Chapter 1 Linear Functions Section 1.2 Linear Functions and Applications

Linear Functions Many situation involve two variables related by a linear equation. When we express the variable y in terms of x, we say that y is a linear function of x. Independent variable: x Dependent variable: y f(x) is used to sometimes denote y.

Linear Function

Example 1 Given a linear function f(x) = 2x – 5, find the following. a.) f(-2) f(-2) = 2 (-2) – 5 = -4 – 5 = -9 b.) f(0) c.) f(4)

Supply and Demand Linear functions are often good choices for supply and demand curves. Typically, there is an inverse relationship between supply and demand in that as one increases, the other usually decreases.

Supply and Demand Graphs While economists consider price to be the independent variable, they will plot price, p, on the vertical axis. (Usually the independent variable is graphed on the horizontal axis.) We will write p, the price, as a function of q, the quantity produced, and plot p on the vertical axis. Remember, though: price determines how much consumers demand and producers supply.

Example 2 Suppose that the demand and price for a certain model of electric can opener are related by p = D(q) = 16 – 5/4 q Demand where p is the price (in dollars) and q is the demand (in hundreds). a.) Find the price when there is a demand for 500 can openers. b.) Graph the function.

Example 2 continued Suppose the price and supply of the electric can opener are related by p = S(q) = 3/4q Supply where p is the price (in dollars) and q is the demand (in hundreds). c.) Find the demand for electric can openers with a price of $9 each. d.) Graph this function on the same axes used for the demand function. NOTE: Most supply/demand problems will have the same scale on both axes. Determine the x-and y-intercepts to decide what scale to use.

Supply and Demand Graph D(q) S(q) Equilibrium point (8,6)

Equilibrium 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 continued p = D(q) = 16 – 5/4 q Demand p = S(q) = 3/4q Supply Use the functions above to find the equilibrium quantity and the equilibrium price for the can openers.

Cost Analysis The cost of manufacturing an item commonly consists of two parts: the fixed cost and the cost per item. The fixed cost is constant (for the most part) and doesn’t change as more items are made. The total value of the second cost does depend on the number of items made.

Marginal Cost In economics, marginal cost is the rate of change of cost C(x) at a level of production x and is equal to the slope of the cost function at x. The marginal cost is considered to be constant with linear functions.

Cost Function

Example 3 Write a linear cost function for each situation below. Identify all variables used. a.) A car rental agency charges $35 a day plus 25 cents a mile. b.) A copy center charges $4.75 to create a flier and 10 cents for every copy made of the flier.

Example 4 Assume that each situation can be expressed as a linear cost function. Find the cost function in each case. a.) Fixed cost is $2000; 36 units cost $8480 b.) Marginal cost is $75; 25 units cost $3770

Break-Even Analysis The revenue R(x) from selling x units of an item is the product of the price per unit p and the number of units sold (demand) x, so that R(x) = p(x). The corresponding profit P(x) is the difference between revenue R(x) and cost C(x). P(x) = R(x) - C(x)

Break-Even Analysis A profit can be made only if the revenue received from its customers exceeds the cost of producing and selling its goods and services. The number of units x at which revenue just equals cost is the break-even quantity; the corresponding ordered pair gives the break-even point.

Break-Even Point As long as revenue just equals cost, the company, etc. will break even (no profit and no loss). R(x) = C(x)

Example 5 The cost function for flavored coffee at an upscale coffeehouse is given in dollars by C(x) = 3x + 160, where x is in pounds. The coffee sells for $7 per pound. a.) Find the break-even quantity. b.) What will the revenue be at that point? c.) What is the profit from 100 pounds? d.) How many pounds of coffee will produce a profit of $500?

Example 6 In deciding whether or not to set up a new manufacturing plant, analysts for a popcorn company have decided that a linear function is a reasonable estimation for the total cost C(x) in dollars to produce x bags of microwave popcorn. They estimate the cost to produce 10,000 bags as $5480 and the cost to produce 15,000 bags as $7780. Find the marginal cost and fixed cost of the bags of microwave popcorn to be produced in this plant, then write the cost function.