Graphs and Properties of Basic Functions

Introduction

This study guide covers the fundamental graphs and properties of functions commonly encountered in a Precalculus course. Understanding these parent functions and their graphical characteristics is essential for analyzing and transforming more complex functions.

Basic Parent Functions

• Linear Function: - Graph: Straight line through the origin with slope 1. - Domain: - Range: - Example: is a linear function shifted and with a different slope.

• Quadratic Function: - Graph: Parabola opening upwards, vertex at (0,0). - Domain: - Range: - Example: is a parabola shifted right and up.

• Cubic Function: - Graph: S-shaped curve passing through the origin. - Domain: - Range: - Example:

• Square Root Function: - Graph: Starts at (0,0) and increases slowly to the right. - Domain: - Range: - Example: is shifted right by 3 units.

• Absolute Value Function: - Graph: V-shaped, vertex at (0,0). - Domain: - Range: - Example:

• Piecewise Functions: Defined by different expressions over different intervals. - Graph: May have jumps, holes, or different shapes in different regions. - Example:

Key Properties of Functions

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can produce.

• Intercepts: Points where the graph crosses the axes. - x-intercept: Where - y-intercept: Where

• Symmetry:

  • Even Function: (symmetric about the y-axis)

  • Odd Function: (symmetric about the origin)

• End Behavior: Describes how the function behaves as or .

Common Graph Types and Their Features

• Parabola: - Opens upward if coefficient is positive, downward if negative. - Vertex is the minimum or maximum point.

• Circle (not a function): - Not a function because it fails the vertical line test.

• Step Function: - Graph has jumps at integer values.

• Piecewise Linear: Combination of line segments, often used to model real-world situations.

Table: Summary of Parent Functions

Graph Transformations

• Vertical Shifts: shifts the graph up by units if , down if .

• Horizontal Shifts: shifts the graph right by units if , left if .

• Reflections: reflects over the x-axis; reflects over the y-axis.

• Stretching/Compressing: stretches vertically if , compresses if .

Piecewise and Step Functions

• Piecewise Functions: Defined by different rules for different intervals of the domain. - Example:

• Step Functions: Functions that increase or decrease abruptly from one constant value to another. - Example: (greatest integer function)

Zeros and Intercepts

• Zero of a Function: Value of where .

• Finding Intercepts:

  • x-intercept: Set and solve for .

  • y-intercept: Set and solve for .

Additional info:

• Some graphs in the images represent non-functions (such as circles) to illustrate the vertical line test.

• Piecewise and step functions are included to demonstrate discontinuities and jumps in graphs.

• Transformations are a key Precalculus topic, though not explicitly labeled in the images, they are inferred from the variety of graph shapes and positions.