Algebra II TRIG Flashcards As the year goes on we will add more and more flashcards to our collection. Bring your cards every TUESDAY for eliminator practice! Your flashcards will be collected on every test day! At the end of the quarter 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 ( Axis of Symmetry) What is it good for? #1

Tells us the x-coordinate of the maximum point Axis of symmetry #1

Quadratic Formula What is it good for? #2

Tells us the roots (x-intercepts). #2

Describe the Steps for “Completing the Square” How does it compare to the quadratic formula? #3

1.) Leading Coeff = 1 (Divide if necessary) 2.) Move ‘c’ over 3.) Half ‘b’ and square (add to both sides) 4.) Factor and Simplify left side. 5.) Square root both sides (don’t forget +/-) 6.) Solve for x. *Same answer as Quadratic Formula. #3

General Form for DIRECT VARIATION Characteristics & Sketch #4

General Form: y = kx Characteristics: y –int = 0 (always!) Sketch: (any linear passing through the origin) #4

Define Inverse Variation #5 Give a real life example

The PRODUCT of two variables will always be the same (constant). xy=c Example: – The speed, s, you drive and the time, t, it takes for you to get to Rochester. #5

State the General Form of an inverse variation equation. Draw an example of a typical inverse variation and name the graph. #6

xy = k or. HYPERBOLA (ROTATED) #6

General Form of a Circle #7

FUNCTIONS BLUE CARD

Define Domain Define Range #8

DOMAIN - List of all possible x-values (aka – List of what x is allowed to be). RANGE – List of all possible y-values. #8

Test whether a relation (any random equation) is a FUNCTION or not? #9

Vertical Line Test Each member of the DOMAIN is paired with one and only one member of the RANGE. #9

Define 1 – to – 1 Function How do you test for one? #10

1-to-1 Function: A function whose inverse is also a function. Horizontal Line Test #10

How do you find an INVERSE Function… ALGEBRAICALLY? GRAPHICALLY? #11

Algebraically: Switch x and y… …solve for y. Graphically: Reflect over the line y=x (look at your table and switch x & y values) #11

1.)What notation do we use for Inverse? 2.) Functions f and g are inverses of each other if _______ and ________! 3.) If point (a,b) lies on f(x)… #12

2.) f(g(x)) = x and g(f(x)) = x 3.) …then point (b,a) lies on 1.) Notation: #12

SHIFTS Let f(x) = x 2 Describe the shift performed to f(x) f(x) + a f(x) – a f(x+a) f(x-a) #13

f(x) + a = shift ‘a’ units upward f(x) – a = shift ‘a’ units down. f(x+a) = shift ‘a’ units to the left. f(x-a) = shift ‘a’ units to the right. #13

COMPLEX NUMBERS YELLOW CARD

Explain how to simplify powers of i #14

Divide the exponent by 4. Remainder becomes the new exponent. #14

Describe How to Graph Complex Numbers #15

x-axis represents real numbers y-axis represents imaginary numbers Plot point and draw vector from origin. #15

How do you evaluate the ABSOLUTE VALUE (Magnitude) of a complex number? |a + bi| |2 – 5i| #16

Pythagorean Theorem |a + bi| = a 2 + b 2 = c 2 |5 – 12i| = 13 #16

How do you identify the NATURE OF THE ROOTS? #17

DISCRIMINANT… #17

#18 POSITIVE, PERFECT SQUARE?

ROOTS = Real, Rational, Unequal Graph crosses the x-axis twice. #18

POSITIVE, NON-PERFECT SQUARE #19

ROOTS = Real, Irrational, Unequal Graph still crosses x-axis twice #19

ZERO #20

ROOTS = Real, Rational, Equal GRAPH IS TANGENT TO THE X-AXIS. #20

NEGATIVE #21

ROOTS = IMAGINARY GRAPH NEVER CROSSES THE X-AXIS. #21

What is the SUM of the roots? What is the PRODUCT of the roots? #22

SUM = PRODUCT = #22

How do you write a quadratic equation given the roots? #23

Find the SUM of the roots Find the PRODUCT of the roots #23

Multiplicative Inverse #24

One over what ever is given. Don’t forget to RATIONALIZE Ex. Multiplicative inverse of 3 + i #24

Additive Inverse #25

What you add to, to get 0. Additive inverse of i is 3 – 4i #25

Inequalities and Absolute Value Green card

Solve Absolute Value … #26

Split into 2 branches Only negate what is inside the absolute value on negative branch. CHECK!!!!! #26

Quadratic Inequalities… #27

Factor and find the roots like normal Make sign chart Graph solution on a number line (shade where +) #27

Solve Radical Equations … #28

Isolate the radical Square both sides Solve CHECK!!!!!!!!! #28

Rational Expressions pink card

Multiplying & Dividing Rational Expressions #29

Change Division to Multiplication flip the second fraction Factor Cancel (one on top with one on the bottom) #29

Adding & Subtracting Rational Expressions #30

FIRST change subtraction to addition Find a common denominator Simplify KEEP THE DENOMINATOR!!!!!! #30

Rational Equations #31

First find the common denominator Multiply every term by the common denominator “KILL THE FRACTION” Solve Check your answers #31

Complex Fractions #32

Multiply every term by the common denominator Factor if necessary Simplify #32

Irrational Expressions

Conjugate #33

Change only the sign of the second term Ex i conjugate 4 – 3i #33

Rationalize the denominator #34

Multiply the numerator and denominator by the CONJUGATE Simplify #34

Multiplying & Dividing Radicals #35

Multiply/divide the numbers outside the radical together Multiply/divide the numbers in side the radical together #35

