Quadratic Models in Business

Business Calculus · Axiom Academy

LESSON Quadratic Models in Business When linear isn't enough: parabolas for revenue and profit optimization Why Do We Need Quadratic Models? Linear models assume relationships stay constant—but real business often shows diminishing returns or accelerating costs . To sell more units, you often must lower the price . Revenue = Price × Quantity, but if price decreases with quantity, revenue isn't linear—it curves! At high production levels, costs may increase faster due to overtime, equipment strain, or inefficiency. Variable cost per unit isn't always constant. a : opens downward (has a maximum) a > 0 : opens upward (has a minimum) When price affects demand, revenue becomes quadratic. Here's how: Step 1: Start with a linear demand function: Step 2: Revenue = Price × Quantity: This is a downward-opening parabola —it has a maximum revenue point! For R(x) = 100x - 2x^2 , there's a "sweet spot" that maximizes revenue. Selling too few units means low volume; selling too many requires prices so low that total revenue drops! Finding the Vertex (Maximum or Minimum) The vertex of a parabola is its highest or lowest point. For optimization, this is exactly what we need! For f(x) = ax^2 + bx + c , the vertex occurs at: This gives the x -value. Plug it back into f(x) to get the maximum/minimum value. For R(x) = 100x - 2x^2 = -2x^2 + 100x : Maximum revenue: R(25) = 100(25) - 2(25)^2 = 2500 - 1250 = \ 1,250

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