Math 3 Flashcards

•As the year goes on we will add more and more flashcards to our collection. You do not need to bring them to class everyday…I will announce ahead of time when you need to bring them.•Your flashcards will be collected at the end of the third and fourth quarters for a grade. The grade received will be equivalent in value to a test grade. Essentially, if you lose your flashcards it will be impossible to pass the quarter.

What will my flashcards be graded on?

• Completeness – Is every card filled out front and back completely?

• Accuracy – This goes without saying. Any inaccuracies will be severely penalized.

• Neatness – If your cards are battered and hard to read you will get very little out of them.

• Order - Is your card #37 the same as my card #37?

Quadratic Equations• Pink Card

Vertex Formula

What is it good for?

#1

Tells us the x-coordinate of the maximum point

Axis of symmetry

a

bx

2

#1

Quadratic Formula

What is it good for?

#2

Tells us the roots

(x-intercepts).

a

acbbx

2

42

#2

Define Inverse Variation

#3

Give a real life example

•The PRODUCT of two variables will always be

the same (constant).• Example:

–The speed, s, you drive and the time, t, it takes for you to get to Rochester.

#3

State the General Form of an inverse variation

equation.

Draw an example of a typical inverse variation

and name the graph.#4

xy = k or . x

ky

HYPERBOLA (ROTATED)

#4

General Form of a Circle

#5

radiusr

Centerkh

rkyhx

),(

222

#5radius

Center

yx

2550

)0,2(

50)2( 22

Identify an Ellipse?

#6

Unequal CoefficientsPlus sign

2 squared terms

cbyax 22

#6

22 3104 yx

Graph an Ellipse?

#7

Set equation = 1(h,k) = center

a = horizontal radiusb = vertical radius

1

2

2

2

2

b

ky

a

hx

#7

Also on back of #7

radiusVertical

radiusHorizontal

Center

yx

yx

2

3

)1,3(

14

1

9

3

36)1(9)3(422

22

Identify Hyperbola&

Sketch Hyperbola

#8

Minus Sign2 Squared Terms

36)1(9)3(4 22 yx

#8

FUNCTIONSBLUE CARD

Define Domain

Define Range

#9

• DOMAIN - List of all possible x-values

(aka – List of what x is allowed to be).

• RANGE – List of all possible y-values.

#9

Test whether a relation (any random equation) is a FUNCTION or not?

#10

Vertical Line Test• Each member of the

DOMAIN is paired with one and only one member of the

RANGE.

#10

Define 1 – to – 1 Function

How do you test for one?

#11

1-to-1 Function: A function whose inverse is also a

function.

Horizontal Line Test

#11

How do you find an INVERSE Function…

ALGEBRAICALLY?

GRAPHICALLY?

#12

Algebraically:Switch x and y…

…solve for y.Graphically:

Reflect over the line y=x

#12

What notation do we use for Inverse?

If point (a,b) lies on f(x)…

#13

)(1 xf

…then point (b,a) lies on )(1 xf

Notation:

#13

TRANSFORMATIONS

GREEN CARD

Define ISOMETRY

#14

•A transformation that preserves distance

•A DILATION is NOT an isometry

#14

Direct Isometry

• List all examples

#15

•Preserves orientation (the order you read

the vertices)

•Translation, rotation

#15

Opposite Isometry

• List all examples

#16

•Does not preserve orientation

•Reflections

#16

f(-x)

•Identify the action

•Identify the result

#17

•Action: Negating x

•Result: Reflection over the y-axis

#17

-f(x)•Identify the action

•Identify the result

#18

•Action: negating y

•Result: Reflection over the x-axis

#18

Instead of memorizing mappings

such as (x,y)→(-y,-x)…

#19

…Just plug the point (4,1) into the mapping and plot the points to identify the transformation

(x,y)→(-y,-x)(4,1) →(-1,-4)

#19

xyr

COMPLEX NUMBERS

YELLOW CARD

Explain how to simplify powers

of i

#20

• Divide the exponent by

4.

Remainder becomes the new

exponent.

ii 3

ii 3

12 i

ii 1

10 i

#20

Describe How to Graph Complex

Numbers

#21

• x-axis represents real numbers

• y-axis represents imaginary numbers

• Plot point and draw vector from origin.

#21

How do you identify the NATURE OF THE

ROOTS?

#22

DISCRIMINANT…

acb 42 #22

#23

acbifWhat 42

POSITIVE,

PERFECT SQUARE?

ROOTS = Real, Rational, Unequal

• Graph crosses the x-axis twice.

