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Pre-Calculus Cheatsheet

Definitions, Symbols, Formulas, and Notes — All in One Place.

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Functions

Term		Definition
Function		A relation between a set of inputs (domain) and a set of possible outputs (range), where each input is related to exactly one output. It is often denoted as $f (x)$ or $y = f (x)$ .
Domain		The set of all possible input values (x-values) for which the function is defined.
Range		The set of all possible output values (y-values) that the function can produce.
Vertical Line Test		A method used to determine if a relation is a function. If any vertical line intersects the graph of the relation at more than one point, then the relation is not a function.
One-to-One Function		A function where each element in the domain corresponds to exactly one element in the range, and vice versa.
Even Function		A function that is symmetric with respect to the y-axis. Mathematically, $f (x) = f (- x)$ for all $x$ in the domain.
Odd Function		A function that is symmetric with respect to the origin. Mathematically, $f (x) = - f (- x)$ for all $x$ in the domain.
Inverse Function		A function that "undoes" the effect of another function. The inverse of a function $f$ , denoted as $f^{- 1}$ , maps outputs of $f$ back to their corresponding inputs.
Composite Function		A function that results from composing (or chaining together) two or more functions. For example, if $f (x)$ and $g (x)$ are functions, then $(f \circ g) (x) = f (g (x))$ is a composite function.
Asymptote		A line that a curve approaches but never intersects as the independent variable approaches positive or negative infinity.

Trigonometric Functions

Term		Definition		Formula
Sine		In a right triangle, the ratio of the length of the side opposite the angle to the length of the hypotenuse.		$\sin (θ) = \frac{Opposite}{Hypotenuse}$
Cosine		In a right triangle, the ratio of the length of the adjacent side to the length of the hypotenuse		$\cos (θ) = \frac{Adjacent}{Hypotenuse}$
Tangent		In a right triangle, the ratio of the length of the side opposite the angle to the length of the adjacent side.		$\tan (θ) = \frac{Opposite}{Adjacent}$
Secant		The reciprocal of the cosine of an angle.		$\sec (θ) = \frac{1}{\cos (θ)}$
Cotangent		The reciprocal of the tangent of an angle.		$\cot (θ) = \frac{1}{\tan (θ)}$
Cosecant		The reciprocal of the sine of an angle.		$\csc (θ) = \frac{1}{\sin (θ)}$
Ray		A straight line that extends infinitely in one direction from a fixed point, called its endpoint.
Radian		A unit of angular measure defined as the angle subtended by an arc of a circle that has the same length as the radius of the circle. One radian is equal to approximately 57.29 degrees.		$π$
Period		The smallest positive value of the independent variable for which a function returns to its initial value. For trigonometric functions like sine and cosine, the period is the length of one complete cycle.
Arc Length		The distance along a curved line or arc.
Area of a Sector		The region bounded by an arc and the two radii that intersect at the endpoints of the arc.

Exponential Functions

Term		Definition		Formula
Exponential Function		A mathematical function of the form $f (x) = a \cdot b^{x}$ , where $a$ and $b$ are constants, and $b$ is the base of the exponential function.
Exponential Growth		A process in which a quantity increases at a rate proportional to its current value over time. It is characterized by a positive base $b > 1$ in the exponential function.
Exponential Decay		A process in which a quantity decreases at a rate proportional to its current value over time. It is characterized by a base $0 < b < 1$ in the exponential function.
Base		In the context of an exponential function, the constant $b$ raised to the power of the variable $x$ . It determines the rate of growth or decay of the function.
Initial Value		The value of an exponential function when the independent variable is zero. It is often denoted as $f (0)$ or $f (x_{0})$ .
Growth Factor		In an exponential function, the constant multiplier $b$ in the form $f (x) = a \cdot b^{x}$ . It determines the rate at which the function increases.
Decay Factor		The reciprocal of the growth factor. For example, if the growth factor is $b$ , the decay factor is $\frac{1}{b}$ .
Doubling Time		The time it takes for a quantity to double in value.		$T = \frac{\ln (2)}{k}$ , where $k$ is the growth rate
Half-Life		The time it takes for a quantity to decrease to half of its initial value.		$T = \frac{\ln (2)}{k}$ , where $k$ is the decay rate

Logarithmic Functions

Term		Definition		Formula
Logarithm		The inverse operation to exponentiation. It is a mathematical function that gives the power to which a fixed number (the base) must be raised to produce a given number.
Common Logarithm		A logarithm with base 10.		$\log (x)$
Natural Logarithm		A logarithm with base e, where e is Euler's number, approximately equal to 2.71828.		$\ln (x)$
Power Rule for Logarithms		States that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.		$\log_{b} (a^{c}) = c \cdot \log_{b} (a)$
Product Rule for Logarithms		States that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers.		$\log_{b} (a \cdot c) = \log_{b} (a) + \log_{b} (c)$
Quotient Rule for Logarithms		States that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers.		$\log_{b} (\frac{a}{c}) = \log_{b} (a) - \log_{b} (c)$
Change of Base Formula		States that the logarithm of a number with base $b$ can be expressed in terms of logarithms with base a and base $c$ , where $a$ and $c$ are any other bases.		$\log_{a} (x) = \frac{\log_{c} (x)}{\log_{c} (a)}$

