Functions, Graphs, and Models

Introduction to Functions

Functions are foundational in calculus and describe relationships between variables. Understanding their properties and graphs is essential for modeling business scenarios.

• Definition: A function is a rule that assigns to each input exactly one output.

• Notation: denotes a function named with input .

• Domain and Range: The domain is the set of all possible inputs; the range is the set of all possible outputs.

• Example: is a linear function with domain and range .

Types of Functions and Their Graphs

Common function types include linear, quadratic, cubic, absolute value, and rational functions. Each has a characteristic graph and algebraic form.

• Linear Functions:

  • Graph: Straight line with slope and -intercept .

• Quadratic Functions:

  • Graph: Parabola opening up if , down if .

• Cubic Functions:

  • Graph: S-shaped curve.

• Absolute Value Functions:

  • Graph: V-shaped, vertex at the origin.

• Rational Functions:

  • Graph: May have vertical/horizontal asymptotes.

Graphing Basic Functions

Graphing involves plotting points and understanding transformations such as shifts, stretches, and reflections.

• Vertical Shift: shifts the graph up by units.

• Horizontal Shift: shifts the graph right by units.

• Reflection: reflects across the -axis.

• Example: is a parabola shifted right 2 units and up 3 units.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example:

• Graph each piece on its respective interval.

Intercepts and Symmetry

Finding intercepts and analyzing symmetry helps in sketching graphs.

• -intercept: Set and solve for .

• -intercept: Evaluate .

• Even Function: (symmetric about -axis).

• Odd Function: (symmetric about origin).

Intervals and Domain

Understanding intervals is crucial for defining domains and ranges.

• Interval Notation:

  • : All such that (open interval).

  • : All such that (closed interval).

  • : (half-open interval).

• Example: The domain of is .

Vertical and Horizontal Asymptotes

Asymptotes describe the behavior of functions as approaches certain values.

• Vertical Asymptote: Occurs where the function grows without bound as approaches a value (e.g., denominator zero in rational functions).

• Horizontal Asymptote: Describes the end behavior as or .

• Example: has a vertical asymptote at and a horizontal asymptote at .

Summary Table: Common Functions and Their Properties

Applications in Business

Functions model cost, revenue, and profit in business. For example, a linear cost function models total cost as a function of units produced.

• Example: If , then the fixed cost is $200.

Additional info:

• Some graphs and examples were inferred from standard calculus curriculum and the visible sketches in the notes.

• Piecewise and rational function examples were expanded for clarity.