#23

POSITIVE,

NON-PERFECT SQUARE

#24

acbifWhat 42

ROOTS = Real, Irrational,

Unequal• Graph still crosses x-axis twice

#24

ZERO

#25

acbifWhat 42

ROOTS = Real, Rational, Equal

•GRAPH IS TANGENT TO THE X-AXIS.

#25

NEGATIVE

#26

acbifWhat 42

ROOTS = IMAGINARY

•GRAPH NEVER CROSSES THE

X-AXIS.

#26

What is the SUM of the roots?

What is the PRODUCT of the roots?

#27

02 cbxax

• SUM =

• PRODUCT =

a

b

#27

a

c

How do you write a quadratic equation

given the roots?

#28

• Find the SUM of the roots

• Find the PRODUCT of the roots

#28

02 productsumxx

Multiplicative Inverse

#29

• One over what ever is given.

• Don’t forget to RATIONALIZE

• Ex. Multiplicative inverse of 3 + i

10

3

3

3

3

13

1

i

i

i

i

i

#29

Additive Inverse

#30

• What you add to, to get 0.

• Additive inverse of -3 + 4i is

3 – 4i

#30

Inequalities and Absolute Value

Pink card

Solve Absolute Value …

#31

• Split into 2 branches

• Only negate what is inside the absolute value on negative branch.

• CHECK!!!!!

#31

Quadratic Inequalities…

#32

• Factor and find the roots like normal

• Make sign chart

• Graph solution on a number line (shade where +)

#32

Solve Radical Equations …

#33

• Isolate the radical

• Square both sides

• Solve

• CHECK!!!!!!!!!#33

Probability and Statistics

blue card

Probability Formula…

#34

At least 4 out of 6

At most 2 out of 6

rnF

rS PPnCr

At least 4 out of 6

4 or 5 or 6

At most 2

2 or 1 or 0#34

Binomial Theorem

#35

nyx )(

Watch your SIGNS!!

#35

nn

nnn

nnn baCbaCbaC )()(...)()()()( 0

011

10

Summation

#36

• "The summation from 1 to 4 of 3n":  

)4(3)3(3)2(3)1(334

1

n

n

#36

Normal Distribution

• What percentage lies within 1 S.D.?

• What percentage lies within 2 S.D.?

• What percentage lies within 3 S.D.?

#37

• What percentage lies within 1 S.D.?

68%

• What percentage lies within 2 S.D.?

95%

• What percentage lies within 3 S.D.?

99%

#37

Rational Expressions

green card

Multiplying &

Dividing Rational Expressions

#38

• Change Division to Multiplication flip the second fraction

• Factor

• Cancel (one on top with one on the bottom)

#38

Adding&

Subtracting Rational Expressions

#39

• FIRST change subtraction to addition

• Find a common denominator

• Simplify

• KEEP THE DENOMINATOR!!!!!!

#39

Rational Equations

#40

• First find the common denominator

• Multiply every term by the common denominator

• “KILL THE FRACTION”

• Solve

• Check your answers#40

Complex Fractions

#41

• Multiply every term by the common denominator

• Factor if necessary

• Simplify

#41

Irrational Expressions

Conjugate

#42

• Change only the sign of the second term

• Ex. 4 + 3i

conjugate 4 – 3i#42

Rationalize the denominator

#43

• Multiply the numerator and denominator by the CONJUGATE

• Simplify

#43

Multiplying &

Dividing Radicals

#44

• Multiply/divide the numbers outside the radical together

• Multiply/divide the numbers in side the radical together

#44

3812423

2412

1563352

Adding &

Subtracting Radicals

#45

• Only add and subtract “LIKE RADICALS”

• The numbers under the radical must be the same.

• ADD/SUBTRACT the numbers outside the radical. Keep the radical #45

272324

1824

Exponents

When you multiply…

the base and

the exponents

#46

• KEEP (the base)

• ADD (the exponents)

#46

853 222

baba xxx

When dividing… the base&

the exponents.

#47

• Keep (the base)

• SUBTRACT (the exponents)

#47

67

33

3

bab

a

xx

x

Power to a power…

#48

• MULTIPLY the exponents

#48

22

4

1

4

2

14

2

1

xxxx

xx abba

Negative Exponents…

#49

• Reciprocate the base

#49

666

66

1)(

22

baab

bb

Ground Hog Rule

#50

4

34 3 xx

xx n

mn m

#50

Exponential Equations

y = a(b)x

Identify the meaning of a & b#51

• Exponential equations occur when the exponent contains a variable

• a = initial amount

• b = growth factor

b > 1 Growth

b < 1 Decay#51

Name 2 ways to solve an

Exponential Equation

#52

1. Get a common base, set the exponents equal

2. Take the log of both sides

5log

7log

7log5log

75

x

x

x

3

22

823

x

x

x

#52

A typical EXPONENTIAL GRAPH looks like…

#53

Horizontal asymptote y = 0y = 2^x

#53

Solving Equations with Fractional

Exponents

#54

• Get x by itself.

