying Polynomials

***Distributive Property: a number multiplied by

something in parenthesis is equal to that number

multiplied by each term inside of the parenthesis.

Expressed as: a(b+c)=ab+ac

Example: (-10+5)(20-3)

Example: (-3b+3)(b+2)

Example: (c-4)(2c-3)

Example: (2x-4)(3x+1)


68

Main Idea

Special Case Polynomials

Squaring binomials

Step 1: Circle the terms with

their signs inside of both sets of

parenthesis.


first term inside of the




second term inside of the



Step 4: Multiply as shown by the

arrows remembering to add

exponents as necessary.

Step 5: Combine like terms.

Step 6: Place each term in

descending order based on the

exponents.





coefficient is 1.


division.

Remember: When squaring a

term, always square both the

variable and the coefficient.

Through


Example: (a+5)2

Example: (3b+4)2

Example: (-2c+3)2

Example: (2x-4)2


69

Main Idea


Multiplying Polynomials in the

form of (a+b)(a-b)

Rule 1: To use this method the

terms of both binomials must be

equal

Step 1: Square the 1st term and

write it down.

Step 3: Square the 2nd term and

write it down.








coefficient is 1.

Through


Example: (a+5) (a-5)

Example: (3b+4) (3b-4)

Example: (-2c+3) (-2c-3)

Example: (2x-4) (2x+4)


70

Main Idea

Creating Factor Trees

Note: It is important to know the

division rules for 2, 3, and 5.

Step 1: Begin by dividing the

number by it smallest factor.

Step 2: Divide the resulting

quotient in step 1 by it smallest

factor.

Step 3: Repeat steps 1 and 2 until

the quotient is a prime number

Step 4: The Prime numbers will

become your bases separate by a

multiplication symbol

Step 5: The Number of times the

prime factor occurs will be the

factors exponent.

Through

Prime Factorization

Definitions: Prime Number-A Number which only

has one and itself as factors.

Factors-The numbers which when multiplied together

yield another number

Composite Number-A number which has factors

other than 1 and itself.

Example:

Answer: 23×3×52

Example: 72

Example: 96

Example: 250

600

300

150

75

25

5 5

2

2

2

3


71

Main Idea

Factoring ax2+bx+c ( + )( + )

Rule 1: The coefficient of the

first term must be 1.

Hint: The sign of the middle term

is the sign of the large factor.

Step 1: Draw 2 pairs of

parenthesis using “+” as the

operation inside the parenthesis.

Step 2: List the factors of the

3rd term’s coefficient in a t-chart.

Step 3: The sum of the factors of

the last term must equal the

middle terms coefficient

Remember: Take the square root

of any variable in the 1st and last

term.

Factoring ax2-bx+c ( - )( - )






parenthesis using “-” as the

operation inside the parenthesis.





middle terms coefficient.

Through

Factoring a=1

Example: Factor (x2+12x+20)

Example: Factor (w2+5w+6)


Example: Factor (y2-15y+14)

Example: Factor (y2-8y+15)

Example: Factor (d2-6d+8)

Large

Factor Small

Factor

Large

Factor Small

Factor


72

Main Idea

Factoring ax2-bx-c ( - )( + )






parenthesis using “+” in one & “-” in the other.








term.

Factoring ax2+bx-c ( + )( - )






parenthesis using “+” in one & “-” in the other.






Through

Factoring a=1

Example: Factor (w2-2w-15)

Example: Factor (p2-5p-14)

Example: Factor (x2-x-20)

Example: Factor (y2+3y-18)

Example: Factor (d2+6d-27)

Example: Factor (x2+2x-15)

Large

Factor Small

Factor

Large

Factor Small

Factor


73

Main Idea

Factoring ax2+bx+c for a≠1

Rule 1: ax2+bx+c ( + )( + )

Rule 2: ax2-bx+c ( - )( - )

Rule 3: ax2-bx-c ( - )( + )

Rule 4: ax2+bx-c ( + )( - )

Step 1: Multiply a × c and let

that value be c

Step 2: Re-write the trinomial

using the new c and making a=1


parenthesis using the rules above

Step 4: List the factors of the 3rd

term’s coefficient in a t-chart.




Step 6: Divide the factors by the

original a (reduce if possible)

Step 7: Slide the denominator in

front of the leading variable





term.

Through

Factoring Trinomials when a≠1

Example: Factor (2w2+5w+2)

Example: Factor (2x2+11x+5)



Large

Factor Small

Factor


74

Main Idea


Rule 1: ax2+bx+c ( + )( + )

Rule 2: ax2-bx+c ( - )( - )

Rule 3: ax2-bx-c ( - )( + )

Rule 4: ax2+bx-c ( + )( - )


that value be c


















term.

Through


Example: Factor (5y2-18y+9)

Example: Factor (3y2-8y+4)

Example: Factor (4d2-17d+4)

Example: Factor (8d2-14d+5)

Large

Factor Small

Factor


75

Main Idea


Rule 1: ax2+bx+c ( + )( + )

Rule 2: ax2-bx+c ( - )( - )

Rule 3: ax2-bx-c ( - )( + )

Rule 4: ax2+bx-c ( + )( - )


that value be c


















term.

