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1.2: Linear Functions and Applications After completing this section, you will be able to do the following: Use function notation. Evaluate business linear models: o supply and demand, o cost function, o revenue, and o profit. Function notation: y = f(x) The f names the function. Linear equation: an equation of two variables that both have an exponent of one y = f(x) = mx +b x = input or independent variable y = output or dependent variable (y changes if you change the value of x) Supply and demand: Supply: amount of product you have Demand: how much of a product is needed/wanted Equilibrium quantity: when supply and demand are equal Ex.) Let the supply and demand function for ice cream be given below. ( ) ( ) Where p is the price in dollars and x is the number of 10-gallon tubs of ice cream. Find the equilibrium quantity and price (equilibrium = equal). S(x) = D(x) ( ) ( ) *When trying to get rid of a fraction next to a variable, multiply by the reciprocal—fraction flipped * 1 x = 125 tubs of ice cream equilibrium quantity ( ) ( ) ( ) ( ) equilibrium price Cost analysis: Marginal cost: the rate the cost changing in relation to production Marginal cost = Rate of change = slope = m C(x) = mx + b Fixed cost: amount of money that is spent no matter how much product is produced (e.g., rent) Fixed cost = y-intercept = b C(x) = mx + b C(0) = m(0) + b C(0) = b fixed cost C(x) = marginal cost * x + fixed cost Ex.) Joanne Ha sells silk-screened T-shirts at community festivals and crafts fairs. Her marginal cost to produce one T-shirt is $3.50. Her total cost to produce 60 T-shirts is $300, and she sells them for $9 each. a. Find the linear cost function for Joanne’s T-shirt production. marginal cost = m = $3.50 C(x) = total cost for x amount of product = C(60) = 300 Plug into the cost function: C(x) = marginal cost * x + fixed cost Plug back into the cost function: 300 = 3.5(60) + b 300 = 210 + b -210 -210 90 = b C(x) = 3.5 x + 90 2 b. What is her fixed cost? Fixed cost = b b = $90 Breakeven: Revenue R(x): how much money is received for a certain amount of product sold R(x) = Price * Quantity Cost C(x): how much money was spend to make the product Breakeven: R(x) = C(x) c. How many T-shirts must she produce and sell in order to break even? Price = $9 R(x) = Price * Quantity = 9x Breakeven: C(x) = 3.5 x + 90 R(x) = C(x) 9x = 3.5x + 90 -3.5x -3.5x 5.5x = 90 divide both sides by 5.5 x = 16.63 They would have to sell 17 T-shirts to break even. (You would not sell part of a T-shirt, so round up.) Profit P(x): the amount of money made after taking away production cost P(x) = R(x) - C(x) d. How many T-shirts must she produce and sell to make a profit of $500? P(x) = 500 P(x) = R(x) - C(x) = 9x - (3.5x +90) always put in parenthesis to help remind us to distribute negative P(x) = 9x - 3.5x – 90 P(x) = 5.5x - 90 profit function 500 = 5.5x – 90 +90 +90 590 = 5.5x divide both sides by 5.5 3 107.27 = x She must sell 108 T-shirts to make a profit of $500. (You would not sell part of a T-shirt, so round up.) 4