• Raise both sides to the reciprocal.

27

9

819

2

32

3

3

2

3

2

x

x

x

Example:

#54

Logarithms

Expand

1) Log (ab)

2) Log(a+b)

#55

1. log(a) + log (b)

2. Done!

#55

Expand

1. log (a/b)

2. log (a-b)

#56

1. log(a) – log(b)

2. DONE!!

#56

Expand

1. logxm

#57

m log x

#57

Convert exponential to log form

23 = 8

#58

#58

Convert log form to exponential form

log28 = 3

#59

Follow the arrows.

823 #59

Log Equations

1. every term has a log

2. not all terms have a log

#60

1. Apply log properties and knock out all the logs

2. Apply log properties condense log equationconvert to exponential and solve

112)4)(32(

)112log()4log()32log(2

2

xxx

xxx

xx

xx

xx

89

1)8)((log

1)8(loglog

21

9

99

#60

What does a typical logarithmic graph look

like?

#61

Vertical asymptote at x = 0

#61

Change of Base Formula

What is it used for?

#62

Used to graph logs

a

xxa log

loglog

#62

Coordinate Geometry

Slope formula

What is it?

When do you use it?

#63

• Used to show lines are PARALLEL (SAME SLOPE)

• Used to show lines are PERPENDICULAR (Slope are opposite reciprocal)

12

12

xx

yym

#63

Distance Formula

What is it?

What is it used for?

#64

Used to show two lines have the same length

212

212 )()( yyxxd

#64

Midpoint Formula

What is it?

What is it used for?

#65

Used to show diagonals bisect each other (THE MIDDLE)

2

,2

2121 yyxxM

#65

EXACT TRIG VALUES

sin 30or

sin #66

6

2

1

#66

sin 60orsin

#67

3

#67

2

3

sin 45orsin

#68

4

#68

2

2

sin 0

#69

0

#69

sin 90or sin

#70

2

1

#70

sin 180or

sin #71

0

#71

sin 270or sin 2

3

#72

-1

#72

sin 360or sin

#73

2

0

#73

cos 30or cos 6

#74

2

3

#74

cos 60or

cos 3

#75

2

1

#75

cos 45or cos 4

#76

2

2

#76

cos 0

#77

1

#77

cos 90or cos 2

#78

0

#78

cos 180 or cos

#79

-1

#79

cos 270 or cos 2

3

#80

0

#80

cos 360or cos 2

#81

1

#81

tan 30or tan 6

#82

3

3

#82

tan 60or tan 3

#83

#83

3

4

tan 45or tan

#84

1

#84

tan 0

#85

0

#85

tan 90or tan 2

#86

D.N.E.or

Undefined

#86

tan 180or tan

#87

0

#87

tan 270or

tan 2

3

#88

D.N.E.

Or

Undefined#88

tan 360or tan 2

#89

0

#89

Trigonometry Identities

Reciprocal Identity

sec =#90

cos

1

#90

Reciprocal Identity

csc =

#91

sin

1

#91

cot =

Reciprocal Identity

#92

sin

cos

tan

1or

#92

Quotient Identity

tan#93

cos

sin

#93

Trig Graphs

Amplitude

#94

Height from the midline

y = asin(fx)y = -2sinxamp = 2

a

#94

Frequency

#95

How many complete cycles between 0 and 2

#95

Period

#96

How long it takes to complete one full cycle

Formula:

fperiod

2

#96

y = sinx

a) graph b) amplitudec) frequency

d) periode) domain

f) range #97

a)

b) 1c) 1d)e) all real numbersf)

2

1

2

11 y

x

y

#97

y = cosx

a) graph b) amplitudec) frequency

d) periode) domain f) range

#98

a)

b) 1c) 1d)e) all real numbersf)

2

1

2

x

y

11 y

#98

y = tan x

a) graphb) amplitude

c) asymptotes at…

#99

a)

b) No amplitude

c) Asymptotes are at odd multiplies of

x

y

2

Graph is always increasing

#99

y = csc x• A) graph

• B) location of the asymptotes

#100

b) Asymptotes are multiples of

x

y

Draw in ghost sketch

#100

y = secx

• A) graph

• B) location of the asymptotes

#101

x

y

• B) asymptotes are odd multiples of 2

Draw in ghost sketch

#101

y=cotx

• A) graph

• B) location of asymptotes

#102

x

y

• B) multiplies of • Always decreasing

#102