Through


Example: Factor (4w2-15w-25)

Example: Factor (3p2-2p-5)

Example: Factor (4x2-15x-25)

Example: Factor (2x2-9x-5)

Large

Factor Small

Factor


76

Main Idea


Rule 1: ax2+bx+c ( + )( + )

Rule 2: ax2-bx+c ( - )( - )

Rule 3: ax2-bx-c ( - )( + )

Rule 4: ax2+bx-c ( + )( - )


that value be c


















term.

Through


Example: Factor (6y2+5y-6)

Example: Factor (2d2+3d-9)

Example: Factor (4x2+15x-25)

Example: Factor (21x2+19x-12)

Large

Factor Small

Factor


77

Main Idea



form of (a+b)(a-b)

Rule 1: The 1st terms of both

binomials must be equal

Rule 2: The 2nd terms of the

binomials must sum to zero


write it down.


write it down.








coefficient is 1.

Through


Product of a Sum and Difference (a+b)(a-b) = a2 - b2

or

(x-y)(x+y) = x2 - y2



Example: (-2c+3) (-2c-3)



78

Main Idea

Factoring Difference of Two Squares

Rule 1: The 1st and 2nd terms must

be perfect squares.

Rule 2: The 1st and 2nd terms must

be able to be written as a

difference.

Step 1: Take the Square root of

the 1st term and make it the 1st

term in each set of parenthesis.


the 2nd term and make it the 2nd


Step 3: Separate the terms in the

1st set of parenthesis by a “+”.

Step 4: Separate the terms in the

2nd set of parenthesis by a “-”.

Note 1: The factors for the

difference of 2-squares follows

the form (a2-b2)=(a+b)(a-b)

Through

Factoring

Difference of Two Squares: One number or

variable is squared and is subtracted from another

number or variable which is also squared.

Such as: (a2-b2) or (92-x2) or (25y2-16s2)

Example: Factor (y2-x2)

Example: Factor (92-x2)

Example: Factor (25y2-16s2)

Example: Factor (49t2-1)

Example: Factor (9t2-64x4)

Example: Factor ( y4-x6)

81 4


79

Main Idea


Rule 1: The terms of both


Squaring Binomials


write it down.

Step 2: Multiply the 1st term by

the second term and multiply the

result by 2 and write it down.


write it down.








coefficient is 1.

Through


Squaring of a Sum

(a+b)2=(a+b)(a+b)=a2+2ab+b2

Example: (a+5)2

Example: (3b+4)2

Squaring of a Difference

(x-y)2=(x-y)(x-y)=x2-2xy+y2

Example: (2x-4)2

Example: (-2c+3)2


80

Main Idea

Factoring Perfect Squares

Rule 1: The first and last terms

must be squares of monomials

Rule 2: The middle term must be

twice the product of the square

roots of the 1st and last term.


parenthesis and determine the

operations using the notes below.


the 1st term and make it the 1st



the 2nd term and make it the 2nd


Note 1: If all of the terms in the

trinomial are separated by “+”, then the monomials of the factors

will be separated by a “+”.

Note 2: If the last term of the

trinomial is separated by a “+” and

the middle term is subtracted, then the monomials of the factors

will be separated by a “-”.

Through

Factoring

Perfect Square Trinomial: The square of a

binomial. Such that:

(a+b)2=(a+b)(a+b)=a2+2ab+b2

or

(x-y)2=(x-y)(x-y)=x2-2xy+y2

Example: Factor (w2+2wt+t2)


Example: Factor (x2-14x+49)

Example: Factor (9a4-24a2b3+4b6)


81

Main Idea

Converting Standard form to

Intercept Form

Quadratic Standard Form: y = ax2

+ bx + c


b=-2ax

c = y-intercept

Quadratic Intercept Form

y=a(x-r1)(x-r2)


r1= root1 and r2= root2

Step 1: Factor the standard form

Step 2: Place “a” in front of the

factors

Line of symmetry equation:

To find “x =”, plug in a & b into the

line of symmetry equation

Definition: Root—Where the

parabola crosses the x-axis.

Synonyms: Solution, x-intercept,

zeros, factor

Remember: A quadratic can have 1,

2, or 0 Roots

Through

Relating Standard Form & Intercept Form

Example:

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1

Example:

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

-4

-5

-6

-7

-8

-9

Example:

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

x = 2a -b

Factor y=2x2+8x+6

Intercept Form

Real Roots : 0, 1, or 2

( , ) ( , )

Line of symmetry

x =

Factor y=-2x2+4x-2

Intercept Form


( , ) ( , )

Line of symmetry

x =

Factor y=2x2-4x+4

Intercept Form


( , ) ( , )

Line of symmetry

x =


82

Main Idea

Discriminant

Discriminant: part of the

quadratic formula which gives the

number and types of roots. 𝒃𝟐 − 𝟒𝒂𝒄

Rule 1: If the discriminant is

greater than zero (positive),

there are 2 real roots. The

parabola crosses the x-axis 2 times

Rule 2: If the discriminant equals

zero, there is only 1 real root. The

vertex is on the x-axis

Rule 3: If the discriminant is less

than zero (negative), there are

NO Real Roots. The parabola does

not cross the x-axis.

Note: There are 2 imaginary roots

Finding the number of Roots

Step 1: Write the equation of

the discriminant

Step 2: Identify a, b and c

Step 3: Plug into the discriminant

𝒃𝟐 − 𝟒𝒂𝒄 and solve.

Step 4: Use the rules to

determine the number of roots.

