Math Antics: Basics of Linear Functions

Introduction

• Presenter: Rob from Math Antics
• Topic: Basics of Linear Functions
• Recommendation: Watch foundational videos on graphing and functions if unfamiliar

Basic Linear Function: y = x

• Equation: y = x
  • 'y' is the output; 'x' is the input
  • Output is the same as input
• Graph: Forms a diagonal line through the origin (0,0)
  • Splits quadrants 1 and 3 equally
  • x and y coordinates are the same

Linear Function with a Slope: y = mx

• Equation: y = mx
  • 'm' is a variable that modifies the slope
• Graph Examples:
  • m = 1: Same as y = x, line passes through origin
  • m = 2: Line steeper than m = 1
  • Larger 'm' values: Steeper lines
  • m < 1: Less steep lines
  • m > 0: Positive slope, line goes uphill from left to right
  • m < 0: Negative slope, line goes downhill from left to right
• Slope: Describes steepness of the line
  • Larger m = steeper slope
  • m = 0: Horizontal line, slope = 0

Slope-Intercept Form: y = mx + b

• Equation: y = mx + b
  • 'b' is the y-intercept
  • Determines where the line intersects the y-axis
• Graph Examples:
  • b = 1: Line shifted up, passes y-axis at 1
  • b = -1: Line shifted down, passes y-axis at -1
• Parameters:
  • 'm': Slope of the line
  • 'b': y-intercept

Determining Linearity

• Linear Functions: Variables are first order (no powers other than 1)
• Non-linear Examples: Equations with squared, cubed variables, etc.

Simplifying to Slope-Intercept Form

• Example: x - 4 = 2(y - 3)
  • Rearrange to y = mx + b form
  • Slope (m) and y-intercept (b) can be identified

Conclusion

• Graphing: y = mx + b can describe any linear function
• Practice: Essential for learning—do exercise problems
• Resource: More learning at www.mathantics.com