1. Linear Functions and Applications Lesson 1.2

2. A Break Even Calculator Consider this web site which helps a business person know when they are breaking even (starting to make money)web site 2 Note that the graph is a line. Quite often, break even analysis involves a linear function.

3. Linear Function A relationship f defined by for real numbers m and b is a linear function The independent variable is x The dependent variable is y 3

4. Supply and Demand Economists consider price to be the independent variable However They choose to plot price, p, on the vertical axis Thus our text will consider p = f(q) That is price is a function of quantity Graph the function (the calculator requires that x be used, not q) 4

5. Supply and Demand The demand for an item can also be represented by a linear function On the same set of axes, graph 5 Note: we are only interested in positive values, Quadrant 1. Reset the window with ♦E

6. Supply and Demand Set window for 0 < x < 3, 0 < y < 5 Use the Trace feature (F3) to note values of quantity and price 6 Demand Supply Quantity Price

7. Supply and Demand What is the price and quantity where the two functions are equal? This is called the point of equilibrium 7 Demand Supply Quantity Price Intersection may be found symbolically or by the calculator.

8. Supply and Demand Surplus is when excess supply exists Shortage is when demand exceeds supply 8 Demand Supply Surplus Shortage

9. Cost Analysis Cost of manufacturing an item usually consists of Fixed cost (rent, utilities, etc.) Cost per item (labor, materials, shipping …) This fits the description of a linear function The slope m is considered the "marginal cost" The y-intercept b is the fixed cost 9

10. Break Even Analysis We compare Cost function with Revenue Function Revenue is price times number sold Usually you must sell a certain number of items to cover the fixed costs … beyond that you are making a profit When R(x) > C(x) The break even point is when R(x) = C(x) 10

11. Break Even Analysis Given Graph both and determine the point of equilibrium 11 R(x) C(x) loss Profit

12. Assignment Lesson 1.2 Page 28 Exercises 1 – 25 odd, 29, 31, 37, 39 12