Linear Functions and Applications
Lesson 1.2
A Break Even Calculator
Consider this web site which helps a business person know when they are breaking even (starting to make money)
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Note that the graph is a line. Quite often, break even analysis involves
a linear function.
Note that the graph is a line. Quite often, break even analysis involves
a linear function.
Linear Function
A relationship f defined by
for real numbers m and b is alinear function
The independent variable is x
The dependent variable is y
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( )y f x mx b
Supply and Demand
Economists consider price to be the independent variableHowever• 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 requiresthat x be used, not q) 4
( ) 1.4 .6p S q q
Supply and Demand
The demand for an item can also be represented by a linear function• On the same set of axes, graph
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( ) 2 3.2p D q q
( ) 1.4 .6p S q q
Note: we are only interested in positive
values, Quadrant 1. Reset the window with ♦E
Note: we are only interested in positive
values, Quadrant 1. Reset the window with ♦E
Supply and Demand
Set window for 0 < x < 3, 0 < y < 5
Use the Trace feature (F3) to note values of quantity and price
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Demand Supply
Quantity
Pric
e
Supply and Demand
What is the price and quantity where the two functions are equal?
This is called the point of equilibrium7
Demand Supply
Quantity
Pric
e
Intersection may be found
symbolically or by the calculator.
Intersection may be found
symbolically or by the calculator.
Supply and Demand
Surplus is when excess supply exists
Shortage is when demand exceeds supply
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DemandSupply
Surplus
Shortage
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
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( )y f x mx b
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)
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( )R x p x
Break Even Analysis
Given
Graph both and determine the point of equilibrium
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( ) 4.95
( ) 525 2.15
R x x
C x x
R(x)
C(x)
loss
Profit
Assignment
Lesson 1.2
Page 28
Exercises 1 – 25 odd,29, 31, 37, 39
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