Note: the line of symmetry equation

is part of the quadratic formula

Through

Quadratic Formula

Quadratic Formula: a formula to find the roots for any

quadratic equation in standard form: ax2 + bx + c = 0

Some quadratics can only be factored using the

quadratic formula

Example: Factor y= x2 + 3x + 1 without using the

quadratic

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

Now use the discriminant 𝒃𝟐 − 𝟒𝒂𝒄 a= b= c=

Example: Find the number of real roots and the line

of symmetry for: y=2x2+2x-3; a= b= c=

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =

x = 2a -b


Line of symmetry

x =


83

Main Idea

Discriminant


















the discriminant








Through

Quadratic Formula




of symmetry for: y=4x2+4x+1; a= b= c=


of symmetry for: y=x2+2x+5; a= b= c=


of symmetry for: y=2x2+2x-3; a= b= c=

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =

x = 2a -b


84

Main Idea

Discriminant


















the discriminant








Through

Quadratic Formula




of symmetry for: y=x2-9; a= b= c=


of symmetry for: y=x2+4; a= b= c=


of symmetry for: y=x2-10x+25; a= b= c=

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =

x = 2a -b


85

Main Idea

Factoring with the Quadratic Formula

Step 1: Place the equation in

standard form: ax2+bx+c=0

Step 2: Write the Quadratic

Formula


Step 4: Plug into the Quadratic

Formula

Step 5: Multiply “2a” and

simplify “-b” if possible and

calculate the discriminant: 𝒃𝟐 − 𝟒𝒂𝒄

Step 6: Finalizing the answer

A. If the discriminant is zero,

simplify the fraction-this is the

answer.(1 Real Root)

B. If the discriminant is negative,

separate the equation into two

fractions-(plus & minus). Place an

“i” outside the radical make the

discriminant positive and stop. (N0 Real Roots; 2-imaginary roots)

C. If the discriminant is positive,


fractions-(plus & minus) (2 Real Roots)

If the square root of the

discriminant has more than 2

decimal places, stop.


discriminant has less than 2

decimals, simplify completely.

Through

Quadratic Formula



Example: Factor 3x2 - 12x = -12 using the quadratic

Example: Factor -4x2 = 20x + 25 using the quadratic

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


86

Main Idea





Formula



Formula






















Through

Quadratic Formula



Example: Factor 0=2x2-4x+4 using the quadratic

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

Example: Factor y= -3x2-18x-29 using the quadratic

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


87

Main Idea





Formula



Formula






















Through

Quadratic Formula



Example: Factor -3 = -2x2 - 4x using the quadratic

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1

Example: Factor 0 = 3x2 - 8x + 2 using the quadratic

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


88

Main Idea





Formula



Formula






















Through

Quadratic Formula



Example: Factor y = -4x2 + 8x + 5 using the quadratic

9

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-1

Example: Factor 6x = 8x2– 119 using the quadratic

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


89

Main Idea

Exponents

54

Rule 1: No negative exponents

are allowed in the solution

Step 1: Write the base as many

times as the exponent indicates.

(expanded form)

Step 2: Multiply the bases

**Note: Every base has an

exponent. (If no exponent is

shown, then the exponent is 1)

Remember: The reciprocal of a

whole number equals 1 divided by

the number

Exponents raised to a negative power

4-3

Step 1: Eliminate negative

exponents by taking the reciprocal

of the bases.

Move bases with negative exponents

in the numerator to the denominator

& make the exponent positive

Move bases with negative exponents

in the denominator to the numerator

& make the exponent positive

Note: Anything raised to the zero

power equals 1

Through

Exponents

Exponent: The number of times the base is to be

used in multiplication.

Example: 54=

Example: 43=

Example: N5=

Remember: 5 = & that the reciprocal is

Example: 4-3=

Example: N-2 =

Example: =

Example: =

Example: =

Base

Base

Exponent

1 5

5

1

Exponent

x3y-4

h-5k3

h-5 1

d-3x5

a-3y5z3


90

Main Idea

Exponents-(Multiplying and Dividing)

Rule 1: No negative exponents

are allowed in the solution.

Rule 2: Add exponents when

multiplying equivalent bases.

Rule 3: Subtract the exponents

when dividing equivalent bases.

Step 1: Eliminate all negative


of the bases

Step 2: Add exponents for all

common bases in the numerator.


common bases in the denominator.

Step 4: Subtract exponents for

bases that are common in both the

numerator and denominator.



the number.

Powers raised to another Power Step 1: Make certain every base

has and exponent.

Step 2: Multiply the base’s

exponent by the power to which it

is raised.


exponents by taking reciprocals.

Through

Exponents

Example Rule 1:

Example Rule 2: y3×z5×a2×z4×a7

Example Rule 2:

Example:

Example:

Example:

Example: (43)5=

Example: (53)-2=

Example: (a-5b2)-3=

x4z-5

xy-3

x4y-4

x-2z-3y-3

x2z3y3

x-4y2

ac-4 bd-3

a4bc5 d3

ab6c2


91

Main Idea

Exponents-(Multiplying and Dividing)



of the bases


common bases in the numerator.


common bases in the denominator.

Step 4: Subtract exponents for

bases that are common in both the




the number




Powers raised to another Power Step 1: Make certain every base

has and exponent.

Step 2: Multiply the base’s

exponent by the power to which it

is raised.


exponents by taking reciprocals.

