b = y-intercept Point, slope

Midpoint
Distance
Circle

r = radius

(h, k) center

r = radius

(h, k) center

Standard Form
vertex at
Factored Form
x = r, s
Vertex Form
a > 0 = opens up

a < 0 = opens down

vertex = (h, k) Factoring

a < 0 = opens down

vertex = (h, k) Factoring

Quadratic Formula
Axis of Symmetry
Discriminant
positive = 2 x-intercepts

zero = 1 x-intercept

negative = 0 x-intercepts Perfect Squares
Difference of Squares

zero = 1 x-intercept

negative = 0 x-intercepts Perfect Squares

Complete the Square for Vertex Form
Complete the Square for Zeros

Sine Law
Cosine Law
Angles
Corresponding = 'F'

Interior Alternate = 'Z'

Exterior Alternate = 'X'

Consecutive Interior = 'C'

Interior Alternate = 'Z'

Exterior Alternate = 'X'

Consecutive Interior = 'C'

Trig Functions

__a = amplitude__

-a = reflection across x-axis

|a| > 1 = vertical stretch

0 < |a| < 1 = vertical compression

__k = period T = 2π/k, tan is π/k__

-k = reflection across y-axis

|k| > 1 = horizontal compression

0 < |k| < 1 = horizontal stretch

__c = phase shift = c/k__

-c = horizontal phase shift left

+c = horizontal phase shift right

__d = vertical shift__

-d = vertical shift down

+d = vertical shift up

-a = reflection across x-axis

|a| > 1 = vertical stretch

0 < |a| < 1 = vertical compression

-k = reflection across y-axis

|k| > 1 = horizontal compression

0 < |k| < 1 = horizontal stretch

-c = horizontal phase shift left

+c = horizontal phase shift right

-d = vertical shift down

+d = vertical shift up

Quotient Identities
Pythagorean Identities
Periodic
Compound Angle (Addition, Subtraction)

(Sum, Difference)

(Sum, Difference)

Cofunction Identities
Q1
Reciprocal Cofunction

Q2
Q3
Q4

Even/Odd
Double Angle Formulas
Half Angle Formulas

Co-Related Angle Q1 → Q2

Co-Related Angle Q1 → Q3

Co-Related Angle Q1 → Q4

Arithmetic

n = term number

t_{n} = term value

d = common difference

a = first term

S_{n} = sum

n = term number

t

d = common difference

a = first term

S

Geometric
r = common ratio
Geometric Sum to Infinity

FV = future value

PV = present value

R = initial amount (or PV)

t = time in years

i = rate / 100%

n = compounding periods per year

Daily: n = 365

Weekly: n = 52

Bi-weekly: n = 26

Monthly: n = 12

Quarterly: n = 4

Semi-annually: n = 2

Annually: n = 1

Simple Interest
Compound Interest

PV = present value

R = initial amount (or PV)

t = time in years

i = rate / 100%

n = compounding periods per year

Daily: n = 365

Weekly: n = 52

Bi-weekly: n = 26

Monthly: n = 12

Quarterly: n = 4

Semi-annually: n = 2

Annually: n = 1

Simple Interest

Annuity (Deposit)
Annuity (Withdrawal)

Symmetry

Even: ƒ(x) = ƒ(-x)

Odd: ƒ(-x) = -ƒ(x)

Neither:

ƒ(x) ≠ ƒ(-x),

ƒ(-x) ≠ -ƒ(x)

ƒ(x) ≠ ƒ(-x),

ƒ(-x) ≠ -ƒ(x)

Sum of Cubes
Difference of Cubes

Perfect Cubes (Think: Pascal's Triangle)

Combinations

(ƒ + g)(x) = ƒ(x) + g(x)

(ƒ − g)(x) = ƒ(x) − g(x)

(ƒ × g)(x) = ƒ(x) × g(x)

Compositions

(ƒ ◦ g)(x) = ƒ(g(x))

(ƒ ◦ g ◦ h)(x) = ƒ(g(h(x)))

(ƒ ◦ ƒ^{-1})(x) = (ƒ^{-1} ◦ ƒ)(x)

(g^{-1} ◦ ƒ^{-1})(x) = (ƒ ◦ g)^{-1}(x)

Vector Addition, Resultant
Direction Vectors
Cartesian 2D Line Equation
'Normal vector' from Cartesian coefficients:
(There is no Cartesian 3D Line Equation)
Cartesian (Scalar) Plane Equation
The Cartesian equation is a scalar.
Planes are parallel when their normals are collinear

Planes are coincident if D_{1} = D_{2}.
Derive 'normal vector' from Cartesian coefficients:
Vector Plane Equation
(x, y, z) = a generic point on the plane

PV = position vector [any point on plane (in relation to origin)]

SM = scalar multiple

Vec 1 & Vec 2 = coplanar vectors Parametric Equations
Symmetric Equation
From the parametric equations...
Magnitude
Dot Product

Perpendicular: Dot Product = 0 Dot Product Example
Dot Product Rule Conventions

Planes are coincident if D

PV = position vector [any point on plane (in relation to origin)]

SM = scalar multiple

Vec 1 & Vec 2 = coplanar vectors Parametric Equations

Perpendicular: Dot Product = 0 Dot Product Example

Scalar Projection
Vector Projection
Direction Cosines

Perpendicular: Direction Cosines = 0

angle with the x-axis

angle with the y-axis

angle with the z-axis
Matrix Determinant
Cross Product Procedure
[Matrix format not shown]
Cross Product Magnitude
Cross Product Conventions
Distance: 2 Points in 2-Dimensions
Distance: 2 Points in 3-Dimensions
Distance: Point and Line in 2-Dimensions
Distance: Point and Line in 3-Dimensions
Where 'N' is the point, and 'P' is a point on the line.
Distance: Point and Plane in 3-Dimensions
Cartesian equation of plane: Ax + By + Cz + D = 0

