Chapter 1: Functions and Graphs

Section 1.1: Graphs of Equations

This section introduces the rectangular coordinate system and the basics of graphing equations. Understanding how to plot points and interpret graphs is foundational for all further study in precalculus.

• Rectangular Coordinate System: A two-dimensional plane defined by the x-axis (horizontal) and y-axis (vertical).

• Graphing Equations: Plotting solutions to equations as points on the coordinate plane.

• Intercepts: Points where the graph crosses the axes. Example: The x-intercept of is found by setting and solving for .

Section 1.2: Graphs and Functions

This section defines functions and explores their graphical representations. It emphasizes the importance of distinguishing functions from general relations.

• Definition of a Function: A relation in which each input (x-value) has exactly one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Graphing Functions: Plotting ordered pairs for various values of .

• Example: is a function; is not a function of .

Section 1.3: Linear Functions and Slope

Linear functions are characterized by constant rates of change. The concept of slope is central to understanding linear relationships.

• Linear Function: A function of the form , where is the slope and is the y-intercept.

• Slope (): Measures the steepness of a line.

• Rate of Change: For linear functions, the rate of change is constant and equal to the slope.

• Example: The line passing through and has slope .

Section 1.4: Equations of Lines

This section covers various forms of linear equations and how to write equations for lines given specific information.

• Slope-Intercept Form:

• Point-Slope Form:

• Parallel and Perpendicular Lines: Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Example: Find the equation of a line passing through with slope :

Section 1.5: Linear Models and Rates of Change

Linear models are used to describe real-world phenomena with constant rates of change. This section also introduces the concept of average rate of change for functions.

• Linear Model: An equation that describes a linear relationship between variables.

• Average Rate of Change: For a function over ,

• Example: If , the average rate of change from to is

Section 1.6: Transformations of Functions

Transformations alter the appearance of function graphs. This section covers vertical and horizontal shifts, stretching, shrinking, and reflections.

• Vertical Shifts: shifts the graph up () or down ().

• Horizontal Shifts: shifts the graph right () or left ().

• Vertical Stretch/Shrink: stretches () or shrinks () the graph vertically.

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

• Example: is shifted right by 2 and up by 3.

Section 1.7: Combinations of Functions

Functions can be combined using algebraic operations and composition. This section explains how to add, subtract, multiply, divide, and compose functions.

• Algebra of Functions: , , ,

• Domain: The set of all input values for which the combined function is defined.

• Composition:

• Example: If and , then

Section 1.8: Inverse Functions

Inverse functions reverse the effect of the original function. This section covers how to find and verify inverses, and the graphical relationship between a function and its inverse.

• Inverse Function: satisfies and

• Finding Inverses: Solve for in terms of , then interchange and .

• One-to-One Functions: Functions that pass the horizontal line test and have inverses.

• Graphical Relationship: The graph of is the reflection of across the line .

• Example: has inverse

Additional info: These notes are based on the objectives and main topics outlined for Chapter 1 in a Precalculus course, focusing on functions and their graphs. The content is organized to provide a comprehensive review for exam preparation.