Through

Exponents

Example

Which expression is equivalent to 𝟐𝟕𝒙−𝟐𝒚𝟔

𝟑𝒙𝟓𝒚𝟐𝒛𝟎

A. 𝟗𝐱𝟕𝐲𝟒

𝐳

B. 𝐲𝟒

𝟗𝒙𝟑

C. 𝟗𝐲𝟒

𝒙𝟕

D. 𝟗𝐲𝟒

𝒙𝟕𝒛

Example Marlena was asked to find an expression that is not

equivalent to 𝟐𝟏𝟐. Which of the following is not

equivalent to the expression?

A. (𝟐𝟐)𝟔

B. (𝟐𝟖)𝟒

C. (𝟐𝟔)(𝟐𝟔)

D. (𝟐𝟑)(𝟐𝟗)


92

Main Idea

Area of a Rectangle

Step 1: Write the area formula

Step 2: Plug into the formula.

Step 3: Solve for Length =

Step 4: Simplify the exponents

1. Eliminate all negative exponents

by taking the reciprocal of the

bases

2. Add exponents for all common

bases in the numerator.

3. Add exponents for all common

bases in the denominator.

4. Subtract exponents for bases

that are common in both the


Area of a Parallelogram

Step 1: Write the area formula

Step 2: Plug into the formula.

Step 3: Solve for base =

Step 4: Simplify the exponents

following the steps above

Note: Bases do not change when

multiplying.

Through

Exponents

Example The area of a rectangle is 𝟏𝟒𝟒𝒂𝟖𝒃𝟒 square units. If

the width of the rectangle is 𝟖𝒂𝟐𝒃𝟐 units, what is the

length in units?

A 𝟏𝟖𝒂𝟔𝒃𝟐 𝒖𝒏𝒊𝒕𝒔

B 𝟏𝟑𝟔𝒂𝟔𝒃𝟐 𝒖𝒏𝒊𝒕𝒔

C 𝟏𝟓𝟐𝒂𝟏𝟎𝒃𝟔 𝒖𝒏𝒊𝒕𝒔

D 𝟏𝟏𝟓𝟐𝒂𝟏𝟎𝒃𝟔 𝒖𝒏𝒊𝒕𝒔

Example The area, A, of a parallelogram is 𝟔𝟒𝒙𝟗𝒚𝟔 square feet.

The height, h, of the parallelogram is 𝟏𝟔𝒙𝟑𝒚𝟐 feet. The

area of a parallelogram can be found using the formula

A=bh. Which of the following best represents the

length of this parallelogram’s base, b?

A. 𝟒𝒙𝟔𝒚𝟒 𝒇𝒕

B. 𝟖𝟎𝒙𝟏𝟐𝒚𝟖𝒇𝒕

C. 𝟒𝒙𝟑𝒚𝟑 𝒇𝒕

D. 𝟒𝟖𝒙𝟔𝒚𝟒 𝒇𝒕


93

Main Idea

Simplifying Square Roots

Step 1: Make a factor tree.

Begin by dividing the number by it

smallest factor.



factor.



Step 4: Circle each pair of

numbers.

Step 5: place any number that is

not paired in parenthesis under the

radical-the square root sign.

Step 6: For every circled pair

write one of the numbers outside

of the radical in parenthesis.

Step 7: Multiply the numbers in

parenthesis under the radical

together.

Step 8: Multiply the numbers

outside in parenthesis outside the

radical together.

Note: square roots undo raising a

number to the 2nd power.

𝟕𝟐 = 𝟒𝟗 = 7 𝟏𝟎𝟐 = 𝟏𝟎𝟎 = 10

Through

Square Roots

Prime Factors: a natural number or counting number

that can only be divided evenly by itself and 1. Ex. 2, 3, 5, 7, 11, 13

Square Root: a factor of some quantity which when

multiplied by itself gives you the original quantity.

Example:

(2)(5)√(𝟐)(𝟑) = 10 𝟔

Example: 𝟖𝟒

Example: 𝟏𝟎𝟖

𝟔𝟎𝟎

300 2

1502

75 2

3 25

5 5


94

Main Idea

Square Roots

Step 1: Make a factor tree.

Begin by dividing the number by it

smallest factor.



factor.



Step 4: Circle each pair of prime

numbers.

Step 5: place any number that is

not paired in parenthesis under the

radical-the square root sign.

Step 6: For every circled pair

write one of the numbers outside

of the radical in parenthesis.

Step 7: Multiply the numbers in

parenthesis under the radical

together.


outside in parenthesis outside the

radical together.

Note: square roots undo raising a

number to the 2nd power.

𝟕𝟐 = 𝟒𝟗 = 7 𝟏𝟎𝟐 = 𝟏𝟎𝟎 = 10

Through

Square Roots


that can only be divided evenly by itself and 1. Ex. 2, 3, 5, 7, 11, 13



Example: 𝟒𝟐𝟎

Example: 5 𝟐𝟕

Example: 9 𝟑𝟐


95

Main Idea


with Variables

Step 1: Make a prime factor tree

of any coefficient.

Step 2: Re-write any variables in

expanded notation


numbers.


variables

Step 5: Place any number or

variable that is not paired in

parenthesis under the radical.

Step 6: Write one of the

numbers/variables outside of the

radical in parenthesis for each pair.

Note: The number of circled pairs

of each variable is the power of

that variable.

Step 7: Multiply the numbers &

variables in parenthesis under the

radical together.


variables in parenthesis outside the

radical together.

