OAE Mathematics 27
Z-score
(x-mean)/SD
Cosine Graph
Period: 2π
Greatest Common Factor (GCF)
The greatest number that will divide evenly into each natural number. Also called Greatest Common Divisor
Observational Studies
The tester does not change or control the variable in any way
Population Growth Formula
f(x)=ae^(rt) a= initial population r= rate t= time
∫(1/x)dx
ln|x|
tan(A)
opp/adj
Absolute Value and Inequality Rules
|ax+b|<c -> -c<ax+b<c |ax+b|>c ->ax+b<-c or ax+b>c
derivative of tan⁻¹(u)
1/(1+u²)du/dx
π/3
60 degrees (1/2, √3/2)
Sector and Area
Area bounded by an arc and two radii A=θr²/2 in radians A=θπr²/360 in degrees
Undefined Fraction
Denominator as 0
Fundamental Theorem of Algebra
The amount of roots is at most the degree of a polynomial
Linear functions
The slope is constant
∫sec(x)tan(x)dx
sec(x)
∫cos(x)dx
sin(x)
Standard Deviation
((x₁-mean)²+...+(xⁿ-mean)²)/n Higher SD the higher the variance
∫e^cxdc
(1/c)e^x
derivative of a^x
(a^x)ln a
cot2θ
(cotθ-tanθ)/2
Interior angle sum formula
(n-2)180
tan(α±β)
(tanα±tanβ)/(1-+tanαtanβ)
Sphere Formula
(x-a)²+(y-b)²+(z-c)²=r² Center (a, b, c)
Ellipse
(x-h)²/a² + (y-k)²/b²=1 Center (h,k) Symmetry lines: horizontal (h+a,k) (h-a,k) and vertical (h, k+b) (h, k-b) Lengths of axes: 2|a| and 2|b|
Hyperbola
(x-h)²/a²-(y-k)²/b²=1 Center (h,k) Vertices: (h-a,k) and (h+a,k) Asymptotes: y=k+b/a(x-h) and y=k-b/a(x-h)
derivative of cos⁻¹(u)
-1/√(1-u²)du/dx
Maximum or minimum
-Take the second derivative >0 minimum <0 maximum
Quadratic Formula
-b±√(b²-4ac)/(2a)
∫sin(x)
-cos(x)
∫csc²(x)dx
-cot(x)
∫csc(x)cot(x)dx
-csc(x)
derivative of cos(x)
-sin(x)
0!
1
Odds against an event
1-P(A)/P(A) If 3:1, P(not occurred)=3/4
cosαcosβ
1/2(cos(α+β)+cos(α-β))
sinαsinβ
1/2(cos(α-β)-cos(α+β))
sinαcosβ
1/2(sin(α+β)+sin(α-β))
cosαsinβ
1/2(sin(α+β)-sin(α-β))
sec(A)
1/cos(A)=hyp/adj
csc(A)
1/sin(A)=hyp/opp
cot(A)
1/tan(A)=adj/opp
derivative of ln(x)
1/x
derivative of sin⁻¹(u)
1/√(1-u²)du/dx
Decagon
10
Dodecagon
12
sin2θ
2sinθcosθ
tan2θ
2tanθ/(1-tan²θ)
Determinant of a Matrix
2x2: ad-bc Larger+_+ and take determinant of smaller matrix
π/6
30 degrees (√3/2, 1/2)
π/4
45 degrees (√2/2, √2/2)
Pentagon
5
Hexagon
6
Normal Distribution rule
68-95-99.7
Heptagon
7
Octagon
8
Nonagon
9
Midpoint of Two Points
=(x₁+x₂)/2, (y₁-y₂)/2)
Golden Ratio
=Phi =(1+√5)/2 When the ratio of longer to shorter is equal to the ration of whole to longer
Rational function
A function which is a fraction constructed by two polynomial expressions -Domain: does not include values where denominator equals 0 -Vertical asymptotes where denom=0 -Num polynomial degree>denom degree → if by exactly one there will be a diagonal asymptote -Denom degree>num degree →x-axis is an asymptote -Num=denom degree →horizontal aympytote
Monotone Function
A function whose graph is constantly increasing or decreasing
Polynomial function
A function with multiple terms and multiple powers of x
Tangent
A line that touches the circle at exactly one point
Secant
A line with two points on the circle (Extension of a chord)
Direct Proportion
A relationship in which quantities increase by a set amount for every increase in the other quantity. Or both decrease
5 number summary
A set of data that gives a very informative picture of the set (minimum, max 3 quartiles)
Circle Area & Circumference
A=πr² C=2πr
f(x±k)
Added=horizontal shift left Subtracted=horizontal shift right
f(x)±k
Added=vertical shift up Subtracted= vertical shift down
Functions
An equation that has exactly one output for each input (vertical line test)
Inverse Proportion
An increase in one quantity is accompanied by a decrease in the other
Coincident
An infinite number of solutions that satisfy both equations. It is represented by a single line, since all points are on both equations
Discontinuities
And graph that has holes or vertical asymptotes
Irrational Number
Any non-terminating, non-repeating number that cannot be expressed as a fraction (Ex: pi)
Rational Numbers
Any number that can be expressed as a fraction with the numerator as an integer and the denominator is a non-zero integer
