The Main Idea 

• Vertex of a Parabola:
  • Maximum point if parabola opens downward
  • Minimum point if parabola opens upward
  • Formula: [latex]h = -\frac{b}{2a}[/latex] for [latex]f(x) = ax^2 + bx + c[/latex]
• Quadratic Function:
  • General form: [latex]f(x) = ax^2 + bx + c[/latex]
  • Graph is a parabola
• Optimization Problems:
  • Often involve area, revenue, or cost
  • Typically have real-world constraints
• Problem-Solving Strategy:
  • Identify the quantity to optimize
  • Express as a quadratic function
  • Find the vertex
  • Interpret the result in context
• Graphical Interpretation:
  • Vertex is the turning point of the parabola
  • [latex]y[/latex]-coordinate of vertex is the maximum/minimum value

The Main Idea 

• Revenue Function:
  • Revenue = Price per unit × Number of units sold
  • Often forms a quadratic relationship
• Price-Demand Relationship:
  • Usually inverse: As price increases, demand decreases
  • Often modeled as a linear function
• Quadratic Revenue Model:
  • Form: [latex]R(p) = ap^2 + bp + c[/latex]
  • [latex]p[/latex] is price, [latex]R[/latex] is revenue
• Maximum Revenue:
  • Occurs at the vertex of the parabola
  • Found using [latex]p = -\frac{b}{2a}[/latex]
• Real-world Applications:
  • Pricing strategies
  • Market analysis
  • Business decision-making