Graphs and Functions

Ordered Pairs and the Coordinate Plane

Understanding ordered pairs and their representation on the coordinate plane is fundamental in college algebra. Ordered pairs are written as (x, y) and indicate a specific location on the plane.

• Graphing Ordered Pairs: Plot each pair by moving horizontally to the x-value and vertically to the y-value.

• Distance Formula: Used to find the distance between two points and .

• Midpoint Formula: Finds the midpoint between two points.

• Equations with x and y: These can be graphed by plotting points or using algebraic techniques.

• Example: The points (2, 3) and (5, 7) have a distance of .

Relations and Functions

A relation is any set of ordered pairs. A function is a special relation where each input (x-value) corresponds to exactly one output (y-value).

• Determining Functions: A relation is a function if no x-value is repeated with different y-values.

• Vertical Line Test: If a vertical line crosses a graph more than once, it is not a function.

• Example: The set {(1,2), (2,3), (3,4)} is a function; {(1,2), (1,3)} is not.

Function Notation

Functions are often written as f(x), where x is the input variable.

• Evaluating Functions: Substitute the input value into the function.

• Example: If , then .

Domain and Range

The domain is the set of all possible input values (x), and the range is the set of all possible output values (y).

• Finding Domain: Identify x-values for which the function is defined (e.g., avoid division by zero or negative square roots).

• Finding Range: Determine possible y-values based on the function's behavior.

• Example: For , domain is , range is .

Increasing, Decreasing, Constant, Even, Odd Functions

Functions can be classified by their behavior and symmetry.

• Increasing: Function values rise as x increases.

• Decreasing: Function values fall as x increases.

• Constant: Function values remain unchanged as x increases.

• Even Function: Symmetric about the y-axis; .

• Odd Function: Symmetric about the origin; .

• Example: is even; is odd.

Operations with Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

• Composition:

• Example: If , , then .

Linear Functions

Forms of Linear Equations

Linear functions can be written in several standard forms.

• Standard Form:

• Slope-Intercept Form:

• Point-Slope Form:

• Example: A line through (2,3) with slope 4:

Basic Concepts of Linear Functions

Linear functions produce straight lines on the coordinate plane. Their slope and intercepts determine their orientation.

• Vertical Line: Undefined slope, equation

• Horizontal Line: Zero slope, equation

• Positive Slope: Line rises left to right

• Negative Slope: Line falls left to right

• Example: is a line with positive slope

Basic Function Graphs

Common Function Types

Several basic functions are frequently graphed in algebra.

• Squared Function: (parabola)

• Cubic Function: (S-shaped curve)

• Absolute Value Function: (V-shaped graph)

• Square Root Function: (starts at origin, rises slowly)

• Piecewise Function: Defined by different expressions for different intervals

• Example:

Graphing Techniques: Transformations

Graphs can be transformed by stretching, shrinking, reflecting, and translating.

• Vertical Stretch/Shrink: Multiply function by a constant ;

• Horizontal Stretch/Shrink: Replace with ;

• Reflection: Reflect across x-axis () or y-axis ()

• Translation: Shift graph horizontally () or vertically ()

• Example: shifted up 3 units:

Quadratic Functions

Properties and Graphing

Quadratic functions are of the form and graph as parabolas.

• Vertex: The highest or lowest point;

• Axis of Symmetry: Vertical line through the vertex;

• Direction: Opens upward if , downward if

• Example: has vertex at ,

Summary Table: Basic Function Graphs

Additional info: These notes expand on the study guide topics to provide definitions, examples, and formulas for key concepts in college algebra, focusing on graphs and functions, linear and quadratic functions, and basic graphing techniques.