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Chapter 1 Linear Functions

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

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Chapter 1 Linear Functions

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  1. Chapter 1Linear Functions Section 1.2 Linear Functions and Applications

  2. 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.

  3. Linear Function

  4. 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)

  5. 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.

  6. 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.

  7. 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.

  8. 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.

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

  10. Equilibrium Point The equilibrium priceof a commodity is the price found at the point where the supply and demand graphs for thatcommodity intersect. The equilibrium quantityis the demand and supply at that same point.

  11. 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.

  12. 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.

  13. 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.

  14. Cost Function

  15. 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.

  16. 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

  17. Break-Even Analysis • The revenueR(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 profitP(x) is the difference between revenue R(x) and cost C(x). P(x) = R(x) - C(x)

  18. 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-evenpoint.

  19. 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)

  20. 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?

  21. 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.

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