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EmSAT Mathematics Core Formulas

25 flashcards reviewing core formulas and principles for EmSAT Achieve preparation.

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Quadratic Formula

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Quadratic Formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

for

a x^{2} + b x + c = 0

Discriminant Analysis (

Δ = b^{2} - 4 a c

)

Δ > 0

: two distinct real roots;

Δ = 0

: one repeated real root;

Δ < 0

: two complex conjugate roots.

Pythagorean Identity

sin^{2} (θ) + cos^{2} (θ) = 1

for all real angle measures.

Vertex Form of a Parabola

y = a (x - h)^{2} + k

, where

(h, k)

is the vertex coordinates.

Slope of a Line

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

for two coordinate points on a Cartesian plane.

Slope of a Line Between Two Points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

(

x_{2} \neq = x_{1}

Logarithm Product Rule

lo g_{b} (x y) = lo g_{b} (x) + lo g_{b} (y)

for positive base and arguments.

Point-Slope Form of Linear Equation

y - y_{1} = m (x - x_{1})

Logarithm Quotient Rule

lo g_{b} (\frac{x}{y}) = lo g_{b} (x) - lo g_{b} (y)

for positive arguments.

Slopes of Parallel and Perpendicular Lines

Parallel:

m_{1} = m_{2}

; Perpendicular:

m_{1} \times m_{2} = - 1 ⟹ m_{2} = - \frac{1}{m _{1}}

Power Rule of Differentiation

\frac{d}{d x} [x^{n}] = n x^{n - 1}

for any real constant exponent

n

Distance and Midpoint Formulas

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

and

Midpoint = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Derivative of Natural Logarithm

\frac{d}{d x} [ln (x)] = \frac{1}{x}

for domain

x > 0

Logarithmic Properties

lo g (ab) = lo g a + lo g b

lo g (a / b) = lo g a - lo g b

, and

lo g (a^{k}) = k lo g a

Derivative of Exponential Function

\frac{d}{d x} [e^{x}] = e^{x}

for all real values of

x

Change of Base Formula for Logarithms

lo g_{b} (a) = \frac{l n ( a )}{l n ( b )} = \frac{l o g _{10} ( a )}{l o g _{10} ( b )}

Circle Area Formula

A = π r^{2}

, where

r

is the radius of the circle.

Power Rule for Differentiation

\frac{d}{d x} (x^{n}) = n x^{n - 1}

for all real

n

Circle Circumference Formula

C = 2 π r

, where

r

is the radius of the circle.

Derivative of

e^{k x}

and

ln (x)

\frac{d}{d x} e^{k x} = k e^{k x}

and

\frac{d}{d x} ln (x) = \frac{1}{x}

(

x > 0

Ohm's Law

V = I R

, relating electrical potential, electric current, and resistance.

Derivative of

sin (x)

and

cos (x)

(sin x)^{'} = cos x

and

(cos x)^{'} = - sin x

Newton's Second Law

F = ma

, where net force equals mass multiplied by acceleration.

Product Rule for Differentiation

(uv)^{'} = u^{'} v + u v^{'}

Coulomb's Law

F = k \frac{∣ q _{1} q _{2} ∣}{r ^{2}}

, governing electrostatic force between point charges.

Quotient Rule for Differentiation

(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Kinetic Energy Formula

K E = \frac{1}{2} m v^{2}

, scalar energy of an object resulting from its speed.

Chain Rule for Composite Functions

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Gravitational Potential Energy

P E = m g h

, energy stored relative to elevation in a uniform gravity field.

Basic Indefinite Integral of

x^{n}

(

n \neq = - 1

)

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

Avogadro's Constant

6.022 \times 1 0^{23} mol^{- 1}

, number of particles in one mole.

Fundamental Theorem of Calculus

\int_{a}^{b} f (x) d x = F (b) - F (a)

, where

F^{'} (x) = f (x)

Ideal Gas Law

P V = n R T

, relating pressure, volume, amount of gas, and absolute temperature.

Pythagorean Trigonometric Identity

sin^{2} (θ) + cos^{2} (θ) = 1

and

1 + tan^{2} (θ) = sec^{2} (θ)

Snell's Law of Refraction

n_{1} sin (θ_{1}) = n_{2} sin (θ_{2})

across an optical boundary interface.

Period of Sinusoidal Function

y = A sin (B x)

Period = \frac{2 π}{∣ B ∣}

;

Amplitude = ∣ A ∣

Boyle's Law

P_{1} V_{1} = P_{2} V_{2}

at constant temperature for a fixed amount of gas.

Mean and Median Definitions

Mean =

\frac{\sum x _{i}}{n}

; Median = middle value of ordered dataset.

Charles's Law

\frac{V _{1}}{T _{1}} = \frac{V _{2}}{T _{2}}

at constant pressure, with temperature in Kelvin.

Probability of Union of Two Events

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Molar Mass Definition

Mass of one mole of a chemical element or substance, measured in

g/mol

Independent Events Probability

P (A \cap B) = P (A) \times P (B)

if and only if

A

and

B

are independent.

Acceleration due to Gravity

g \approx 9.8 m/s^{2}

near the surface of Earth in free fall.

Conditional Probability Formula

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

for

P (B) > 0

Speed of Light in Vacuum

c \approx 3.0 \times 1 0^{8} m/s

, maximum speed of electromagnetic waves.

Permutations vs Combinations

_{n} P_{r} = \frac{n !}{( n - r )!}

(order matters);

_{n} C_{r} = \frac{n !}{r ! ( n - r )!}

(order does not matter).

Arithmetic Mean

Sum of all numerical observations divided by the total number of observations.

Sum of Finite Geometric Series

S_{n} = a_{1} \frac{1 - r ^{n}}{1 - r}

for

r \neq = 1

Standard Deviation Definition

Statistical metric measuring the degree of dispersion of data points from the mean.

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