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Linear Functions and Line Equations

Sep 10, 2025

Overview

This lecture introduces linear functions, covering their forms, how to find intercepts, the concept of slope, equations of lines, and examples in supply and demand.

Linear Functions

  • A linear function is of the form y = ax + b where a and b are constants.
  • The x-intercept is where y = 0; solve for x.
  • The y-intercept is where x = 0; solve for y.
  • A function can have multiple x-intercepts but at most one y-intercept.
  • The number of possible x-intercepts equals the highest power of x.

Slope and Line Equations

  • The slope (m) measures rate of change: m = (y₂ - y₁) / (x₂ - x₁).
  • Parallel lines have equal slopes (m₁ = m₂).
  • Perpendicular lines have slopes that are negative reciprocals (m₁ = -1/m₂).
  • Slope-intercept form: y = mx + b, where m is slope, b is y-intercept.
  • Point-slope form: y - y₁ = m(x - x₁), used with a known point and slope.
  • Standard form: ax + by + c = 0, where a, b, c are integers.
  • Vertical line: x = a (a real number); horizontal line: y = b (a real number).

Example: Phone Company Cost Function

  • Monthly bill includes a rate per minute (converted to dollars) and a fixed base charge.
  • Cost equation: y = 0.0833x + 18.36, where x is minutes, y is total monthly charge in dollars.

Example: Supply and Demand Equations

  • Demand points: (10, 25) and (5, 50); slope = -5; demand function: y_D = -5x + 75.
  • Supply points: (3, 16) and (11, 64); slope = 6; supply function: y_S = 6x - 2.
  • Equilibrium found by setting demand equal to supply: -5x + 75 = 6x - 2.
  • Solving gives x = 7; substituting back gives y = 40.
  • Equilibrium point: 7 units at $40.

Key Terms & Definitions

  • Linear function — Equation of the form y = ax + b.
  • Intercept — Point where curve crosses axis; x-intercept (y=0), y-intercept (x=0).
  • Slope (m) — Rate of change between two points on a line.
  • Parallel lines — Lines with equal slopes.
  • Perpendicular lines — Lines with slopes that are negative reciprocals.
  • Slope-intercept form — y = mx + b.
  • Point-slope form — y - y₁ = m(x - x₁).
  • Standard form — ax + by + c = 0.
  • Equilibrium point — Where supply equals demand.

Action Items / Next Steps

  • Practice finding intercepts and slopes from given points.
  • Rewrite equations in different linear forms.
  • Solve supply and demand problems for equilibrium points.

Full transcript