Chapter 1 Linear Functions Section 1.2 Linear Functions and Applications.

Chapter 1Chapter 1Linear FunctionsLinear Functions

Section 1.2Section 1.2

Linear Functions and ApplicationsLinear Functions and Applications

Linear FunctionsLinear Functions

Many situation involve two variables related Many situation involve two variables related by a linear equation. by a linear equation.

When we express the variable When we express the variable yy in terms of in terms of xx, we say that , we say that yy is a is a linear functionlinear function of of xx..

Independent variableIndependent variable: : xx Dependent variableDependent variable: y: y f(x)f(x) is used to sometimes denote is used to sometimes denote yy..

Linear FunctionLinear Function

Example 1Example 1

Given a linear function f(Given a linear function f(xx) = 2) = 2xx – 5, find – 5, find the following.the following.

a.) f(-2)a.) f(-2)

f(-2) = 2 (-2) – 5 = -4 – 5 = -9f(-2) = 2 (-2) – 5 = -4 – 5 = -9

b.) f(0)b.) f(0)

c.) f(4)c.) f(4)

Supply and DemandSupply and Demand

Linear functions are often good choices for Linear functions are often good choices for supply and demand curvessupply and demand curves..

Typically, there is an inverse relationship Typically, there is an inverse relationship between supply and demand in that as between supply and demand in that as one increases, the other usually one increases, the other usually decreases.decreases.

Supply and Demand GraphsSupply and Demand Graphs

While economists consider price to be the While economists consider price to be the independent variable, they will plot price, independent variable, they will plot price, pp, on , on the vertical axis. (Usually the independent the vertical axis. (Usually the independent variable is graphed on the horizontal axis.)variable is graphed on the horizontal axis.)

We will write We will write pp, the price, as a function of , the price, as a function of qq, the , the quantity produced, and plot quantity produced, and plot pp on the vertical on the vertical axis.axis.

Remember, though: Remember, though: price determines how price determines how much consumers demand and producers much consumers demand and producers supply.supply.

Example 2Example 2 Suppose that the demand and price for a certain Suppose that the demand and price for a certain

model of electric can opener are related by model of electric can opener are related by pp = = D(q)D(q) = 16 – 5/4 = 16 – 5/4 qq DemandDemand

where where pp is the price (in dollars) and is the price (in dollars) and qq is the demand is the demand (in hundreds). (in hundreds).

a.) Find the price when there is a demand for a.) Find the price when there is a demand for 500 can openers.500 can openers.

b.) Graph the function.b.) Graph the function.

Example 2 continuedExample 2 continued Suppose the price and supply of the electric can Suppose the price and supply of the electric can

opener are related byopener are related bypp = = S(q)S(q) = 3/4 = 3/4qq SupplySupply

where where pp is the price (in dollars) and is the price (in dollars) and qq is the demand is the demand (in hundreds). (in hundreds).

c.) Find the demand for electric can openers with a c.) Find the demand for electric can openers with a price of $9 each.price of $9 each.

d.) Graph this function on the same axes used d.) Graph this function on the same axes used for the demand function.for the demand function.

NOTE: Most supply/demand problems will have the same NOTE: Most supply/demand problems will have the same scale on both axes. Determine the scale on both axes. Determine the xx-and -and yy-intercepts to -intercepts to decide what scale to use.decide what scale to use.

Supply and Demand GraphSupply and Demand Graph

D(q)

S(q)Equilibrium point

(8,6)

Equilibrium PointEquilibrium Point

The equilibrium price of a commodity is the price found at the point where the supply and demand graphs for that commodity intersect.

The equilibrium quantity is the demand and supply at that same point.

Example 2 continuedExample 2 continued

pp = = D(q)D(q) = 16 – 5/4 = 16 – 5/4 qq DemandDemand

pp = = S(q)S(q) = 3/4 = 3/4qq SupplySupply

Use the functions above to find the equilibrium Use the functions above to find the equilibrium quantity and the equilibrium price for the can quantity and the equilibrium price for the can openers.openers.

Cost AnalysisCost Analysis

The cost of manufacturing an item The cost of manufacturing an item commonly consists of two parts: the commonly consists of two parts: the fixed fixed costcost and the and the cost per itemcost per item..

