Precalculus Dictionary
Essential Precalculus Definitions

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Asymptotes

An Asymptote is simply a line that a given function y=f(x) approaches but never crosses. y=f(x) gets ever closer and closer to this asymptote line as x gets larger and larger on its way to infinity, but never actually touches the asymptote. The simplest example of an Asymptote is the function y=1/x. As x gets larger and larger, y gets smaller and smaller, in this case closer and closer to 0, but never actually reaches 0. The line y=0 is said to be a horizontal Asymptote of the function y=1/x.

Complex Numbers

In our section on the Real Number System we defined all numbers that are commonly used in arithmetic and general math. In algebra and precalculus we must add one additional type of number which we call Imaginary or Complex Numbers. A Complex or Imaginary Number is defined as any number which contains a multiple of the square root of (-1) which is defined as lower case "i". There are specific rules for carrying out mathematical operations with complex or imaginary numbers.

Dividing Polynomials

Dividing Polynomials is a common method of finding the zeros or roots of a polynomial. For example the linear polynomial (x-2) can be divided into the polynomial x²-3x+2. The result is (x-1) with no remainder. This means that both x=2 and x=1 are zeros or roots of the polynomial x²-3x+2. Dividing Polynomials is a technique used to carryout this process which is similar to Long Division for numbers.

Exponential Functions

Exponential Functions look very similar to quadratic and polynomial functions such as y=x² with one very important difference - the variable x is the exponent. Examples of exponential functions include y=2^x, y=e^x and y=10^x.

Functions

Functions are algebra equations in which there are two variables such as y = 2*x + 3. From these equations we see that the variable y changes as x changes. When x = 1 then y = 2*1 + 3 which is 5. When x = 2 then y = 2*2 +3 which is 7. Y is said to vary as a function of x and is sometimes written as y = f(x) [read as "y = a function of x"].

Hyperbola

A Hyperbola is the general shape of a double quadratic function (one which has both x² and y²). The simplest hyperbolas have the general formula y²/a² + x²/b² = 1. A Hyperbola looks like two "boomerangs" with their tips facing each other and is the result of the intersection of a plane with two cones stacked on top of one another point-to-point.

Imaginary Numbers

Imaginary Numbers - See Complex Numbers

Laws of Logarithms

Logarithms, being exponents, follow some specific and unique rules when involved in the standard math operations like addition, subtraction, multiplication and division. These rules are referred to as the Laws of Logarithms. One example of these rules is log_a(A*B) = log_a(A) + log_a(B).

Logarithmic Functions

A Logarithmic Function is defined as a function y=f(x) in which at least one term contains log_ax. Examples of Logarithmic Functions include y=log₂x, y=log₁₀x and y=log_ex.

Logarithm Properties

There are four unique Logarithm Properties. The simplest two Logarithm Properties are log_a(1)=0 and log_a(a)=1.

Logarithms

Logarithms are actually a Code for Exponents. The logarithm of a given number X is the exponent or power to which a specified number "b" called the logarithm base is raised. For example log₁₀(100)=2 because the logarithm base "10" raised to the power 2 (squared) = 100.

Parabola

A Parabola is the general shape of a quadratic function the simplest of which is y=x². The Parabola looks like a "boomerang" and in geometry is the result of the intersection of a plane and a cone.

Polynomials

Polynomials are algebra expressions with more than two terms. Examples of Polynomials include: x² - 2*x + 1 and x³ + 2 * x² + 3 * x + 5. Polynomials are defined by their order or degree, which is the highest power of x present in the polynomial.

A polynomial of zero order or degree 0 is simply a constant number. A polynomial of the first order or degree 1 is a linear equation (containing only x to the first power) such as y = ax + b where a and b are constants. A polynomial of the second order or degree 2 is a quadratic equation (containing only x to the first and second powers) such as ax² + bx + c where a, b and c are constants.

Quadratic Functions

Quadratic Functions, sometimes referred to as Quadratic Equations, are polynomials of the second order that contain only x to the first and second power and have a general form of ax² + bx + c = 0 where a, b and c are constants.

Rational Functions

Just as Rational Numbers are Ratios or fractions of integers so Rational Functions are Ratios or fractions of polynomials.

These Rational Functions take on the general form of f(x)=P(x)/Q(x) where P(x) and Q(x) are polynomials. An example of a Rational Function is f(x)=(x-2)/(x²-3x+2).

Synthetic Division is a short hand method of Dividing Polynomials. For dividing (x-2) into x²-3x+2 you would divide -2 into [1 -2 2] using a specific series of arithmetic steps.

Transformations

All functions y = f(x) can be expressed as a geometric figure in a graph by locating each unique value of x and y of the function and placing a point at that location on the Cartesian Plane. The shape of this geometric figure or graph can be changed by carrying out standard Transformations on the function. These standard transformations include shifting up, down, left or right; stretching or shrinking horizontally or vertically, and reflecting through a straight line usually either the x or y axis. There are specific arithmetic operations that define each of these Transformations.

Zero of a Polynomial

The zero or root of a polynomial y=P(x) is simply a value of x which results in y=0. For example, the two zeros or roots of the polynomial y=x²-3x+2 are x=1 and x=2. This can be shown by substituting 1 and 2 for x which results in y=1²-3*1+2=1-3+2=0 and y=2²-3*2+2=4-6+2=0.

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