Remember: Add exponents when

multiplying variables with like bases

Through

Square Roots


that can only be divided evenly by itself and 1. Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…



Example:

(2)(3)(a)(a)(a)(c)(c)√(𝟐)(𝟓)(𝒂)(𝒃) = 6a3c2 𝟏𝟎𝒂𝒃

Example: 𝟏𝟖 𝒙𝟓 𝒛𝟐

Example: -√𝟔 𝒙 𝒚𝟐𝒛

√𝟑𝟔𝟎 𝒂𝟕 𝒃 𝒄𝟒

45 2

180 2

2 90

3 15

3 5

a

a a

a a

a

a

c c

c c

b


96

Main Idea


with Variables


of any coefficient.


expanded notation


numbers.


variables







Note: The number of circled pairs

of each variable is the power of

that variable.



radical together.



radical together.



Through

Square Roots





Example: -4𝒑√𝟏𝟕 𝒑𝟏𝟑

Example: 5𝒙𝒚𝟐√𝟑𝟔 𝒙𝟐𝟓𝒚𝟑𝟔 𝒛𝟏𝟒

Example: -8𝒙𝒚𝟑√𝟒𝟗 𝒙𝟏𝟖 𝒚𝟐𝒛𝟒𝟒


97

Main Idea

Simplifying Square Roots of

Negative Numbers

Rule 1: The square root of a

negative number is imaginary (i)


for the number under the radical.


expanded notation


numbers.


variables







Note: If the number under the

radical is negative, make it

positive and place an “i” outside the radical.



radical together.



radical together.



Through

Square Roots





Example: −𝟗𝟎

Example: −𝟔𝟒

Example: 5 −𝟏𝟖𝟎

Example: 6 𝟐𝟕


98

Main Idea

Simplifying Square Roots of

Negative Numbers

Rule 1: The square root of a

negative number is imaginary (i)


for the number under the radical.


expanded notation


numbers.


variables








radical is negative, make it

positive and place an “i” outside the radical.



radical together.



radical together.



Through

Square Roots





Example: √−𝟏𝟔𝒙𝟖 𝒚𝟐

Example: 5𝒘𝟒𝒙√−𝟏𝟑 𝒙𝟐𝒚𝟒 𝒛𝟖

Example: -3√𝟒𝟗 𝒙𝒚𝒛


99

Main Idea


with Variables

Simplified Steps

If the variable’s exponent is Even

1. Write the variable outside the

radical.

2. Divide the exponent by 2-the

result is the variable’s exponent

Note: If the exponent is even, the

variable will not be under the

radical.


radical is negative make it positive

& place an “i” outside the radical.

If the variable’s exponent is Odd

1. Write the variable under the

radical

2. If the exponent is > “1” Write the variable outside

radical.

Subtract 1 from the exponent

Divide the resulting exponent

by two-this is the variable’s

exponent outside of the radical

Through

Square Roots

Example: (a)(a)(a)(c)(c)√(𝒂)(𝒃)

= a3c2 𝒂𝒃

Example: 𝒙𝟒 𝒛𝟐

Example: -√ 𝒚𝟐𝟒

Example: −𝒅𝟑

Example: √ −𝒇𝟗𝒅𝟐𝟕

Example: -4 √ 𝒇𝟓𝒅𝟏𝟏

c c

c c

a

a a

a a

a

a

√ 𝒂𝟕 𝒃 𝒄𝟒

b


100

Main Idea


with Variables

Simplified Steps

If the variable’s exponent is Even

1. Write the variable outside the

radical.

2. Divide the exponent by 2-the


Note: If the exponent is even, the

variable will not be under the

radical.




If the variable’s exponent is Odd

1. Write the variable under the

radical

2. If the exponent is > “1” Write the variable outside

radical.


Divide the resulting exponent

by two-this is the variable’s


Through

Square Roots

Example: −𝒙𝟒 𝒛𝟑

Example: 𝒙𝟐𝒛𝟑√ 𝒙𝟏𝟓 𝒚 𝒛𝟐

Example: -𝟓𝒚𝒛√ 𝒚𝟐𝟒 𝒛

Example: 𝟔𝒅𝟐 −𝒅𝟑

Example: −𝟕𝒇𝒅𝟕√ 𝒇𝟗𝒅𝟐𝟕

Example: 4 𝒅𝟐√− 𝒇𝟑𝒅𝟖

Example: −𝟐𝒇𝒈𝟐𝒅√ 𝒇𝒈𝟕𝒅𝟑


101

Main Idea


with Variables & Numbers

Simplified Steps

1. Separate into 2 parts

Coefficients

Variables

2. Simplifying the Coefficients

Make a factor tree

Circle every common pair

Place unpaired numbers under

the radical & multiply

Place 1 number from each pair

outside the radical & multiply




3. Simplifying the Variables

If the variable’s exponent is Even Write the variable outside the

radical.

Divide the exponent by 2-the


If the variable’s exponent is Odd Write the variable under the

radical

If the exponent is > “1” Write the variable outside

radical.


