The Main Idea

• Definition of Direct Variation:
  • A relationship where one variable is a constant multiple of another
  • Expressed as [latex]y = kx^n[/latex], where [latex]k[/latex] is the constant of variation
• Characteristics:
  • As one variable increases, the other increases proportionally
  • The ratio between variables remains constant: [latex]k = \frac{y}{x^n}[/latex]
• Types of Direct Variation:
  • Linear ([latex]n = 1[/latex]): [latex]y = kx[/latex]
  • Quadratic ([latex]n = 2[/latex]): [latex]y = kx^2[/latex]
  • Cubic ([latex]n = 3[/latex]): [latex]y = kx^3[/latex]
  • And so on for higher powers
• Graphical Representation:
  • All direct variation graphs pass through the origin [latex](0, 0)[/latex]
  • Shape depends on the power n (linear, parabola, cubic, etc.)
• Applications:
  • Used in real-world situations where quantities vary consistently
  • Examples: sales commissions, simple interest, distance-time relationships

The Main Idea

• Definition of Inverse Variation:
  • A relationship where one variable decreases as the other increases
  • Expressed as [latex]y = \frac{k}{x^n}[/latex], where [latex]k[/latex] is the constant of variation
• Characteristics:
  • As one variable increases, the other decreases proportionally
  • The product of the variables remains constant: [latex]k = x^n \cdot y[/latex]
• Types of Inverse Variation:
  • Simple inverse ([latex]n = 1[/latex]): [latex]y = \frac{k}{x}[/latex]
  • Inverse square ([latex]n = 2[/latex]): [latex]y = \frac{k}{x^2}[/latex]
  • Inverse cube ([latex]n = 3[/latex]): [latex]y = \frac{k}{x^3}[/latex]
  • And so on for higher powers
• Graphical Representation:
  • Hyperbola for simple inverse variation
  • Asymptotic to both axes (never touches x or y axis)
  • Not defined when [latex]x = 0[/latex] (division by zero)
• Applications:
  • Used in physics (Boyle’s Law, gravitational force)
  • Economics (supply and demand)
  • Engineering (gear ratios)

The Main Idea

• Definition of Joint Variation:
  • A relationship where a variable depends on two or more other variables
  • Can involve both direct and inverse variations simultaneously
• General Forms:
  • Direct joint variation: [latex]x = kyz[/latex] ([latex]x[/latex] varies directly with both [latex]y[/latex] and [latex]z[/latex])
  • Mixed joint variation: [latex]x = \frac{ky}{z}[/latex] ([latex]x[/latex] varies directly with [latex]y[/latex] and inversely with [latex]z[/latex])
• Characteristics:
  • Only one constant (k) is used in the equation
  • Can involve different powers of variables
• Key Formula: [latex]x = k \frac{y^a z^b}{w^c}[/latex]
  Where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers (positive for direct variation, negative for inverse variation)
• Applications:
  • Physics (force, pressure, work)
  • Engineering (stress and strain relationships)
  • Economics (production functions)