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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