Download - Algebra II TRIG Flashcards
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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.
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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?
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Quadratic Equations• Pink Card
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Vertex Formula(Axis of Symmetry)
What is it good for?
#1
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Tells us the x-coordinate of the maximum point
Axis of symmetry
a
bx
2
#1
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Quadratic Formula
What is it good for?
#2
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Tells us the roots (x-intercepts).
a
acbbx
2
42
#2
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Describe the Steps for “Completing the
Square”• How does it compare to the quadratic
formula?
#3
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1.) Leading Coeff = 1 (Divide if necessary)
2.) Move ‘c’ over3.) 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
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General Form for DIRECT VARIATION
Characteristics & Sketch
#4
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General Form: y = kx
Characteristics: y –int = 0 (always!)Sketch: (any linear passing through the origin)
#4
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Define Inverse Variation
#5
Give a real life example
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•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
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State the General Form of an inverse variation equation.
Draw an example of a typical inverse variation
and name the graph.#6
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xy = k or . x
ky
HYPERBOLA (ROTATED)
#6
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General Form of a Circle
#7
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radiusr
Centerkh
rkyhx
),(
222
#7radius
Center
yx
2550
)0,2(
50)2( 22
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FUNCTIONSBLUE CARD
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Define Domain
Define Range
#8
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• DOMAIN - List of all possible x-values (aka – List of what x is allowed to be).
• RANGE – List of all possible y-values.
#8
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Test whether a relation (any random equation) is a FUNCTION or not?
#9
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Vertical Line Test• Each member of the DOMAIN
is paired with one and only one member of the RANGE.
#9
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Define 1 – to – 1 Function
How do you test for one?
#10
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1-to-1 Function: A function whose inverse is also a function.
Horizontal Line Test
#10
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How do you find an INVERSE Function…
ALGEBRAICALLY?
GRAPHICALLY?
#11
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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
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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
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)(1 xf
2.) f(g(x)) = x and g(f(x)) = x
3.) …then point (b,a) lies on
1.) Notation:
#12
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SHIFTSLet f(x) = x2
Describe the shift performed to f(x)• f(x) + a• f(x) – a• f(x+a)• f(x-a)
#13
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• 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
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COMPLEX NUMBERSYELLOW CARD
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Explain how to simplify
powers of i
#14
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• Divide the exponent by
4.
Remainder becomes the
new exponent.
ii 3
ii 3
12 i
ii 1
10 i
#14
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Describe How to Graph Complex Numbers
#15
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• x-axis represents real numbers
• y-axis represents imaginary numbers
• Plot point and draw vector from origin.
#15
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How do you evaluate the ABSOLUTE VALUE (Magnitude) of a
complex number?
|a + bi||2 – 5i|
#16
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Pythagorean Theorem|a + bi| = a2 + b2 = c2
|5 – 12i| = 13
#16
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How do you identify the NATURE OF THE ROOTS?
#17
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DISCRIMINANT…
acb 42 #17
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#18
acbifWhat 42
POSITIVE,
PERFECT SQUARE?
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ROOTS = Real, Rational, Unequal
• Graph crosses the x-axis twice.
#18
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POSITIVE, NON-PERFECT SQUARE
#19
acbifWhat 42
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ROOTS = Real, Irrational, Unequal
• Graph still crosses x-axis twice
#19
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ZERO
#20
acbifWhat 42
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ROOTS = Real, Rational, Equal
•GRAPH IS TANGENT TO THE X-AXIS.
#20
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NEGATIVE
#21
acbifWhat 42
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ROOTS = IMAGINARY
•GRAPH NEVER CROSSES THE
X-AXIS.
#21
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What is the SUM of the roots?
What is the PRODUCT of the roots?
#22
02 cbxax
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• SUM =
• PRODUCT =
a
b
#22
a
c
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How do you write a quadratic equation given
the roots?
#23
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• Find the SUM of the roots
• Find the PRODUCT of the roots
#23
02 productsumxx
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Multiplicative Inverse
#24
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• 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
#24
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Additive Inverse
#25
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• What you add to, to get 0.
• Additive inverse of -3 + 4i is 3 – 4i
#25
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Inequalities and Absolute Value
Green card
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Solve Absolute Value …
#26
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• Split into 2 branches• Only negate what is inside the
absolute value on negative branch.
• CHECK!!!!!
#26
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Quadratic Inequalities…
#27
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• Factor and find the roots like normal
• Make sign chart
• Graph solution on a number line (shade where +)
#27
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Solve Radical Equations …
#28
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• Isolate the radical
• Square both sides
• Solve
• CHECK!!!!!!!!!#28
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Rational Expressions
pink card
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Multiplying &
Dividing Rational Expressions
#29
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• Change Division to Multiplication flip the second fraction
• Factor
• Cancel (one on top with one on the bottom)
#29
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Adding&
Subtracting Rational Expressions
#30
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• FIRST change subtraction to addition
• Find a common denominator
• Simplify
• KEEP THE DENOMINATOR!!!!!!