The fixed cost is constant (for the most The fixed cost is constant (for the most part) and doesn’t change as more items part) and doesn’t change as more items are made.are made.

The total value of the second cost does The total value of the second cost does depend on the number of items made.depend on the number of items made.

Marginal CostMarginal Cost

In economics, In economics, marginal costmarginal cost is the rate of is the rate of change of cost C(change of cost C(xx) at a level of ) at a level of production production x x and is equal to the slope of and is equal to the slope of the cost function at the cost function at xx. .

The marginal cost is considered to be The marginal cost is considered to be constant with linear functions.constant with linear functions.

Cost FunctionCost Function

Example 3Example 3

Write a linear cost function for each Write a linear cost function for each situation below. Identify all variables used.situation below. Identify all variables used.a.) A car rental agency charges $35 a day a.) A car rental agency charges $35 a day

plus 25 cents a mile.plus 25 cents a mile.

b.) A copy center charges $4.75 to create b.) A copy center charges $4.75 to create a flier and 10 cents for every copy a flier and 10 cents for every copy made of the flier.made of the flier.

Example 4Example 4 Assume that each situation can be expressed Assume that each situation can be expressed

as a linear cost function. Find the cost as a linear cost function. Find the cost function in each case.function in each case.

a.) Fixed cost is $2000; 36 units cost $8480a.) Fixed cost is $2000; 36 units cost $8480

b.) Marginal cost is $75; 25 units cost $3770b.) Marginal cost is $75; 25 units cost $3770

Break-Even AnalysisBreak-Even Analysis The The revenuerevenue RR((xx) from selling ) from selling xx units of an units of an

item is the product of the price per unit item is the product of the price per unit pp and and the number of units sold (demand) the number of units sold (demand) xx, so that , so that

RR((xx) = ) = pp((xx).).

The corresponding The corresponding profitprofit PP((xx) is the difference ) is the difference between revenue between revenue RR((xx) and cost ) and cost CC((xx). ).

PP((xx) = ) = RR((xx) - ) - CC((xx) )

Break-Even AnalysisBreak-Even Analysis A profit can be made only if the revenue A profit can be made only if the revenue

received from its customers exceeds the received from its customers exceeds the cost of producing and selling its goods and cost of producing and selling its goods and services.services.

The number of units The number of units x x at which revenue at which revenue just equals cost is the just equals cost is the break-even break-even quantityquantity; the corresponding ordered pair ; the corresponding ordered pair gives the gives the break-evenbreak-even pointpoint..

Break-Even PointBreak-Even Point

As long as revenue just equals cost, the As long as revenue just equals cost, the company, etc. will break even (no profit company, etc. will break even (no profit and no loss).and no loss).

RR((xx) = ) = CC((xx) )

Example 5Example 5 The cost function for flavored coffee at an The cost function for flavored coffee at an

upscale coffeehouse is given in dollars by upscale coffeehouse is given in dollars by CC((xx) = 3) = 3xx + 160, where + 160, where xx is in pounds. The is in pounds. The coffee sells for $7 per pound.coffee sells for $7 per pound.

a.) Find the break-even quantity. a.) Find the break-even quantity.

b.) What will the revenue be at that point?b.) What will the revenue be at that point?

c.) What is the profit from 100 pounds?c.) What is the profit from 100 pounds?

d.) How many pounds of coffee will produce a d.) How many pounds of coffee will produce a

profit of $500?profit of $500?

Example 6Example 6 In deciding whether or not to set up a new In deciding whether or not to set up a new

manufacturing plant, analysts for a popcorn manufacturing plant, analysts for a popcorn company have decided that a linear function company have decided that a linear function is a reasonable estimation for the total cost is a reasonable estimation for the total cost CC((xx) in dollars to produce ) in dollars to produce xx bags of bags of microwave popcorn. microwave popcorn. They estimate the cost to produce 10,000 They estimate the cost to produce 10,000 bags as $5480 and the cost to produce bags as $5480 and the cost to produce 15,000 bags as $7780. 15,000 bags as $7780. Find the marginal cost and fixed cost of the Find the marginal cost and fixed cost of the bags of microwave popcorn to be produced bags of microwave popcorn to be produced in this plant, then write the cost function.in this plant, then write the cost function.