Divide the resulting exponent by

two-this is the variable’s


4. Combine Coefficients &

Variables

Through

Square Roots

Example:

−𝟑𝟔𝟎

𝒂𝟕 𝒃 𝒄𝟐

Example: −√ 𝟐𝟓𝟐𝒙 𝒚𝟏𝟐𝒛𝟖

Example: √−𝟐𝟖𝟖𝒙𝟑 𝒚𝟕𝒛𝟐

(𝟐)(𝟑)𝒊√(𝟐)(𝟓) = 𝟔𝒊 𝟏𝟎

√−𝟑𝟔𝟎 𝒂𝟕 𝒃 𝒄𝟐

15

180 2

45 2

2 90

3

5 3 a

a a a a

c c

b

a a

𝒂𝟑𝒄 𝒂 𝒃

𝟔𝐚𝟑𝐜𝒊 𝟏𝟎𝐚𝐛


102

Main Idea


with Variables & Numbers

Simplified Steps

1. Separate into 2 parts

Coefficients

Variables

2. Simplifying the Coefficients

Make a factor tree

Circle every common pair

Place unpaired numbers under

the radical & multiply

Place 1 number from each pair

outside the radical & multiply




3. Simplifying the Variables

If the variable’s exponent is Even Write the variable outside the

radical.

Divide the exponent by 2-the


If the variable’s exponent is Odd Write the variable under the

radical

If the exponent is > “1” Write the variable outside

radical.


Divide the resulting exponent by

two-this is the variable’s


4. Combine Coefficients &

Variables

Through

Square Roots

Example: -𝟑𝒙𝟐𝒚𝒛𝟑√ 𝟐𝟓𝟐𝒙 𝒚𝟏𝟐𝒛𝟖

Example: 5𝒚𝒛𝟑√ 𝟐𝟖𝟖𝒙𝟐 𝒚𝟐𝒛

Example: 2𝒙𝟐𝒛𝟑√−𝟏𝟔𝟐𝒙 𝒚𝟏𝟓𝒛𝟖


103

Main Idea

Multiplying Square Roots


outside of the radical


under the radical-the radicand

Step 3: Simplify the radicand by

making a factor tree

Rule 1: “i” times “i” equals “-1”

Rule 2: (-1) times i equals “-i ”

Through

Square Roots

Definition: Radicand-the number under the radical

Example: 𝟐 𝟔 × 𝟕 𝟏𝟓

Example: −𝟑 𝟕 × 𝟒 𝟏𝟒

Example: 𝟓 𝟐 × 𝟒 −𝟑

Example: − 𝟏𝟓 × 𝟐 −𝟏𝟎

Example: 𝟑 −𝟕 × 𝟔 −𝟏𝟒


104

Main Idea

Dividing Square Roots

Rule 1: No Square roots in the

denominator of the solution

Rule 2: Square roots undo squares.

Ex. 𝟑𝟔 = 𝟔𝟐 = 6

Step 1: Write the square roots as

a fraction.

Step 2: Reduce the numbers

outside of the radical if possible.

Step 3: Multiply the numerator

and the denominator by the value

of the radical in the denominator.

Note: this removes the radical

from the denominator.



Step 5: Simplify the numerator by




Remember: “i” times “i” equals “-1”

Remember: (-1) times i equals “-i ”

Through

Square Roots


Example: 𝟐𝟕 𝟔 ÷ 𝟏𝟐 𝟏𝟓 = 𝟐𝟕 𝟔

𝟏𝟐 𝟏𝟓 =

𝟗 𝟔

𝟒 𝟏𝟓 ×

𝟏𝟓

𝟏𝟓

= 𝟗 𝟗𝟎

𝟒(𝟏𝟓) =

𝟗 𝟗𝟎

𝟔𝟎 =

𝟑 𝟗𝟎

𝟐𝟎 = 𝟑(𝟑) 𝟏𝟎

𝟐𝟎 =

𝟗 𝟏𝟎

𝟐𝟎= 𝟗 𝟏𝟎

𝟐𝟎

𝟗𝟎 = 𝟑 𝟏𝟎

Example: −𝟑 𝟕 ÷ 𝟒 𝟏𝟒

Example: 𝟓 𝟐 ÷ 𝟒 −𝟑


105

Main Idea





Ex. 𝟑𝟔 = 𝟔𝟐 = 6


a fraction.
















Through

Square Roots


Example: − 𝟏𝟓 ÷ 𝟐 −𝟏𝟎

Example: 𝟑 −𝟕 × 𝟔 𝟏𝟒

Example: 𝟏𝟎 𝟔 ÷ 𝟒 −𝟑


106

Main Idea





Ex. 𝟑𝟔 = 𝟔𝟐 = 6


a fraction.
















Through

Square Roots


Example: 𝟗 −𝟔

𝟒 𝟏𝟓

Example: 𝟗 𝟏𝟐

𝟒 −𝟑

Example: −𝟓 𝟔

𝟑 −𝟏𝟎


107

Main Idea

Adding & Subtracting Square

Roots

Rule 1: To add or subtract Square

Roots the radicands-the number

under the radical-must be the same

Note: To combine like terms, the

variables must be the same

Note: To combine square roots,

the radicands must be the same.

Radicands are the same

Step 1: Add the numbers outside

the radicals.

Step 2: Simplify the resulting

radicand by creating a factor tree.

Radicands are Different

Step 1: Simplify the radicands by

creating a factor tree.

Step 2: If the numbers outside of

the radicals are the same, add or

subtract the numbers outside of

the radicals.