#30
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Rational Equations
#31
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• First find the common denominator
• Multiply every term by the common denominator
• “KILL THE FRACTION”
• Solve
• Check your answers #31
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Complex Fractions
#32
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• Multiply every term by the common denominator
• Factor if necessary
• Simplify
#32
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Irrational Expressions
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Conjugate
#33
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•Change only the sign of the second term
• Ex. 4 + 3i conjugate 4 – 3i
#33
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Rationalize the denominator
#34
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• Multiply the numerator and denominator by the CONJUGATE
• Simplify
#34
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Multiplying &
Dividing Radicals
#35
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• Multiply/divide the numbers outside the radical together
• Multiply/divide the numbers in side the radical together
#35
3812423
2412
1563352
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Adding &
Subtracting Radicals
#36
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• 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
272324
1824
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Exponents
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When you multiply… the base and the exponents
#37
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• KEEP (the base)
• ADD (the exponents)
#37
853 222
baba xxx
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When dividing… the base&
the exponents.
#38
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• Keep (the base)
• SUBTRACT (the exponents)
#38
67
33
3
bab
a
xx
x
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Power to a power…
#39
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•MULTIPLY the exponents
#39
22
4
1
4
2
14
2
1
xxxx
xx abba
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Negative Exponents…
#40
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•Reciprocate the base
#40
666
66
1)(
22
baab
bb
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Ground Hog Rule
#41
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4
34 3 xx
xx n
mn m
#41
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Exponential Equations
y = a(b)x
Identify the meaning of a & b
#42
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• Exponential equations occur when the exponent contains a variable• a = initial amount• b = growth factor
b > 1 Growthb < 1 Decay #42
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Name 2 ways to solve an Exponential
Equation
#43
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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
#43
![Page 96: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/96.jpg)
A typical EXPONENTIAL GRAPH looks like…
#44
![Page 97: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/97.jpg)
Horizontal asymptote y = 0y = 2^x
#44
![Page 98: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/98.jpg)
Solving Equations with Fractional Exponents
#45
![Page 99: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/99.jpg)
• Get x by itself.
• Raise both sides to the reciprocal.
27
9
819
2
32
3
3
2
3
2
x
x
x
Example:
#45
![Page 100: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/100.jpg)
Logarithms
![Page 101: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/101.jpg)
Expand
1) Log (ab)
2) Log(a+b)
#46
![Page 102: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/102.jpg)
1. log(a) + log (b)
2. Done!
#46
![Page 103: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/103.jpg)
Expand
1. log (a/b)
2. log (a-b)
#47
![Page 104: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/104.jpg)
1. log(a) – log(b)
2. DONE!!
#47
![Page 105: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/105.jpg)
Expand
1. logxm
#48
![Page 106: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/106.jpg)
m log x
#48
![Page 107: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/107.jpg)
Convert exponential to log form
23 = 8
#49
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#49
![Page 109: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/109.jpg)
Convert log form to exponential form
log28 = 3
#50
![Page 110: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/110.jpg)
Follow the arrows.
823 #50
![Page 111: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/111.jpg)
Log Equations
1. every term has a log
2. not all terms have a log
#51
![Page 112: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/112.jpg)
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
#51
![Page 113: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/113.jpg)
What does a typical logarithmic graph look
like?
#52
![Page 114: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/114.jpg)
Vertical asymptote at x = 0
#52
![Page 115: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/115.jpg)
Change of Base Formula
What is it used for?
#53
![Page 116: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/116.jpg)
Used to graph logs
a
xxa log
loglog
#53
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Probability and Statistics
![Page 118: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/118.jpg)
Probability Formula…
#54
At least 4 out of 6
At most 2 out of 6
![Page 119: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/119.jpg)
rnF
rS PPnCr
At least 4 out of 6
4 or 5 or 6
At most 2
2 or 1 or 0#54
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Binomial Theorem
#55
nyx )(
![Page 121: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/121.jpg)
Watch your SIGNS!!
#55
nn
nnn
nnn baCbaCbaC )()(...)()()()( 0
011
10
![Page 122: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/122.jpg)
Summation
#56
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• "The summation from 1 to 4 of 3n":
)4(3)3(3)2(3)1(334
1
n
n
#56
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Normal Distribution
• What percentage lies within 1 S.D.?
• What percentage lies within 2 S.D.?
• What percentage lies within 3 S.D.?