Real Number
Any number that can fall into the above categories
Complex Number
Any number that contains the imaginary number i, where i= square root of negative 1
Absolute Inequality
Any real number as the value for the variable to make the condition true, while there is no real number value that can make the condition false
Inscribed Angle and Arc
Arc measurement is double the angle ARC=2Angle
Standard Form of Line
Ax + By = C Slope= -A/B Y-intercept=C/B
Objective Probability
Based on mathematical formulas and documented evidence
Subjective Probability
Based on personal or professional feelings and judgements
Box and Whiskers plot
Box end points: 1st and 3rd quartile Vertical lines of box: median Line to minimum and maximum
Dilation
By a factor of k K 0 0 K So that all matrix values have a scalar K
Percent between two values
Calculate two z-scores, percent is the sum z=(lower-mean)/2 z=(upper-mean)/2
Orthocenter of Triangle
Concurrent point of 3 altitudes
Centroid of Triangle
Concurrent point of 3 medians
Bivariate Data
Date from 2 different variables
Sampling Distribution of the Mean
Derived from random samples of a given size 1. Mean of the sampling distribution=mean of the population that was sampled 2. Standard Error: Assuming the SD is non-zero, the SD of the sampling distribution of the mean= SD of the sampling population/√(sample size) 3. As sample sizes get larger, the sampling distribution of the mean gets closer to the normal distribution
Correlation Studies
Determine how one variable is affected by changes in a second variable
Convex Polygon
Diagonals all line within polygon
Concave Polygon
Diagonals can lie outside polygon
Limit with higher exponent in denominator
Divide top and bottom by (1/xⁿ)
One-to-One
Each element in domain is mapped to one element in range, and each element in range is mapped to one element in domain (passes vertical and horizontal line test) -They are invertible
Intersect
Exactly 1 solution that satisfies both equations. Represented by a single point where two lines intersect on a graph
Empirical Probability
Experimental probability, so actual outcomes used instead of possible outcomes
Velocity
First derivative
Parabola
For ax²+bx+c (quadratic) a is positive: parabola opens upward a is negative: parabola opens downward Axis of symmetry= -b/2a Vertex= (-b/2a, 4ac-b²/4a) Vertex form: a(x-h)²+k where v(h,k) Standard form: (x-h)²=4c(y-k)
Rational Expression
Fractions with polynomials in the numerator and denominator (denominator cannot be zero)
Survey Study
Gathering info from a small group in an attempt to gain enough info to make accurate general assumptions about the population
Mathematical Induction Proof
Given P(k) is true, show P(1) is true and that P(k+1) is true
Range
Greatest-Lowest
Cartesian Coordinate Plane Quadrants
II I III IV
Proper Subset
If the subset is not equal to the set, denoted A⊂B Not a proper subset, the to sets are equal so it's denoted A ⊆B
Compound Interest
Interest that is paid multiple times a year P=P₀(1+r/n)ⁿ^t P= total value of investment P₀= initial value t= time in years r= interest rate n= number of times per year
Simple Interest
Interest that is paid once per year I=Prt I= amount of interest P= the principal (original amount) r= annual interest rate t= amount of time in years
Limit when numerator and denominator come out to 0/0 or ∞/∞
L'Hospitals -take the derivative of top and bottom
Altitude of a Triangle
Line segment drawn from one vertex perpendicular to opposite side -Concurrent, intersect at a single point
Median of Triangle
Line segment from vertex to opposite sides midpoint
Chord of a circle
Line segment with both points on the circle
Line of Best Fit
Line that shows the trends of the data
Distance from a line to a point not on the line
Line: Ax+By+C=0 Point: (x₁,y₁) d= |Ax₁+By₁+C|/√(A²+B²)
Least Common Multiple
Lowest number that is a multiple of each number
Quartiles
Make up quarter sections of the data 1st: 25th percentile 2nd: 50th percentile...
