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Linear equations are one of the fundamental building blocks in algebra. They come in the standard form of \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. In simple terms, the slope \( m \) tells us how steep the line is, while \( b \) indicates where the line crosses the y-axis.
Linear functions are straightforward because every change in \( x \) results in a direct and proportional change in \( y \). This relationship makes them easy to work with and interpret.

• The graph of a linear equation is always a straight line.
• The degree of a linear polynomial is 1.
• They are expressed in the simplest polynomial form.

For example, the equation \( y = 3x - 2 \) is a linear equation where the slope \( m = 3 \) and the y-intercept \( b = -2 \). This equation tells us that for every increase of 1 in \( x \), \( y \) will increase by 3, and the line will cross the y-axis at -2.

Interval notation is a mathematical notation used to describe the domain and range of functions. It provides a concise way to express intervals on the number line.
In interval notation, a domain is expressed as an interval of numbers, showcasing all possible values that x can take. Parentheses \(( )\) denote that the endpoints are not included, while brackets \([ ]\) are used when they are.

• A closed interval such as \([a, b]\) includes all numbers between \( a \) and \( b \), including \( a \) and \( b \) themselves.
• An open interval like \((a, b)\) includes all numbers between \( a \) and \( b \) but not \( a \) and \( b \) themselves.
• \(( -\infty, +\infty )\) represents all real numbers available for use.

In the case of a linear function like \( y = 3x - 2 \), the domain is all real numbers, represented as \( (-\infty, \infty) \). This notation indicates that \( x \) can be any real number, reflecting the idea that linear functions have no restrictions on their domain.

Polynomial functions are expressions involving variables raised to whole number powers. They can include constants, coefficients, and operations like addition and subtraction.
These functions can be classified by degree, which indicates the highest power of the variable within the function.

• Constant functions are polynomial functions of degree 0.
• Linear functions, like \( y = 3x - 2 \), are polynomials of degree 1.
• Quadratic functions are of degree 2, involving terms like \( x^2 \).

Linear functions are simple polynomial functions. Understanding them makes it easier to learn about more complex polynomials, like quadratic and cubic functions.
Linear polynomial functions, with their degree of 1, are always graphed as straight lines and have the most straightforward domain of all real numbers, effective for elementary algebraic operations.