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)
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
View Demo

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)

5. Supply and Demand
The demand for an item can also be represented by a linear function
On the same set of axes, graph
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
Supply
Demand
Price
Quantity

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

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

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)

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

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