Adding & Subtracting Radicals #36

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 #36

Exponents

When you multiply… the base and the exponents #37

KEEP (the base) ADD (the exponents) #37

When dividing… the base & the exponents. #38

Keep (the base) SUBTRACT (the exponents) #38

Power to a power… #39

MULTIPLY the exponents #39

Negative Exponents… #40

Reciprocate the base #40

Ground Hog Rule #41

Exponential Equations y = a(b) x Identify the meaning of a & b #42

Exponential equations occur when the exponent contains a variable a = initial amount b = growth factor b > 1 Growth b < 1 Decay #42

Name 2 ways to solve an Exponential Equation #43

1. Get a common base, set the exponents equal 2. Take the log of both sides #43

A typical EXPONENTIAL GRAPH looks like… #44

Horizontal asymptote y = 0 #44

Solving Equations with Fractional Exponents #45

Get x by itself. Raise both sides to the reciprocal. Example: #45

Logarithms

Expand 1) Log (ab) 2) Log(a+b) #46

1. log(a) + log (b) 2. Done! #46

Expand 1. log (a/b) 2. log (a-b) #47

1. log(a) – log(b) 2. DONE!! #47

Expand 1. logx m #48

m log x #48

Convert exponential to log form 2 3 = 8 #49

Convert log form to exponential form log 2 8 = 3 #50

Follow the arrows. #50

Log Equations 1. every term has a log 2. not all terms have a log #51

1. Apply log properties and knock out all the logs 2. Apply log properties condense log equation convert to exponential and solve #51

What does a typical logarithmic graph look like? #52

Vertical asymptote at x = 0 #52

Change of Base Formula What is it used for? #53

Used to graph logs #53

Probability and Statistics

Probability Formula… #54 At least 4 out of 6 At most 2 out of 6

At least 4 out of 6 4or5or6 At most 2 2or1 or0 #54

Binomial Theorem #55

Watch your SIGNS!! #55

Summation #56

"The summation from 1 to 4 of 3n": #56

Normal Distribution What percentage lies within 1 S.D.? What percentage lies within 2 S.D.? What percentage lies within 3 S.D.? #57

What percentage lies within 1 S.D.? 68% What percentage lies within 2 S.D.? 95% What percentage lies within 3 S.D.? 99% #57

Permutation or combination #58

Permutation – order is important ex: position, placement Combination: order is not important ex: teams, #58

Mean & Standard deviation #59

= mean. Stat/1 var stats Population standard deviation sample standard deviation #59

Varience #60

Standard deviation squared #60

EXACT TRIG VALUES

sin 30 or sin #61

sin 60 or sin #62

sin 45 or sin #63

sin 0 #64

0

sin 90 or sin #65

1

sin 180 or sin #66

0

sin 270 or sin #67

#67

sin 360 or sin #68

0

cos 30 or cos #69

cos 60 or cos #70

cos 45 or cos #71

cos 0 #72

1

cos 90 or cos #73

0

cos 180 or cos #74

#74

cos 270 or cos #75

0

cos 360 or cos #76

1

tan 30 or tan #77

tan 60 or tan #78

tan 45 or tan #79

1

tan 0 #80

0

tan 90 or tan #81

D.N.E. or Undefined #81

tan 180 or tan #82

0

tan 270 or tan #83

D.N.E. Or Undefined #83

tan 360 or tan #84

0

Trig Graphs

Amplitude #85

Height from the midline y = asin(fx) y = -2sinx amp = 2 #85

Frequency #86

How many complete cycles between 0 and #86

Period #87

How long it takes to complete one full cycle Formula: #87

y = sinx a) graph b) amplitude c) frequency d) period e) domain f) range #88

a) b) 1 c) 1 d) e) all real numbers f) #88

y = cosx a) graph b) amplitude c) frequency d) period e) domain f) range #89

a) b) 1 c) 1 d) e) all real numbers f) #89

y = tan x a) graph b) amplitude c) asymptotes at… #90

a) b) No amplitude c) Asymptotes are at odd multiplies of Graph is always increasing #90

y = csc x A) graph B) location of the asymptotes #91

b) Asymptotes are multiples of Draw in ghost sketch #91

y = secx A) graph B) location of the asymptotes #92

B) asymptotes are odd multiples of Draw in ghost sketch #92

y=cotx A) graph B) location of asymptotes #93

B) multiplies of Always decreasing #93

Vertical Shifts f(x) = asin(fx) + c #94

* Identify the vertical shift. Draw a ghost sketch of the midline. midline amplitudeFreq = 1 1 cycle till 2pi #94

Horizontal Shift f(x) = asin(fx+b) + c #95

Horizontal Shifts go in the opposite direction STEPS: Ignore the shift, make a ghost sketch then apply the shift! #95 Graph y = cos(x-pi) st graph y = cosx + 3 Now shift your graph over pi and redraw! y = cos(x-pi) + 3

y = sin -1 x or y = arcsinx a)Sketch graph b)State domain #96

Domain #96 Quadrants I & IV

y = tan -1 x or y = arctanx a)State domain b)Sketch graph #97

Domain #97 Quadrants I & IV

y = cos -1 x or y = arccosx a)State domain b)Sketch graph #98

Domain Quadrants I & II

Trigonometry Identities

Reciprocal Identity sec = #99

Reciprocal Identity csc = #100

cot = Reciprocal Identity #101

Quotient Identity #102