1.2 Linear functions & Applications

• Linear Functions: y=mx+b, y-y1=m(x-x1)• Supply and Demand Functions• Equilibrium Point• Cost, Revenue, and Profit Functions• Break-even Point (quantity, price)

•If price of an item increases, then consumers less likely to buy so the demand for the item decreases

•If price of an item increases, producers see profit and supply of item increases.

Linear functions - good for supply and demand curves.

Linear Function f defined by

bmxxfy )((for real numbers m and b)x=independent variabley=dependent variable

Cranberry example and explanation of quantity (x-axis), price (y-axis)

See Page 18 of e-text Cranberry example of late 1980’s

early 1990’s.http://view.ebookplus.pearsoncmg.com/ebook/launcheText.do?values=bookID::6852::platform::1030::bookPageNumber::17::invokeType::lms::launchState::goToEBook::platform::1030::userID::682191::scenario::11::scenarioid::scenario11::courseid::nix38175::pageid::17::sessionID::832966585424454542013::smsUserID::7818337::hsid::b3c380b26cf0d98ec719cb0d3dc8af02

Explanation of why price is on the y-axis.

Demand Function•defined by p = D(q)

•The function that gives the relationship between the number of units (q) that customers are willing to purchase at a given price (p). •The graph of a demand function is typically decreasing.

( ) 0.04 72p D q q If

is the relationship between p, the price per unit in dollars and q, the quantity demanded, what is the demand when the price is $50 per unit?

EXAMPLE

ANSWER p = -0.04q + 7250 = -0.04q +72-22 = -0.04q550 = q

EXAMPLE: Find the price when the level

of demand is 500.

Answer: p = -0.04q + 72 p = -0.04 (500) +72 p = -20 + 72 p = 52

Supply Function defined by p = S(q)

gives the relationship between the number of units (q) that suppliers are willing to produce at a given price (p).

The graph of a supply function is typically increasing.

If p = 5 + 0.04q is the relationship between the price (p) per unit and the quantity (q) supplied, When the price is set at $73 per unit, what quantity will be supplied?

EXAMPLE

Answer p = 5 + 0.04q 73 = 5 + 0.04q 68 = 0.04q1700 = q

Example 2 page 22Part c shows (6, $4.50) as the

intersection of the supply and the demand curve.

If the price is > $4.50, supply will exceed demand and a surplus will occur.

If the price is < $4.50, demand will exceed supply and a shortage will occur.

Graph of example 2

The price at the point where the supply and demand graphs intersect is called the equilibrium price.

The quantity at the point where the supply and demand graphs intersect is called the equilibrium quantity.

Equilibrium Point

To find the equilibrium quantity algebraically, set the supply and the demand functions equal and solve for quantity.

Using demand function p = 74 - .08q supply function p = .02q + 3 to find…(a) the equilibrium quantity(b) the equilibrium price(c) the equilibrium point

Example

Answera) 74 – 0.08q = 0.02q + 3 71 = 0.10q 710 = q c) (710, $17.20)b) p = 0.02q + 3 p = 0.02(710) + 3 p = 17.2

costs that remain constant regardless of the business’s level of activity.

Examples rental fees salaries insurance rent

Fixed costs (or overhead)

costs that vary based on the number of units produced or sold.

Examples wages cost of raw materials taxes

Variable Costs

Cost FunctionTotal cost C(x)= variable cost + fixed

cost

A company determines that the cost to make each unit is $5 and the fixed cost is $1200. Find the total cost function

Example

Answer

C(x) = 5x + 1200

Marginal Cost is the rate of change of cost C(x) at a production level of x units and is equal to the slope of the cost function at x (in linear functions)

It approximates the cost of producing one additional item.

The marginal cost to make x capsules of a certain drug is $15 per batch, while it cost $2000 to make 40 batches. Find the cost function, given that it is linear.

Example

AnswerUseand slope = 15, point (40, $2000)

y – 2000 = 15 (x - 40) y = 15x + 1400

bmxyorxxmyy )( 11

Revenue, R(x)

money from the sale of x unitsR(x) = p xp is price per unitx is number of units

the difference between the total revenue realized and the total cost incurred:

P(x)= R(x) – C(x)

Profit, P(x)

If the revenue from the sale of x units of a product is R(x) = 90x and the cost of obtaining x units is

(a)determine the profit function.(b)find the profit from selling 300 units.

80050)( xxC

Example

Answera) P(x) = R(x) – C(x) P(x) = 90x – (50x + 800) P(x) = 40x – 800b) P(300) = 40 (300) – 800 P(300) = $11,200

P(x)= R(x) – C(x) when R(x) > C(x) then P(x)> 0 or a

gain. If R(x) < C(x) then P(x) < 0 or a

loss.

Review of Profit, Revenue, and Cost

Finding breakeven quantity

If R(x) = C(x), then P(x) = 0. Where this happens is the breakeven point

To find the breakeven quantity (x-value of the break even point) either use a or b below.a)Set R(x)=C(x) and solve for x.b)Set P(x)=0 and solve for x.Always round the breakeven quantity up to the next whole number.

(a) find the cost function (b) find the revenue function (c) find the profit function (d) the break-even quantity (e) the profit from producing 250 units.(f) number of units for profit of $1000.

A manufacturer can produce x units for (240 + 0.18x) dollars. They can sell the product for $3.59 per unit.