Two secants intersect outside circle
Measure of the angle is equal to half the difference of the two arcs that lie between the secants
Two secants intersect inside circle
Measure of the vertical angles is equal to half the sum of the two intercepted arcs
Deductive Reasoning
Method uses logic to determine a true conclusion
Inductive Reasoning
Method uses to make a conjecture based on patterns and observations. Conclusion may be true or false
Median
Middle value
Matrix Translation
Moves along x or y axis x x x x1 x2 xn y y y +y1 y2 yn
Rotation
Multiplied by cosθ sinθ -sinθ cosθ
Expected Value
Multiply weights to their probability
Parallel
No solutions satisfy both equations
Monotonic Sequence
Non increasing or nondecreasing
Transcendental Function
Not algebraic function, so it can include logs, trig, variables as exponents
Statistic
Numerical value that gives info about that sample
Parameter
Numerical value that gives information about the population (mean, median, mode, etc.)
Extraneous Variables
Outside influences that can affect the outcome of the study
Compound Event Probability
P(A or B)= P(A) + P(B) - P(A and B)
Odds in favor
P(A)/(1-P(A)) If 2:5, you can expect the event to occur two times for every 5 times that it does not occur So, P=2/7
P(A and B)
P(A)×P(B) for independent events P(A)×P(B|A) for dependent events
Conditional Probability
P(A|B)= probability of B given A has already occurred
Rhombus
Parallelogram with 4 equal sides
Sine Graph
Period: 2π
Tangent Graph
Period: π
Trapezoid and Area
Quadrilateral with one pair of parallel sides A=(1/2)h(b₁+b₂)
Vector
Quantity with magnitude and direction Unit vector in direction of x-axis: i with arrow over it In direction of y-axis: j with arrow over it
Matrix Reflection
Reflects over x-axis 1 0 multiplied so that x1 x2 xn 0 -1 -y1 -y2 -yn Reflects over y axis -1 0 0 1
Triangle Theorems
SSS, SAS, ASA, AAS
Central Angle and Arc
Same angle measurements
Concentric Circle
Same center and different radii
Permutation formula
Specific number of a set of objects in a specific order, r items given a set of n items =n!/(n-r)!
Combination formula
Specific number of a set of objects in any order =n!/r!(n-r)!