Point: (x, y, z)

Magnitude of vector of plane:

Perpendicular: Direction Cosines = 0

angle with the x-axis

angle with the y-axis

angle with the z-axis

Point: (x, y, z)

Magnitude of vector of plane:

Sum & Difference Rule
Product Rule

Constant (C) Multiple Rule
Quotient Rule

Power Rule

Where m & n are integers, and n ≠ 0

Where m & n are integers, and n ≠ 0

Average Rate of Change
(Approximate) Instantaneous Rate of Change
For small values of 'h'
Instantaneous Rate of Change (Difference Quotient)
Instantaneous Velocity
[where s(t) is position]
Average Velocity
[where s(t) is position]
Instantaneous Acceleration

Intervals

Positive interval: f(x) > 0

Negative interval: f(x) < 0

Slopes
Increasing: f'(x) > 0

Decreasing: f'(x) < 0

Local Max/Min (Critical/Stationary)
Point at 'x' where: f'(x) = 0

'Jump' Discontinuity
Oblique Asymptote

Degree Numerator = Degree Denominator + 1 Point of Inflection (POI) at 'c'

- Vertical POI at 'x' when f'(x) = undefined

- same slopes (limit, or derivative) on either side of 'x' Concavity

Degree Numerator = Degree Denominator + 1 Point of Inflection (POI) at 'c'

When opposite concavity (sign) on:

f''(c^{-}) & f''(c^{+})

f''(c

- Vertical POI at 'x' when f'(x) = undefined

- same slopes (limit, or derivative) on either side of 'x' Concavity

Concave up (slope increasing) at c: f''(c) > 0

Local minimum at c: f''(c) > 0, and f'(c) = 0

Neither concave up nor concave down: f''(x) = 0

Concave down (slope decreasing) at c: f''(c) < 0

Local maximum at c: f''(c) < 0, and f'(c) = 0

Horizontal Point of Inflection

- f'(x) = 0 at the point

- opposite slopes either side

- same concavity on either side Vertical Point of Inflection

- f'(x) Does Not Exist (DNE) at the point

- same slopes either side

- opposite concavity on either side Oblique Point of Inflection

- f'(x) defined at the point

- same slopes either side

- opposite concavity on either side Cusp Points

- f'(x) Does Not Exist (DNE) at the point

- opposite slopes either side

- same concavity on either side Corner Points

- f'(x) Does Not Exist (DNE) at the point

- opposite slopes either side

- f'(x) = 0 at the point

- opposite slopes either side

- same concavity on either side Vertical Point of Inflection

- f'(x) Does Not Exist (DNE) at the point

- same slopes either side

- opposite concavity on either side Oblique Point of Inflection

- f'(x) defined at the point

- same slopes either side

- opposite concavity on either side Cusp Points

- f'(x) Does Not Exist (DNE) at the point

- opposite slopes either side

- same concavity on either side Corner Points

- f'(x) Does Not Exist (DNE) at the point

- opposite slopes either side

Power Rule
Product Rule
Chain Rule (Substitution)
Chain Rule (Leibniz)
Chain Rule (Composition)
Quotient Rule

Displacement (d) & Position (s)
Velocity (v) & Position (s)
Acceleration (a) & Velocity (v)
Jerk (j) & Acceleration (a)
Marginal Revenue Function
Marginal Profit Function

Exponential
Natural Exponential
Logarithmic

Common Trig

Integration as Antiderivative
Rules
a, b, n, c = constants

Integration By Substitution
[This formula is a general guide, but not a precise procedure. Your steps will differ based on the question.]
Integration By Parts
(Shown in several different forms)
Definite Integrals

Mean
Variance
Standard Deviation

Combination

(Binomial Coefficient) "n choose k"
Equivalent notations
Binomial Formula
Binomial Expansion

(Binomial Coefficient) "n choose k"

Bernoulli Probability
Events that are binary

(one, or the other)
Binomial Probability
'n' trials of Bernoulli events
Geometric
perform trials until first 'success'

(one, or the other)

Permutation

Venn:
Non-disjoint
Sum of Event and its Complement

Complement can be shown as: A' or A^{c}
Union
'A' union 'B' equals either 'A', or 'B', or both.
Intersection
'A' intersect 'B' equals both 'A' and 'B'.
Addition Rule of Combined Events
Non-disjoint only

Complement can be shown as: A' or A

Conditional Probability
P('A given B'); when outcome 'A' is dependent on outcome 'B'

E.g.) Selecting cards *without* replacement.
De Morgan's Laws
Other Stuff
You don't have to memorize these.

You can work them out by looking at the Venn diagram.

E.g.) Selecting cards *without* replacement.

You can work them out by looking at the Venn diagram.

Independent Events
The __outcome of one event doesn't affect the chance of another event__.

E.g.) Selecting cards *with* replacement.
Product Rule for Independent Events
Product Rule for Dependent Events
General Rules About Reversibility

Intersections are (equivalent when) reversible

Conditionals are not (equivalent when) reversible Mutually Exclusive (Disjoint)

Two event outcomes__cannot occur at the same time__.

E.g.) In cards, the events of selecting an 'Ace' and a 'King'

E.g.) Selecting cards *with* replacement.

Intersections are (equivalent when) reversible

Conditionals are not (equivalent when) reversible Mutually Exclusive (Disjoint)

Two event outcomes

E.g.) In cards, the events of selecting an 'Ace' and a 'King'

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