Linear Equations

Definition and Properties

Linear equations are fundamental in algebra and represent equations of the first degree, meaning the highest power of the variable is one. - Definition: A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and x is the variable. - Example: - Solving Linear Equations: Isolate the variable using inverse operations.

Solving Linear Equations: Examples

- Example 1: - Example 2:

Graphs

Distance and Midpoint Formulas

Graphs are visual representations of equations and relationships between variables. The distance and midpoint formulas are essential for analyzing points in the coordinate plane. - Distance Formula: Provides a method for computing the distance between two points and : - Example: Find the distance between and : - Midpoint Formula: The midpoint of the segment from to is:

Intercepts of a Graph

Intercepts are points where a graph crosses the axes. - x-intercept: The x-coordinate where the graph crosses the x-axis. - y-intercept: The y-coordinate where the graph crosses the y-axis. - Example: For the graph shown, x-intercepts are and y-intercept is .

Finding Intercepts from Equations

- To find the x-intercept, set and solve for . - To find the y-intercept, set and solve for . - Example: x-intercepts: y-intercept:

Symmetry in Graphs

Symmetry helps identify properties of graphs. - Symmetry with respect to x-axis: If for every , is also on the graph. - Symmetry with respect to y-axis: If for every , is also on the graph. - Symmetry with respect to origin: If for every , is also on the graph.

Testing for Symmetry

- Replace with for x-axis symmetry, with for y-axis symmetry, and both for origin symmetry. - Example: Test for symmetry.

Slope of a Line

The slope measures the steepness of a line. - Positive slope: Line rises from left to right. - Negative slope: Line falls from left to right. - Zero slope: Line is horizontal. - Undefined slope: Line is vertical. - Formula:

Equations of Lines

- Vertical line: - Horizontal line: - Point-slope form:

Slope-Intercept Form

- Equation: where is the slope and is the y-intercept.

Functions and Their Graphs

Definition of Functions

Functions are a special type of relation where each input has exactly one output. - Domain: The set of all possible input values (x-values). - Range: The set of all possible output values (y-values). - Function: A relation from set X to set Y such that each element of X corresponds to exactly one element of Y.

Identifying Functions

- If each input has only one output, the relation is a function. - Example: Menu items and prices: Each item has one price, so this is a function.

Domain and Range from Graphs

- Example: is a function. is not a function.

Function Notation

- Notation: represents the value of the function at . - Example:

Difference Quotient

The difference quotient is used to measure the average rate of change of a function. - Definition: , - Example: For ,

Finding Domain of Functions

- The domain is the set of real numbers for which the function is defined. - Exclude values that make denominators zero or radicands negative (for even roots). - Example: Domain:

Operations on Functions

Functions can be added, subtracted, multiplied, or divided. - Example: ,

Graphs of Functions and Non-Functions

- Example: is a function; is not a function.

Properties of Functions: Even and Odd

- Even Function: for all in the domain. Graph is symmetric with respect to the y-axis. - Odd Function: for all in the domain. Graph is symmetric with respect to the origin.

Increasing, Decreasing, and Constant Functions

- Increasing: Function rises as increases. - Decreasing: Function falls as increases. - Constant: Function remains unchanged as increases.

Local Maxima and Minima

- Local maximum: Occurs where the graph changes from increasing to decreasing. - Local minimum: Occurs where the graph changes from decreasing to increasing.

Evaluating Functions

- Substitute values into the function to find outputs. - Example:

Summary Table: Types of Symmetry

Summary Table: Forms of Linear Equations