Overview
This lecture covers solving quadratic equations, graphing quadratics, and applying these concepts to real-world problems involving revenue, cost, and profit functions.
Quadratics and Methods of Solving
- A quadratic in one variable has the form ( ax^2 + bx + c = 0 ) with ( a \neq 0 ).
- If ( a = 0 ), the equation is linear, not quadratic.
- Methods to solve quadratics: factoring, extracting roots, and quadratic formula.
- The vertex of ( y = ax^2 + bx + c ) is at ( x = -b/(2a) ).
- The parabola opens up if ( a > 0 ) and down if ( a < 0 ).
Graphing Quadratics
- The vertex is the maximum or minimum point of the graph.
- The axis of symmetry is the vertical line ( x = -b/(2a) ).
Application: Airborne Pollution and Wind Speed
- Given ( P = 81 - 0.01s^2 ), set ( P = 0 ) to solve for ( s ).
- Solving gives ( s = \pm 90 ); only the positive value makes sense, so ( s = 90 ).
- ( P = 0 ) means recorded pollution is zero at starting measurement, not that all pollution is gone.
Revenue Maximization Problem
- Demand: ( P = 175 - 0.2x ); revenue: ( R = Px ).
- Revenue function: ( R = -0.2x^2 + 175x ).
- Maximum revenue occurs at ( x = -b/(2a) = 437.5 ).
- To find price that maximizes revenue, use demand function: ( P = 87.50 ).
Cost, Revenue, and Profit Functions (Company Example)
- Fixed cost: $50,400; variable cost per unit: ( \frac{4}{9}x + 222 ).
- Cost function: ( C(x) = \frac{4}{9}x^2 + 222x + 50,400 ).
- Selling price per unit: ( 2050 - \frac{5}{9}x ); Revenue: ( R(x) = 2050x - \frac{5}{9}x^2 ).
- Profit function: ( P(x) = R(x) - C(x) = -x^2 + 1828x - 50,400 ).
- Break-even points: ( x = 28 ) and ( x = 1800 ).
- Maximum revenue at ( x = 1845 ), ( R(1845) = 1,891,125 ).
- Maximum profit at ( x = 914 ), ( P(914) = 784,996 ).
- Price for maximum profit: ( P(914) = $1,542.22 ).
Key Terms & Definitions
- Quadratic equation — An equation of the form ( ax^2 + bx + c = 0 ).
- Vertex — The point ((x, y)) where a parabola reaches its max or min.
- Axis of symmetry — Vertical line through the vertex, ( x = -b/(2a) ).
- Break-even point — The value(s) of ( x ) where profit is zero.
Action Items / Next Steps
- Review the quadratic formula and its application to break-even analysis.
- Practice solving quadratic equations using extracting roots and the quadratic formula.
- Prepare for regression/next lesson as referenced.