Odd Function
Symmetric with respect to origin Satisfy f(x)=-f(-x) Has an odd degree
Skewness
Symmetrical- no skew Median→Mean: Positively skewed or right skewed
Percentiles
Tell what percentage of the data in the set fall below a specific point
Discriminant
The portion of the quadratic formula that is under the square root. If d=0, 1 root d=+, 2 real roots d=-, no real roots
Many-to-One
The relation is a function, but the inverse is not a function. The range may be mapped to multiple domains. (passes, vertical line test, but does no pass horizontal line test)
Sample space
Total set of all possible results
Similar Triangle
Triangles whose corresponding angles are equal and corresponding sides are proportional (not equal)
Experimental Studies
Try to prove or disprove a cause-effect relationship Control and Treatment groups
Supplementary
Two angles add up to 180
Complementary
Two angles add up to 90
Inferential Statistics
Uses samples to make predictions about the entire population
Simple Regression
Using an equation to rep a relation between an independent and dependent variable
Measure of Central Tendency
Value that gives a general tendency for the center of a group of data
Correlation Coefficient
Value that indicates how strong the relationship between the two variables of a linear regression equation
Mode
Value that occurs the most
Weighted Mean
Weighted values (w₁, w₂, w₃,...) assigned to members of the set (x₁, x₂, x₃,...) =(w₁x₁ + ... + wⁿxⁿ)/(w₁ +..+ wⁿ)
Proportional Relationship
When a linear graph passes through the origin
cos(A)
adj/hyp
Pythagorean Theorem
a²+b²=c² where c is the hypotenuse of a right triangle
derivative of sin(x)
cos(x)
cos2θ
cos²θ-sin²θ 2cos²θ-1 1-2sin²θ
cos(α±β)
cosαcosβ±sinαsinβ
Distance, Rate, Time
d=rt r=d/t t=d/r
f(x)=sqrt(ax+b)
domain: all real numbers that have ax+b≥0 range: 0 to infinity
Absolute value funtion
domain: all reals range: 0 to infinity f(x)= ax+b if ax+b≥0 and -(ax+b) if ax+b<0
Chain Rule: f(g(x))'
f'(g(x))×g'(x)
(f°g)(x)
f(g(x)) inverse of this is (g°f)(x)
Exponential Growth Function
f(x)=a(1+r)^x a= current count r= rate x= time
Even Function
f(x)=axⁿ where a is any real number and n is a positive even integer Symmetric with respect to y-axis Satisfy f(x)=f(-x)
Inverse of log and exponential
f(x)=b^x and f⁻¹(x)=log↓b(x)
Exponential Function
f(x)=bⁿ where b>0 and b≠1
Logarithmic Functions
f(x)=log₀x where the 0=b Base b can be any number but 1 (most Common base are 10 and e) log↓e(x)=ln(x) log(x) has base 10
derivative of f(x)g(x)
first times the derivative of the second plus the second times the derivative of the first
k×f(x)
k is a whole number=vertical stretch k is a fraction=vertically compressed k is negative=reflected x-axis
f(k×x)
k is whole number=compressed horizontally k is a fraction=stretched horizontally k is negative= reflected y-axis
Log rules
logb(1)=0 logb(b)=1 logb(b^p)=p logb(MN)=logb(M)+logb(N) logb(M/N)=logb(M)-logb(N) logb(M^p)=plogb(M)
derivative of f(x)/g(x)
low d high minus high d low all above the square of whats below
Slope
m=y₂-y₁/x₂-x₁
Number of diagonals
n(n-3)/2
sin(A)
opp/hyp
Arc Length Formula
s=πrθ/180 in degrees s=rθ in radians
derivative of tan(x)
sec²x
Tan(x)
sin/cos
Pythagorean Theorem Calc
sin²θ+cos²θ=1 →1+cot²θ=csc²θ →tan²θ+1=sec²θ
sin(α±β)
sinαcosβ±cosαsinβ
tan(θ/2)
sinθ/1+cosθ
cot(θ/2)
sinθ/1-cosθ
Geometric Sequence
sⁿ=a₁(1-rⁿ)/(1-r)
∫sec²(x)dx
tan(x)
Intercept Form
x/x₁ + y/y₁=1
Sum of Two Cubes
x³+y³= (x+y)(x²-xy+y²)
Difference between Two Cubes
x³-y³= (x-y)(x²+xy+y²)
Two Point Form
y-y₁/x-x₁=y₂-y₂/x₂-x₁
Point Slope Form
y-y₁=m(x-x₁) m= slope (x₁,y₁) is a point
Log and exponential relationship
y=b^x then x=logb(y)
Slope Intercept Form
y=mx+b
cos(θ/2)
±√((1+cosθ)/2)
sin(θ/2)
±√((1-cosθ)/2)
Mean symbol
µ for population ⁻x⁻ for statistic
Standard Deviation symbol
σ for population s for statistic
Variance
√(SD)
Distance Formula
√(x₂-x₁)²+(y₂-y₁)²
Integration by Parts
∫udv=uv-∫vdu