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