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1.2: Linear Functions and Applications
After completing this section, you will be able to do the following:
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
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:
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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 *
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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
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b. What is her fixed cost?
Fixed cost = b
b = $90
Breakeven:
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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
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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.)
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