Through

Square Roots

Adding and subtracting square roots is very similar

to the concept of combining like terms. Combining

Like Terms-If the variables are the same just add or

subtract the coefficients. If the variables are

different, they cannot be simplified

Ex. 3x + 5y cannot be simplified

However, 3x + 5x = 8x

Ex. 3 𝟏𝟑 + 5 𝟏𝟕 cannot be simplified

However, 3 𝟏𝟑 + 5 𝟏𝟑 = 8 𝟏𝟑

Example: 𝟓 𝟏𝟖 - 𝟖 𝟏𝟖

Example: -𝟑 𝟕 + 𝟗 𝟕

Example: - 𝟓 - 𝟒 𝟓

Example: −𝟑 𝟏𝟖 + 𝟒 𝟐

Example: 𝟑 𝟕 + 𝟔 𝟐𝟖


108

Main Idea

Adding & Subtracting Square

Roots

Rule 1: To add or subtract Square

Roots the radicands-the number

under the radical-must be the same

Radicands are the same

Step 1: Add the numbers outside

the radicals.

Step 2: Simplify the resulting

radicand by creating a factor tree.

Radicands are Different

Step 1: Simplify the radicands by

creating a factor tree.

Step 2: If the numbers outside of

the radicals are the same, add or

subtract the numbers outside of

the radicals.

Through

Square Roots

Example: 𝟔 𝟗𝟖 - 𝟐 𝟗𝟖

Example: 𝟒 𝟓𝟎 + 𝟗 𝟓𝟎

Example: - 𝟓 + 𝟓

Example: 𝟑 𝟏𝟖 + 𝟑 𝟑

Example: 𝟏𝟖 𝟑 + 𝟔 𝟐𝟕

Example: 𝟗 𝟑 + 𝟔 𝟏𝟓


109

Main Idea

Difference of 2 squares

Definition: Conjugate-Changing of

the signs between 2 terms


form of (a+b)(a-b)




write it down.


write it down.








coefficient is 1.

Through

Multiplying Conjugates


or

(x-y)(x+y) = x2 - y2



Example: (2c+3) (2c-3)



110

Main Idea





form of (a+b)(a-b)




write it down.


write it down.








coefficient is 1.

Through


Product of a Sum and Difference (a+ 𝒃)(a- 𝒃) = a2 - b

or

(x-√𝒚)(x+√𝒚) = x2 - y

Example: (a+ 𝟓) (a- 𝟓)

Example: (3b+ 𝟐) (3b- 𝟐)

Example: (-2c+5 𝟑) (-2c-5 𝟑)

Example: (3x-4 𝟕) (2x+𝟒 𝟕)


111

Main Idea

Rationalizing Denominators


form of (a+b)(a-b)






and the denominator by the

conjugate of the denominator.



Step 2: Simplify the denominator

Step 3: Simplify all square roots

in the numerator by making a

factor tree


combining like terms.

Step 5: Reduce if possible



number outside the radical and

remove the radical.

Through


Definition: Conjugate-Changing the signs between the

2 terms of a binomial

Example: 𝟑

𝟓+ 𝟕

Example: 𝟑 𝟑

𝟓−𝟐 𝟑

Example: 𝟒−𝟐 𝟑

𝟒+𝟐 𝟑

Example: 𝟐+𝟐 𝟑

𝟐−𝟐 𝟑


112

Main Idea



form of (a+b)(a-b)







conjugate of the denominator.



Step 2: Simplify the denominator

Step 3: Simplify all square roots

in the numerator by making a

factor tree


combining like terms.




number outside the radical and

remove the radical.

Through


Definition: Conjugate-Changing the signs between the

2 terms of a binomial

Example: 𝟓 𝟑−𝟐 𝟐

𝟓 𝟑+𝟐 𝟐

Example: 𝟔−𝟐 𝟓

𝟕+𝟒 𝟓

Example: 𝟑 𝟐+𝟒 𝟓

𝟓 𝟐−𝟐 𝟓


113

Main Idea





Formula



Formula




Step 6: Simplify the radical



















Through

Quadratic Formula



Example: Factor 3x2 - 12x = -12 using the quadratic

Example: Factor -4x2 = 20x + 25 using the quadratic

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

-3

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


114

Main Idea





Formula



Formula























Through

Quadratic Formula



Example: Factor 0=2x2-4x+4 using the quadratic

13

12

11

10

9

8

7

6

5

4

3

2

1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-1

-2

Example: Factor y= -3x2-18x-29 using the quadratic

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


115

Main Idea





Formula



Formula























Through

Quadratic Formula



Example: Factor -3 = -2x2 - 4x using the quadratic

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-8

-1

Example: Factor 0 = 3x2 - 8x + 2 using the quadratic

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


116

Main Idea





Formula



Formula























Through

Quadratic Formula



Example: Factor y = -4x2 + 8x + 5 using the quadratic

9

8

7

6

5

4

3

2

1

-8 -7 -6 -5 -4 -3 -2-1

1 2 3 4 5 6 7 8

-2

-3

-4

-5

-6

-7

-1

Example: Factor 6x = 8x2– 119 using the quadratic

−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂 x =


117

Main Idea

Simplifying Exponents of Imaginary Numbers

Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1

Step 1: Divide the exponent by 4

Step 2: Determining the value of

“i” to a power

If the remainder is 1-the decimal

is .25-the value is “i”


is .50-the value is “-1”


is .75-the value is “-i”


is .00 -the value is “1”




Through


Imaginary number: any number that can be written as

the product of a real number and the imaginary unit “i”

where “i” represents the −𝟏.

