Math Antics: Basics of Linear Functions

Introduction

• Focus: Basics of Linear Functions in Algebra
• Prerequisite: Understanding of graphing and functions

Basic Linear Function: y = x

• Equation: y = x
  • Output (y) is equal to input (x)
  • E.g., if x = 1, then y = 1; if x = 2, then y = 2
• Graph:
  • Forms a diagonal line on the coordinate plane
  • Passes through the origin (0,0)
  • Splits quadrants 1 and 3

Linear Function with a Slope: y = mx

• Equation: y = mx
  • 'm' is a new variable that makes equation versatile
  • Different values for 'm' yield different lines
• Graph:
  • Passes through the origin (0,0)
  • Slope (steepness) changes with 'm'

Examples of Slopes

• m = 1: y = 1x (same as y = x)
• m = 2: y = 2x (line is steeper)
• m = 3: y = 3x (even steeper)

Steepness and Slope

• Larger 'm': Steeper line
• m approaching infinity: Line approaches vertical
• Negative values: Lines mirror positive slopes

Horizontal and Vertical Lines

• Horizontal Line: m = 0, y = 0 (flat, slope = 0)
• Vertical Line: Never truly vertical; slope approaches infinity

Slope-Intercept Form: y = mx + b

• Equation: y = mx + b
  • 'b' shifts the line up or down (y-intercept)
  • 'm' determines the slope

Shifting the Line

• Positive 'b': Line shifts up on the y-axis
• Negative 'b': Line shifts down on the y-axis

Graphing with Slope-Intercept Form

• y-intercept: Where line crosses y-axis (x = 0)
• Any linear function can be expressed as y = mx + b

Rearranging Equations

• Linear equations contain only first-order variables
• Example: x - 4 = 2(y - 3)
  • Rearrange to y = mx + b
  • Slope: 1/2, y-intercept: 1

Conclusion

• y = mx + b allows graphing of any linear function
• Practice problems to reinforce learning
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