Polynomial and Rational Functions

Term		Definition
Polynomial Functions		Functions that can be expressed as the sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power
Rational Functions		Functions that can be expressed as the ratio of two polynomial functions
Axis of Symmetry		A vertical line that divides a graph into two symmetric halves.
Degree		The highest power of a variable in a polynomial function.
Factor Theorem		States that if a polynomial function $f (x)$ has a factor $x - c$ , then $f (c) = 0$ . In other words, if $c$ is a root (or zero) of the polynomial, then $x - c$ is a factor of the polynomial.
Descartes' Rule of Signs		A method used to determine the possible number of positive and negative real roots of a polynomial function by examining the signs of its coefficients.
Multiplicity		The number of times a particular root or zero of a polynomial function is repeated. If a root has a multiplicity of $m$ , it means that the factor corresponding to that root is repeated $m$ times in the factored form of the polynomial.
Turning Point		A point where the function changes direction, either from increasing to decreasing or from decreasing to increasing. For a polynomial function, turning points occur at local maxima or minima.
Rational Zero Theorem		States that if a polynomial function has a rational root (or zero), then it can be expressed as the ratio of a factor of the constant term to a factor of the leading coefficient.
End Behavior		Refers to the behavior of a function as the independent variable $x$ approaches positive or negative infinity.
Division Algorithm		A method used to divide one polynomial by another polynomial. It states that any polynomial $P (x)$ can be divided by another polynomial $D (x)$ to yield a quotient $Q (x)$ and a remainder $R (x)$ , such that $P (x) = D (x) \cdot Q (x) + R (x)$ , where the degree of $R (x)$ is less than the degree of $D (x)$ .
Synthetic Division		A shortcut method used to perform polynomial division by linear divisors of the form $x - c$ , where $c$ is a constant.
Leading Coefficient		The coefficient of the term with the highest power of the variable. It determines the behavior of the polynomial function as $x$ approaches positive or negative infinity.

Complex Numbers

Term		Definition
Complex Number		A number of the form $a + b i$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ( $i^{2} = - 1$ )
Modulus		The non-negative real number $\| z \| = \sqrt{a^{2} + b^{2}}$ given a complex number $z = a + b i$ . It represents the distance of the complex number from the origin in the complex plane.
Polar Form		The representation of a complex number $z = a + b i$ in terms of its modulus $\| z \|$ and argument $θ$ , written as $z = \| z \| (\cos θ + i \sin θ)$ .
Complex Conjugate		The number $a - b i$ . It is obtained by changing the sign of the imaginary part of a complex number $a + b i$ .

Trigonometric Identities

Term		Formula
Pythagorean Identities		$\sin^{2} (θ) + \cos^{2} (θ) = 1$ $1 + \tan^{2} (θ) = \sec^{2} (θ)$ $1 + \cot^{2} (θ) = \csc^{2} (θ)$
Sum and Difference Identities		$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double Angle Identities		$\sin (2 θ) = 2 \sin θ \cos θ$ $\cos (2 θ) = \cos^{2} θ - \sin^{2} θ$ $\tan (2 θ) = \frac{2 \tan θ}{1 - \tan^{2} θ}$
Half Angle Identities		$\sin (\frac{θ}{2}) = \pm \sqrt{\frac{1 - \cos θ}{2}}$ $\cos (\frac{θ}{2}) = \pm \sqrt{\frac{1 + \cos θ}{2}}$ $\tan (\frac{θ}{2}) = \frac{\sin θ}{1 + \cos θ} = \frac{1 - \cos θ}{\sin θ}$
Product to Sum Identities		$\sin A \sin B = \frac{\cos (A - B) - \cos (A + B)}{2}$ $\cos A \cos B = \frac{\cos (A - B) + \cos (A + B)}{2}$ $\sin A \cos B = \frac{\sin (A + B) + \sin (A - B)}{2}$
Sum to Product Identities		$\sin A + \sin B = 2 \sin (\frac{A + B}{2}) \cos (\frac{A - B}{2})$ $\sin A - \sin B = 2 \cos (\frac{A + B}{2}) \sin (\frac{A - B}{2})$ $\cos A + \cos B = 2 \cos (\frac{A + B}{2}) \cos (\frac{A - B}{2})$ $\cos A - \cos B = - 2 \sin (\frac{A + B}{2}) \sin (\frac{A - B}{2})$
Law of Sines		$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Law of Cosines		$c^{2} = a^{2} + b^{2} - 2 a b \cos C$ $a^{2} = b^{2} + c^{2} - 2 b c \cos A$ $b^{2} = a^{2} + c^{2} - 2 a c \cos B$

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