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
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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
•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)