#57
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• 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
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Permutation or
combination
#58
![Page 127: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/127.jpg)
Permutation – order is importantex: position, placement
Combination: order is not importantex: teams,
#58
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Mean
&
Standard deviation
#59
![Page 129: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/129.jpg)
= mean. Stat/1 var statsx
Population standard deviation
S sample standard deviation
#59
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Varience
#60
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Standard deviation squared
#60
![Page 132: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/132.jpg)
EXACT TRIG VALUES
![Page 133: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/133.jpg)
sin 30or
sin
#61
6
![Page 134: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/134.jpg)
2
1
#61
![Page 135: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/135.jpg)
sin 60orsin
#62
3
![Page 136: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/136.jpg)
#62
2
3
![Page 137: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/137.jpg)
sin 45orsin
#63
4
![Page 138: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/138.jpg)
#63
2
2
![Page 139: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/139.jpg)
sin 0
#64
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0
#64
![Page 141: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/141.jpg)
sin 90or sin
#652
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1
#65
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sin 180or
sin
#66
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0
#66
![Page 145: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/145.jpg)
sin 270or sin 2
3
#67
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-1
#67
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sin 360or sin
#68
2
![Page 148: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/148.jpg)
0
#68
![Page 149: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/149.jpg)
cos 30or cos 6
#69
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2
3
#69
![Page 151: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/151.jpg)
cos 60or
cos
3
#70
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2
1
#70
![Page 153: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/153.jpg)
cos 45or cos
4
#71
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2
2
#71
![Page 155: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/155.jpg)
cos 0
#72
![Page 156: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/156.jpg)
1
#72
![Page 157: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/157.jpg)
cos 90or cos
2
#73
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0
#73
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cos 180 or cos
#74
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-1
#74
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cos 270 or cos
2
3
#75
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0
#75
![Page 163: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/163.jpg)
cos 360or cos 2
#76
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1
#76
![Page 165: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/165.jpg)
tan 30or tan
6
#77
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3
3
#77
![Page 167: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/167.jpg)
tan 60or tan 3
#78
![Page 168: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/168.jpg)
#78
3
![Page 169: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/169.jpg)
4
tan 45or tan
#79
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1
#79
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tan 0
#80
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0
#80
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tan 90or tan 2
#81
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D.N.E.or
Undefined
#81
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tan 180or tan
#82
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0
#82
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tan 270or
tan 2
3
#83
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D.N.E. Or
Undefined#83
![Page 179: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/179.jpg)
tan 360or tan 2
#84
![Page 180: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/180.jpg)
0
#84
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Trig Graphs
![Page 182: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/182.jpg)
Amplitude
#85
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Height from the midline
y = asin(fx)y = -2sinxamp = 2
a
#85
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Frequency
#86
![Page 185: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/185.jpg)
How many complete cycles between 0 and 2
#86
![Page 186: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/186.jpg)
Period
#87
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How long it takes to complete one full cycle
Formula:
fperiod
2
#87
![Page 188: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/188.jpg)
y = sinxa) graph
b) amplitudec) frequency
d) periode) domain
f) range #88
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a)
b) 1c) 1d)e) all real numbersf)
2
1
2
11 y
x
y
#88
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y = cosxa) graph
b) amplitudec) frequency
d) periode) domain
f) range
#89
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a)
b) 1c) 1d)e) all real numbersf)
2
1
2
x
y
11 y
#89
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y = tan xa) graph
b) amplitudec) asymptotes at…
#90
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a)
b) No amplitude
c) Asymptotes are at odd multiplies of
x
y
2
Graph is always increasing
#90
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y = csc x• A) graph• B) location of the asymptotes
#91
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b) Asymptotes are multiples of
x
y
Draw in ghost sketch
#91
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y = secx
• A) graph• B) location of the asymptotes
#92
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x
y
• B) asymptotes are odd multiples of 2
Draw in ghost sketch
#92
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y=cotx
• A) graph • B) location of asymptotes
#93
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x
y
• B) multiplies of • Always decreasing
#93
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Vertical Shifts
f(x) = asin(fx) + c
#94
![Page 201: Algebra II TRIG Flashcards](https://reader036.vdocument.in/reader036/viewer/2022062322/56814edf550346895dbc74a2/html5/thumbnails/201.jpg)
* Identify the vertical shift.
Draw a ghost sketch of the midline.
3sin2)( xxf
x
y
midline
amplitude Freq = 11 cycle till 2pi
#94
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Horizontal Shift
f(x) = asin(fx+b) + c
#95
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• Horizontal Shifts go in the opposite directionSTEPS: Ignore the shift, make a ghost sketch then apply the shift!
#95
x
y
x
y
Graph y = cos(x-pi) + 3
1st graph y = cosx + 3
Now shift your graph over pi and redraw! y = cos(x-pi) + 3
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y = sin-1xor
y = arcsinxa)Sketch graph b)State domain
#96
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Domain
#96
x
y
22
x
Quadrants I & IV
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y = tan-1xor
y = arctanxa)State domainb)Sketch graph
#97
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x
y
Domain 22
x
#97Quadrants I & IV
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y = cos-1xor
y = arccosx
a)State domainb)Sketch graph
#98
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#98
x0Domain
x
y
Quadrants I & II
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Trigonometry Identities
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Reciprocal Identity
sec =#99
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cos
1
#99
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Reciprocal Identity
csc =
#100
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sin
1
#100
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cot =
Reciprocal Identity
#101
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sin
cos
tan
1or
#101
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Quotient Identity
tan#102
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cos
sin
#102