Functions and Their Graphs

Domain of a Function

The domain of a function is the set of all possible input values (usually x-values) for which the function is defined.

• Key Point: For rational functions, exclude values that make the denominator zero.

• Example: For , the domain is all real numbers except .

Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and the basis of derivatives in calculus.

• Definition: The difference quotient for a function is .

• Example: For , compute , subtract , and divide by .

Analyzing Graphs of Functions

Graphs provide visual insight into the behavior of functions, including domain, range, intervals of increase/decrease, and symmetry.

• Domain and Range: The domain is the set of x-values shown; the range is the set of y-values the graph attains.

• Intercepts: Points where the graph crosses the axes.

• Intervals of Increase/Decrease: Where the graph rises or falls as x increases.

• Even/Odd Functions: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• Example: Use the provided graph to answer questions about these properties.

Linear and Quadratic Functions

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Key Point: Analyze each piece separately for domain, intercepts, and graphing.

• Example:

Transformations of Functions

Transformations include shifting, stretching, compressing, and reflecting graphs of functions.

• Shifting: Moves the graph horizontally or vertically.

• Stretching/Compressing: Changes the steepness or width of the graph.

• Reflecting: Flips the graph over an axis.

• Example: is a vertical stretch and horizontal shift of the basic absolute value function.

Graphing Linear Functions

Linear functions have the form , where is the slope and is the y-intercept.

• Slope: Rate of change; rise over run.

• Y-intercept: The value of when .

• Example: has a slope of -2 and y-intercept of -3.

Quadratic Functions and Their Properties

Quadratic functions have the form and graph as parabolas.

• Vertex: The highest or lowest point of the parabola.

• Axis of Symmetry: Vertical line through the vertex, .

• Intercepts: Points where the graph crosses the axes.

• Direction: Opens upward if , downward if .

• Example:

Quadratic Formula

The quadratic formula solves for .

• Formula:

• Application: Use to find zeros (roots) of quadratic functions.

Applications of Functions

Linear Models and Break-Even Analysis

Linear functions can model real-world scenarios such as cost, revenue, and break-even points.

• Key Point: Set up equations based on given information and solve for unknowns.

• Example: If the cost per item is $1.50 and total cost for 60 items is $100, set up a linear equation to find the break-even point.

Quadratic Applications: Projectile Motion

Quadratic functions model the height of objects in projectile motion.

• Formula: , where is initial velocity and is initial height.

• Key Point: Solve for when (object hits the ground) or specific value.

• Example: A ball thrown upward with initial velocity; find when it hits the ground and when it is above a certain height.

Summary Table: Function Properties

*Additional info: The study guide covers foundational Precalculus topics including functions, their domains and ranges, graphing, transformations, linear and quadratic models, and applications. These are essential for success in early chapters of a Precalculus course.*