Graphs and Functions

Introduction to Graphs and Functions

This study guide covers foundational concepts in precalculus, focusing on the graphical representation of functions, function notation, and the analysis of basic algebraic and piecewise functions. Understanding these concepts is essential for success in more advanced topics in mathematics.

Function Notation and Evaluation

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

• Function Notation: Functions are commonly written as , where is the input variable.

• Evaluating Functions: To evaluate a function, substitute the given value into the function's formula.

• Example: If , then .

Basic Types of Functions and Their Graphs

• Linear Functions: Graph is a straight line with slope and y-intercept .

• Quadratic Functions: Graph is a parabola opening up if and down if .

• Square Root Functions: Graph starts at the origin and increases slowly to the right.

• Absolute Value Functions: Graph is a 'V' shape with vertex at the origin.

• Piecewise Functions: Defined by different expressions over different intervals of the domain.

Graphing Functions

• Plotting Points: To graph a function, plot several points by evaluating the function at selected x-values, then connect the points smoothly.

• Intercepts:

  • x-intercept: Set and solve for .

  • y-intercept: Evaluate .

• Symmetry:

  • Even functions: (symmetric about the y-axis).

  • Odd functions: (symmetric about the origin).

• Transformations:

  • Vertical shifts: shifts up/down.

  • Horizontal shifts: shifts right/left.

  • Reflections: reflects over x-axis, reflects over y-axis.

Special Functions and Notation

• Greatest Integer Function (Floor Function): Returns the greatest integer less than or equal to .

• Ceiling Function: Returns the smallest integer greater than or equal to .

• Piecewise Functions: Functions defined by different rules for different parts of their domain.

• Example Table:

Analyzing Graphs

• Identifying Functions from Graphs: Use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.

• Domain and Range:

  • Domain: Set of all possible input values ().

  • Range: Set of all possible output values ().

• Example: The domain of is ; the range is .

Piecewise and Step Functions

• Piecewise Functions: Defined by different expressions for different intervals.

• Step Functions: Functions that increase or decrease abruptly from one constant value to another, such as the floor function.

• Example:

Summary Table: Common Function Properties

Additional info:

• Some graphs in the file correspond to step functions and piecewise functions, which are important in understanding discontinuities and non-linear behavior in functions.

• The review also includes answer keys, which suggest this is a practice exam or review worksheet for Precalculus students.