Notes on Linear Functions

Definition and Basics

• A linear function represents a straight line on the coordinate plane.
  • Example: y = 3x - 2
• General form: f(x) = mx + b
  • 'm' = slope of the line
  • 'b' = y-intercept
  • 'x' = independent variable
  • 'y' or f(x) = dependent variable

Features of Linear Functions

• Algebraic function involving only algebraic operations.
• Parent function: f(x) = x (line through origin).
• Real-life examples:
  • Movie streaming service costs (monthly fee + per movie fee).
  • T-shirt printing costs (base fee + per t-shirt fee).

Finding a Linear Function

• Use slope-intercept form or point-slope form.
• Example calculation: Given points (-1, 15) and (2, 27), find:
  • Slope (m) = (27 - 15) / (2 - (-1)) = 4
  • Equation: y = 4x + 19

Identifying Linear Functions

• Graph: If a graph is a line, it's linear.
• Algebraic form: If of the form f(x) = mx + b, it's linear.
• Table data: Constant ratio of change in y-values to x-values.

Graphing Linear Functions

• Increasing line for m > 0
• Decreasing line for m < 0
• Horizontal line for m = 0
• Two methods:
  • Finding two points: Choose random x-values, calculate y.
  • Using slope and y-intercept: Plot intercept, use slope to find second point.

Domain and Range

• Both domain and range are all real numbers (R).
• Horizontal line: Domain = R, Range = {b}.

Inverse of a Linear Function

• Inverse represented as f⁻¹(x).
• Steps:
  1. Replace f(x) with y.
  2. Interchange x and y.
  3. Solve for new y.
  4. Replace y with f⁻¹(x).
• Example: f(x) = 3x + 5, f⁻¹(x) = (x - 5)/3.

Piecewise Linear Function

• Linear function defined in pieces over its domain.
• Example: Different linear expressions defined over domain parts.

Important Notes

• Horizontal lines are constant functions with no inverse.
• Two parallel linear functions have equal slopes.
• If the product of slopes of two lines is -1, they are perpendicular.
• Vertical lines are not linear functions.

Examples and Practice

• Converting Celsius to Fahrenheit: F = (9/5)C + 32.
• Renting cost example: C(x) = 30x + 20.

Additional Concepts

• Linear function calculator and worksheets available for practice.
• Examples of linear vs. nonlinear differences.
• Graphing techniques and identifying linear functions from a table.

FAQs

• Definitions, formulas, and distinguishing between linear and nonlinear functions.

Related Topics

• Linear Function Calculator
• Quadratic Function
• Graphing Functions

Note: These points summarize key concepts and formulas related to linear functions, beneficial for understanding and solving problems involving them.