Equations Basics

Study Notes

An equation is a statement that says two expressions are equal.
It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).
The goal is to find the value of the variable(s) that makes the equation true.
Types of equations:
- Linear equation: degree of the variable(s) is 1 (e.g., 2x + 3 = 5)
- Quadratic equation: degree of the variable(s) is 2 (e.g., x^2 + 4x + 4 = 0)
- Exponential equation: involves exponential functions (e.g., 2^x = 8)
Solving equations:
- Addition/Subtraction property: add/subtract the same value to both sides
- Multiplication/Division property: multiply/divide both sides by the same non-zero value
- Distributive property: expand and simplify expressions

A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
It is denoted by f(x) or g(x), where x is the input or independent variable.
Notation:
- f(x) = expression: defines the function
- f(a) = output: evaluates the function at input a
Types of functions:
- Linear function: f(x) = mx + b, where m is the slope and b is the y-intercept
- Quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants
- Exponential function: f(x) = a^x, where a is the base
Function operations:
- Domain and range: find the set of inputs and outputs
- Composition: combine two or more functions
- Inverse: find the function that "reverses" the original function

The graph of a function is a visual representation of the relationship between the input and output values.
Coordinates:
- x-axis: horizontal axis, represents the input values
- y-axis: vertical axis, represents the output values
Graph types:
- Linear graph: straight line
- Quadratic graph: parabola (opens upward or downward)
- Exponential graph: curved line that increases/decreases rapidly
Graphing techniques:
- Plotting points: find and plot points on the graph
- Using intercepts: find the x-intercept (where the graph crosses the x-axis) and y-intercept (where the graph crosses the y-axis)
- Using symmetry: identify symmetries about the x-axis, y-axis, or origin

An equation consists of two expressions separated by an equal sign (=) with the goal of finding the value of the variable(s) that makes the equation true.
There are different types of equations, including:
- Linear equations, where the degree of the variable(s) is 1 (e.g., 2x + 3 = 5)
- Quadratic equations, where the degree of the variable(s) is 2 (e.g., x^2 + 4x + 4 = 0)
- Exponential equations, which involve exponential functions (e.g., 2^x = 8)
To solve equations, properties such as:
- Addition/Subtraction property can be used to add/subtract the same value to both sides
- Multiplication/Division property can be used to multiply/divide both sides by the same non-zero value
- Distributive property can be used to expand and simplify expressions

A function is a relation between a set of inputs (domain) and a set of possible outputs (range) denoted by f(x) or g(x).
Functions can be defined using notation such as f(x) = expression, which defines the function, and f(a) = output, which evaluates the function at input a.
There are different types of functions, including:
- Linear functions, where f(x) = mx + b, with m as the slope and b as the y-intercept
- Quadratic functions, where f(x) = ax^2 + bx + c, with a, b, and c as constants
- Exponential functions, where f(x) = a^x, with a as the base
Function operations include:
- Finding the domain and range of a function
- Composing two or more functions
- Finding the inverse of a function

The graph of a function is a visual representation of the relationship between the input and output values.
The graph has a:
- x-axis, which represents the input values
- y-axis, which represents the output values
Different types of graphs include:
- Linear graphs, which are straight lines
- Quadratic graphs, which are parabolas (opening upward or downward)
- Exponential graphs, which are curved lines that increase/decrease rapidly
Graphing techniques include:
- Plotting points to find and plot points on the graph
- Using intercepts to find the x-intercept and y-intercept
- Using symmetry to identify symmetries about the x-axis, y-axis, or origin