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Linear Functions
Chapter 1
Linear Functions
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1.2 Linear Functions and Applications
Equations of Lines
Equation
Description
ax  by = c
Standard form: a  0, b  0, x-intercept = c/a,
y-intercept = c/b, m = –a/b.
y = mx + b
Slope-intercept form: slope m, y-intercept b
y – y1 = m(x – x1) Point-slope form: slope m, line passes
through (x1,y1)
Slopes of Lines
Undefined slope (Vertical line)
m=0
(Horizontal line)
1.2 Applications of Linear
Functions
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Many situations involve two variables, x
(independent variable) and y (dependent
variable), with a linear relationship.
If y is expressed in terms of x, then y is a
linear function of x.
For any value of x, the corresponding value of
y can be calculated.
Linear Functions
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Examples
y = 3x + 7
 12x + 6y = 24  y = -2x + 4
f(x) Notation: Letters can be used to
name functions.
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if f is used to name y = 3x + 7
f(x) = 3x + 7
f(2) = 3(2) + 7 = 13
f(-2) = 1, f(0) = 7, f(1) = 10...
Linear Functions
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A relationship f defined by
y = f(x) = mx + b,
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for real numbers m and b, is a linear
function
Applications of Linear Functions
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Basic business relationships:
 Revenue = Price  Quantity
 Total Cost = Fixed Costs + Variable Costs
 Average Cost = Total Cost ÷ Quantity
 Profit = Revenue – Total Costs
Linear Functions
y  mx  b
y is the dependent variable.
y
x is the independent variable.
Therefore, changes in x result
in changes in y.
x
Supply and Demand
Economists normally plot
price (p) on the vertical
axis, and quantity (q) on
the horizontal axis.
Demand function D(q):
p  m  qd   b
p b
qd 
m
Quantity
Supply and Demand
Economists normally plot
price (p) on the vertical
axis, and quantity (q) on
the horizontal axis.
Supply function S(q):
p  m  qs   b
p b
qs 
m
Quantity
Supply and Demand
Example
Quantity demanded D(q):
p = – ¾ q + 60
(20, 45)

Calculate quantity
demanded at $45 and $18
p  b 45  60
 20

qd 
m
3 4
(20, 45)
Quantity
Supply and Demand
Example
Quantity demanded D(q):
p = – ¾ q + 60
(20, 45)
Calculate quantity
demanded at $45 and $18

(56, 18)

p  b 18  60
 56

qd 
m
3 4
(56, 18)
Quantity
Supply and Demand
Example
p = – ¾q + 60
(80, 60)

(20, 45)
Quantity supplied: S(q)
p = ¾q
Calculate quantity supplied
at $60 and $12

(56, 18)

p  b 60  0
 80

qS 
34
m
(80, 60)
Quantity
Supply and Demand
Example
p = – ¾q + 60
(80, 60)

(20, 45)
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p=¾q
(56, 18)
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(16, 12)
Quantity supplied: S(q)
p = ¾q
Calculate quantity supplied
at $60 and $12
p  b 12  0
 16

qS 
34
m
(12, 16)
Quantity
Market Equilibrium
Equilibrium
p = 60 – ¾ q
p=¾q

(40, 30)
Quantity
The market is in
equilibrium when qd = qs
Equilibrium quantity:
60 – ¾ q = ¾ q
240 – 3q = 3q
240 = 6q
(40, 30)
40 = q
Equilibrium price:
p = 3/4 (40)
p = 30
Market Equilibrium
S
Surplus
Equilibrium
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Shortage
Quantity
D
Short-Run Cost Analysis
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Fixed costs
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Variable costs
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Constant in the short run
Do not change with changes in output
Per item costs (labor, materials, etc)
Change with changes in output.
Marginal cost
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The change in total cost from a one-unit change in
output.
Short-Run Cost Analysis
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The cost of mowing lawns:
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C(x) = 8x + 325
C(x) is the cost in dollars to mow x lawns.
The cost of mowing 0 lawns is
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C(0) = 8(0) + 325 = $325.00
$325.00 = fixed costs
C(1) = 8(1) + 325 = $333.00
C(2) = 8(2) + 325 = $341.00
Short-Run Cost Analysis
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C(x) = 8x + 325
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m=8
Marginal cost = $8.00
COST FUNCTION
In a cost function of the form C(x) = mx + b, the m
represents the marginal cost per item and b the
fixed cost. Conversely, if the fixed cost of producing
an item is b and the marginal cost is m, then the
cost function C(x) for producing x items is
C(x) = mx + b
Short-Run Cost Analysis
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Assume that the marginal cost to make x
items of a product is $90, and that 150 items
cost $16,000 to produce. Find the cost
function given it is linear.
C  x   90 x  b
16, 000  90 150   b
16, 000  13,500  b
2,500  b
C  x   90x  2500
Short-Run Cost Analysis
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Given the cost function
C(x) = 90x + 2500,
Calculate the average cost (per-item cost) of
producing 1000 items.
C 1000 
C  x
C 1000  
C  x 
1000
x
Short-Run Cost Analysis
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Given the cost function
C(x) = 90x + 2500,
Calculate the average cost (per-item cost) of
producing 1000 items.
90 1000   2500
C 1000  
 $92.50
1000
Break-Even Analysis
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Total revenue = Price(p)  Quantity(x)
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Profit = total revenue – total cost
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R(x) = px
P(x) = R(x) – C(x)
Break-even quantity: The output at which
total revenue = total cost.
Break-even point: The corresponding
ordered pair (x, p).
Break-Even Analysis
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The cost function C(x) for a firm that
produces x units of poultry feed is:
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C(x) = 20x + 100
The price of the feed is $24 per unit
How many units must be sold for the firm
to break even? What is total revenue and
total cost at the break-even quantity?
Break-Even Analysis
C  x   20 x  100
Total revenue: R  x   p  x
Break-even point: R  x   C  x 
Total cost:
p  $24
R  x   24 x
24x  20x 100
x  25
(break-even quantity)
Break-Even Analysis
R  x  C  x
R  x   24 x  24  25   600
C  x   20 x  100
 20  25   100  600
Total cost at break-even
quantity
Break-even point: (25, 600)
Now You Try
The profit (in millions of dollars) from the sale of x
million units of Blue Glue is given by P(x) = .7x – 25.5.
The cost is given by C(x) = .9x + 25.5
a) Find the revenue equation
b) What is the revenue from selling 10 million units?
c) What is the break-even point?
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Chapter 1
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