Example: i17=

Example: i22=

Example: i7=

Example: i36=

Example: i0=

Example: i23=

Example: i125=

Example: i46=

Example: i54=

Example: i68=

Example: i235=


118

Main Idea

Simplifying Exponents of Complex Numbers

Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1

Step 1: Multiply the real numbers

Step 2: Sum the exponents of “i”

Step 3: Divide the exponent by 4


“i” to a power


is .25-the value is “i”


is .50-the value is “-1”


is .75-the value is “-i”


is .00 -the value is “1”

Step 5: Place the value of “i” in

parentheses and multiply by the

real number




Through

Simplifying Exponents of Complex Numbers




Example: 3i17× 6i17 =

Example: -3i9× 4i13 =

Example: 6i8× -6i17 =

Example: -5i17× -7i18 =

Example: 5i2× 6i22 =

Example: -3i54× 8i33 =

Example: -i22× -7i10 =

Example: -12i7× -6i10 =

Example: -3i17× i17 =


119

Main Idea


Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1

Rule 5: No Imaginary numbers in

the denominator.

Step 1: Divide the exponent(s) of

“i” by 4


“i” to a power

If the decimal is .25-then “i”

If the decimal is .50-then “-1”

If the decimal is .75-then “-i”

If the decimal is .00-then “1”

Step 3: If “i” is in the denominator,

multiply the fraction by 𝒊

𝒊 .


“i” by 4. Simplify using rules above.

Step 5: Reduce the fraction




Through





Example: 𝟑

𝟓𝒊𝟑𝟗=

Example: 𝟒

𝟕𝒊𝟓𝟎𝟓=

Example: 𝟗𝒊𝟏𝟕

𝟐𝟏𝒊𝟑𝟗=

Example: 𝟐𝟏𝒊𝟏𝟑

𝟏𝟎𝟓𝒊𝟏𝟒=


120

Main Idea


Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1


the denominator.


“i” by 4


“i” to a power





Step 3: If “i” is in the denominator,

multiply the fraction by 𝒊

𝒊 .


“i” by 4. Simplify using rules above.

Step 5: Reduce the fraction




Through





Example: 𝟏𝟕𝒊𝟐𝟐

𝟓𝟏𝒊𝟓=

Example: −𝟒𝒊

𝟕𝒊𝟐𝟕=

Example: 𝟗𝒊𝟏𝟕

−𝒊𝟓𝟏=

Example: 𝟗𝒊𝟑𝟓

𝟒𝟐𝒊𝟏𝟗=


121

Main Idea


Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1


the denominator.

Rule 6: Imaginary numbers cannot

be added or subtracted with real

numbers.


“i” by 4


“i” to a power





Step 3: Combine the imaginary

numbers those with “i” by + or -.

Step 4: Combine the real numbers

those without “i” by + or -.




Through

Adding Complex Numbers




Example: 3i + 8i - 5 =

Example: 13i - 27i + 17 =

Example: 13i25 - 27i27 - i =

Example: 21i11 + 17i13 - i =

Example: -18i - i – i22 =


122

Main Idea





form of (a+b)(a-b)




write it down.


write it down.







write it down.


write it down.




Remember: If a variable or “i”

does not have a number in front of

it, the coefficient is 1.

Through



or

(x-y)(x+y) = x2 - y2


Example: (2c-3) (2c+3)

Product of a Sum and Difference (a+ 𝒃)(a- 𝒃) = a2 - b

or

(x-√𝒚)(x+√𝒚) = x2 – y

Example: (3b+ 𝟐) (3b- 𝟐)

Example: (3x-4 𝟕) (2x+𝟒 𝟕)


123

Main Idea

Simplifying Imaginary

Numbers

Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1


the denominator.



numbers.

Step 1: Write the conjugate.

Step 2: Square the 1st & 2nd terms.

Step 3: Replace i2 with -1.

Step 4: Distribute the negative

and combine any like terms.




Through

Conjugates of Imaginary Numbers




Product of a Sum and Difference

(a+bi)(a-bi) = a2 + b2 (demonstrate by foil)

Multiply each of the following by their Conjugates

Example: (3x-2i)

Example: (4x+7i)

Example: (2xi - 5i)

Example: (9xi - i)


124

Main Idea


Numbers

Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1


the denominator.



numbers.



Conjugate of the denominator. (Use FOIL or the Product of a Difference)

Note: the above step should create

an i2 in the denominator.


Step 3: Distribute negatives and

Combine like terms





Through

Rationalizing Denominators Containing “i”




Example: 𝟒

𝟐𝒙−𝟑𝒊

Example: 𝟓𝒊

𝟕𝒙+𝟐𝒊

Example: −𝟗𝒊

𝒙−𝟒𝒊

Example: 𝒊

𝟑𝒙−𝒊


125

Main Idea


Numbers

Rule 1: i1 = i

Rule 2: i2 = -1

Rule 3: i3 = -i

Rule 4: i4 = 1 and i0 = 1


the denominator.



numbers.



Conjugate of the denominator. (Use FOIL or the Product of a Difference)

Note: the above step should create

an i2 in the denominator.


Step 3: Distribute negatives and

Combine like terms





Through

Rationalizing Denominators Containing “i”




Example: 𝟐𝒙+𝟑𝒊

𝟐𝒙−𝟑𝒊

Example: 𝒙+𝟓𝒊

𝟕𝒙+𝟐𝒊

Example: 𝟑𝒙+𝟐𝒊

𝟓𝒙−𝒊

Documents

A Matrix is named by the Definition: Matrix A · A soil analysis of Hector’s lawn determined that it ne Step 1: Write the Let Statements defining the variables Step 2: Make a Table